From 2f5106964ad5641581bbd5b01269a38ad02fb5f2 Mon Sep 17 00:00:00 2001 From: pangara Date: Mon, 15 Nov 2021 09:25:23 -0800 Subject: [PATCH 01/28] initial commit --- demonstrations/tutorial_dqva_mis.py | 380 ++++++++++++++++++++++++++++ 1 file changed, 380 insertions(+) create mode 100644 demonstrations/tutorial_dqva_mis.py diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py new file mode 100644 index 0000000000..c03bca6337 --- /dev/null +++ b/demonstrations/tutorial_dqva_mis.py @@ -0,0 +1,380 @@ +r""".. _trapped_ions: + +Solving MIS with the DQVA Ansatz +================================ + +.. meta:: + :property="og:description": Description and assessment of trapped ion quantum computers + :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png + +.. related:: + tutorial_pasqal Quantum computation with neutral atoms + +*Author: PennyLane dev team. Posted: 10 November 2021. Last updated: 10 November 2021.* + +DQVA ansatz intro + +.. container:: alert alert-block alert-info + + Alert block + +""" + +############################################################################## +# +# QAOA +# ~~~~~~~~~~~~~~~~~~ +# +# Something about QAOA +# +# .. figure:: ../demonstrations/trapped_ions/confining.png +# :align: center +# :width: 70% +# +# .. +# +# Confining potential +# +# What are the limitations for the general QAOA? +# +# .. figure:: ../demonstrations/trapped_ions/saddle_potential.png +# :align: center +# :width: 70% +# +# .. +# +# Image caption +# + + + +import pennylane as qml +from pennylane import qaoa +import numpy as np +from matplotlib import pyplot as plt +import networkx as nx + +G = nx.Graph() +G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)]) + +nx.draw(graph, with_labels=True) +plt.show() + +def hamming_weight(bitstr): + return sum([1 for bit in bitstr if bit == '1']) + +def mixer_layer(alpha, init_state, G, mixer_order): + """ + Apply the mixer unitary U_M(alpha) to circ + + Input + ----- + alpha : list[float] + The angle values of the parametrized gates + init_state : str + The current initial state for the ansatz, bits which are "1" are hit + with an X-gate at the beginning of the circuit and their partial mixers + are turned off. Bitstring is little-endian ordered. + G : NetworkX Graph + The graph we want to solve MIS on + mixer_order : list[int] + The order that the partial mixers should be applied in. For a list + such as [1,2,0,3] qubit 1's mixer is applied, then qubit 2's, and so on + """ + + # Apply partial mixers V_i(alpha_i) + if mixer_order is None: + mixer_order = list(G.nodes) + print('Mixer order:', mixer_order) + + # Pad the given alpha parameters to account for the zeroed angles + pad_alpha = [None]*len(init_state) + next_alpha = 0 + for qubit in mixer_order: + bit = list(reversed(init_state))[qubit] + if bit == '1' or next_alpha >= len(alpha): + continue + else: + pad_alpha[qubit] = alpha[next_alpha] + next_alpha += 1 + + print('init_state: {}, alpha: {}, pad_alpha: {}'.format(init_state, + alpha, pad_alpha)) + + for qubit in mixer_order: + if pad_alpha[qubit] == None or not G.has_node(qubit): + # Turn off mixers for qubits which are already 1 + continue + + neighbors = list(G.neighbors(qubit)) + anc_idx = 4 + + qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors)) + + # apply an X rotation controlled by the state of the ancilla qubit + qml.CRX(2*pad_alpha[qubit], wires = [anc_idx, qubit]) + + qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors)) + +def cost_layer(gamma, G): + """ + Apply a parameterized Z-rotation to every qubit + """ + for qb in G.nodes: + qml.RZ(2*gamma, wires=qb) + +def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None): + nq = len(G.nodes) + print(nq) + + # Step 1: Jump Start + # Run an efficient classical approximation algorithm to warm-start the optimization + if init_state is None: + init_state = '0'*nq + + # Step 2: Mixer Initialization + for qb, bit in enumerate(reversed(init_state)): + if bit == '1': + qml.PauliX(wires = qb) + + # parse the variational parameters + # The dqva ansatz dynamically turns off partial mixers for qubits in |1> + # and adds extra mixers to the end of the circuit + num_nonzero = nq - hamming_weight(init_state) + assert (len(params) == (nq + 1) * P), "Incorrect number of parameters!" + alpha_list = [] + gamma_list = [] + last_idx = 0 + for p in range(P): + chunk = num_nonzero + 1 + cur_section = params[p*chunk:(p+1)*chunk] + alpha_list.append(cur_section[:-1]) + gamma_list.append(cur_section[-1]) + last_idx = (p+1)*chunk + + # Add the leftover parameters as extra mixers + alpha_list.append(params[last_idx:]) + + for i in range(len(alpha_list)): + print('alpha_{}: {}'.format(i, alpha_list[i])) + if i < len(gamma_list): + print('gamma_{}: {}'.format(i, gamma_list[i])) + + # Construct the dqva ansatz + #for alphas, gamma in zip(alpha_list, gamma_list): + for i in range(len(alpha_list)): + alphas = alpha_list[i] + mixer_layer(alphas, init_state, G, mixer_order) + + + if i < len(gamma_list): + gamma = gamma_list[i] + cost_layer(gamma, G) +############################################################################## +# Text in between code goes here + +dev = qml.device("default.qubit", wires=1) + + +@qml.qnode(dev) +def ion_hadamard(state): + + if state == 1: + qml.PauliX(wires=0) + + """We use a series of seemingly arbitrary pulses that will give the Hadamard gate. + Why this is the case will become clear later""" + + qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0) + qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0) + qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) + qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0) + qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) + + return qml.state() + +#For comparison, we use the Hadamard built into PennyLane +@qml.qnode(dev) +def hadamard(state): + + if state == 1: + qml.PauliX(wires=0) + + qml.Hadamard(wires=0) + + return qml.state() + +#We confirm that the values given by both functions are the same up to numerical error +print(np.isclose(1j * ion_hadamard(0), hadamard(0))) +print(np.isclose(1j * ion_hadamard(1), hadamard(1))) + +############################################################################## +# Note that the desired gate was obtained up to a global phase factor. +# A similar exercise can be done for the :math:`T` gate: + + +@qml.qnode(dev) +def ion_Tgate(state): + + if state == 1: + qml.PauliX(wires=0) + + qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0) + qml.QubitUnitary(evolution(np.pi / 2, np.pi / 4 / Omega), wires=0) + qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) + + return qml.state() + + +@qml.qnode(dev) +def tgate(state): + + if state == 1: + qml.PauliX(wires=0) + + qml.T(wires=0) + + return qml.state() + + +print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(0), tgate(0))) +print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(1), tgate(1))) + +############################################################################## +# Text + +import matplotlib.pyplot as plt + + +@qml.qnode(dev) +def evolution_prob(t): + + qml.QubitUnitary(evolution(0, t / Omega), wires=0) + + return qml.probs(wires=0) + + +t = np.linspace(0, 4 * np.pi, 101) +s = [evolution_prob(i)[1].numpy() for i in t] + +fig1, ax1 = plt.subplots(figsize=(9, 6)) + +ax1.plot(t, s, color="#9D2EC5") + +ax1.set( + xlabel="time (in units of 1/Ω)", + ylabel="Probability", + title="Probability of measuring the excited state" +) +ax1.grid() + +plt.show() + + +# +# References +# ---------- +# +# .. [#DiVincenzo2000] +# +# D. DiVincenzo. (2000) "The Physical Implementation of Quantum Computation", +# `Fortschritte der Physik 48 (9–11): 771–783 +# `__. +# (`arXiv `__) +# +# .. [#Paul1953] +# +# W. Paul, H. Steinwedel. (1953) "Ein neues Massenspektrometer ohne Magnetfeld", +# RZeitschrift für Naturforschung A 8 (7): 448-450. +# +# .. [#CiracZoller] +# +# J. Cirac, P. Zoller. (1995) "Quantum Computations with Cold Trapped Ions". +# Physical Review Letters 74 (20): 4091–4094. +# +# .. [#Malinowski] +# +# M. Malinowski. (2021) "Unitary and Dissipative Trapped-​Ion Entanglement Using +# Integrated Optics". PhD Thesis retrieved from `ETH thesis repository +# `__. +# +# .. [#NandC2000] +# +# M. A. Nielsen, and I. L. Chuang (2000) "Quantum Computation and Quantum Information", +# Cambridge University Press. +# +# .. [#Hughes2020] +# +# A. Hughes, V. Schafer, K. Thirumalai, et al. (2020) +# "Benchmarking a High-Fidelity Mixed-Species Entangling Gate" +# `Phys. Rev. Lett. 125, 080504 +# `__. +# (`arXiv `__) +# +# .. [#Bergou2021] +# +# J. Bergou, M. Hillery, and M. Saffman. (2021) "Quantum Information Processing", +# Springer. +# +# .. [#Molmer1999] +# +# A. Sørensen, K. Mølmer. (1999) "Multi-particle entanglement of hot trapped ions", +# `Physical Review Letters. 82 (9): 1835–1838 +# `__. +# (`arXiv `__) +# +# .. [#Brown2019] +# +# M. Brown, M. Newman, and K. Brown. (2019) +# "Handling leakage with subsystem codes", +# `New J. Phys. 21 073055 +# `__. +# (`arXiv `__) +# +# .. [#Monroe2014] +# +# C. Monroe, R. Ruassendorf, A Ruthven, et al. (2019) +# "Large scale modular quantum computer architecture with atomic memory and photonic interconnects", +# `Phys. Rev. A 89 022317 +# `__. +# (`arXiv `__) +# +# .. [#QCCD2002] +# +# D. Kielpinski, C. Monroe, and D. Wineland. (2002) +# "Architecture for a large-scale ion-trap quantum computer", +# `Nature 417, 709–711 (2002). +# `__. +# +# .. [#Amini2010] +# +# J. Amini, H. Uys, J. Wesenberg, et al. (2010) +# "Toward scalable ion traps for quantum information processing", +# `New J. Phys 12 033031 +# `__. +# (`arXiv `__) +# +# +# .. [#Pino2021] +# +# J. Pino, J. Dreiling, J, C, Figgatt, et al. (2021) +# "Demonstration of the trapped-ion quantum CCD computer architecture". +# `Nature 592, 209–213 +# `__. +# (`arXiv `__) +# +# .. [#Blumel2021] +# +# R. Blumel, N. Grzesiak, N. Nguyen, et al. (2021) +# "Efficient Stabilized Two-Qubit Gates on a Trapped-Ion Quantum Computer" +# `Phys. Rev. Lett. 126, 220503 +# `__. +# (`arXiv `__) +# +# .. [#Niffenegger2020] +# +# R. Niffenegger, J. Stuart, C.Sorace-Agaskar, et al. (2020) +# "Integrated multi-wavelength control of an ion qubit" +# `Nature volume 586, pages538–542 +# `__. +# (`arXiv `__) From 7cce680e9bcb2c62c8aa48ff01bc03769267dfd9 Mon Sep 17 00:00:00 2001 From: pangara Date: Fri, 19 Nov 2021 18:19:41 -0800 Subject: [PATCH 02/28] init --- demonstrations/tutorial_dqva_mis.py | 213 +--------------------------- 1 file changed, 6 insertions(+), 207 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index c03bca6337..0408cf2594 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -10,7 +10,7 @@ .. related:: tutorial_pasqal Quantum computation with neutral atoms -*Author: PennyLane dev team. Posted: 10 November 2021. Last updated: 10 November 2021.* +*Author: * DQVA ansatz intro @@ -33,17 +33,10 @@ # # .. # -# Confining potential +# Image caption # # What are the limitations for the general QAOA? # -# .. figure:: ../demonstrations/trapped_ions/saddle_potential.png -# :align: center -# :width: 70% -# -# .. -# -# Image caption # @@ -57,7 +50,7 @@ G = nx.Graph() G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)]) -nx.draw(graph, with_labels=True) +nx.draw(G, with_labels=True) plt.show() def hamming_weight(bitstr): @@ -173,208 +166,14 @@ def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None): ############################################################################## # Text in between code goes here -dev = qml.device("default.qubit", wires=1) - - -@qml.qnode(dev) -def ion_hadamard(state): - - if state == 1: - qml.PauliX(wires=0) - - """We use a series of seemingly arbitrary pulses that will give the Hadamard gate. - Why this is the case will become clear later""" - - qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0) - qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0) - qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) - qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0) - qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) - - return qml.state() - -#For comparison, we use the Hadamard built into PennyLane -@qml.qnode(dev) -def hadamard(state): - - if state == 1: - qml.PauliX(wires=0) - - qml.Hadamard(wires=0) - - return qml.state() - -#We confirm that the values given by both functions are the same up to numerical error -print(np.isclose(1j * ion_hadamard(0), hadamard(0))) -print(np.isclose(1j * ion_hadamard(1), hadamard(1))) - -############################################################################## -# Note that the desired gate was obtained up to a global phase factor. -# A similar exercise can be done for the :math:`T` gate: - - -@qml.qnode(dev) -def ion_Tgate(state): - - if state == 1: - qml.PauliX(wires=0) - - qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0) - qml.QubitUnitary(evolution(np.pi / 2, np.pi / 4 / Omega), wires=0) - qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0) - - return qml.state() - - -@qml.qnode(dev) -def tgate(state): - - if state == 1: - qml.PauliX(wires=0) - - qml.T(wires=0) - - return qml.state() - - -print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(0), tgate(0))) -print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(1), tgate(1))) - -############################################################################## -# Text - -import matplotlib.pyplot as plt - - -@qml.qnode(dev) -def evolution_prob(t): - - qml.QubitUnitary(evolution(0, t / Omega), wires=0) - - return qml.probs(wires=0) - - -t = np.linspace(0, 4 * np.pi, 101) -s = [evolution_prob(i)[1].numpy() for i in t] - -fig1, ax1 = plt.subplots(figsize=(9, 6)) - -ax1.plot(t, s, color="#9D2EC5") - -ax1.set( - xlabel="time (in units of 1/Ω)", - ylabel="Probability", - title="Probability of measuring the excited state" -) -ax1.grid() - -plt.show() # # References # ---------- # -# .. [#DiVincenzo2000] -# -# D. DiVincenzo. (2000) "The Physical Implementation of Quantum Computation", -# `Fortschritte der Physik 48 (9–11): 771–783 -# `__. -# (`arXiv `__) -# -# .. [#Paul1953] -# -# W. Paul, H. Steinwedel. (1953) "Ein neues Massenspektrometer ohne Magnetfeld", -# RZeitschrift für Naturforschung A 8 (7): 448-450. -# -# .. [#CiracZoller] -# -# J. Cirac, P. Zoller. (1995) "Quantum Computations with Cold Trapped Ions". -# Physical Review Letters 74 (20): 4091–4094. -# -# .. [#Malinowski] -# -# M. Malinowski. (2021) "Unitary and Dissipative Trapped-​Ion Entanglement Using -# Integrated Optics". PhD Thesis retrieved from `ETH thesis repository -# `__. -# -# .. [#NandC2000] -# -# M. A. Nielsen, and I. L. Chuang (2000) "Quantum Computation and Quantum Information", -# Cambridge University Press. -# -# .. [#Hughes2020] -# -# A. Hughes, V. Schafer, K. Thirumalai, et al. (2020) -# "Benchmarking a High-Fidelity Mixed-Species Entangling Gate" -# `Phys. Rev. Lett. 125, 080504 -# `__. -# (`arXiv `__) -# -# .. [#Bergou2021] -# -# J. Bergou, M. Hillery, and M. Saffman. (2021) "Quantum Information Processing", -# Springer. -# -# .. [#Molmer1999] -# -# A. Sørensen, K. Mølmer. (1999) "Multi-particle entanglement of hot trapped ions", -# `Physical Review Letters. 82 (9): 1835–1838 -# `__. -# (`arXiv `__) -# -# .. [#Brown2019] -# -# M. Brown, M. Newman, and K. Brown. (2019) -# "Handling leakage with subsystem codes", -# `New J. Phys. 21 073055 -# `__. -# (`arXiv `__) -# -# .. [#Monroe2014] -# -# C. Monroe, R. Ruassendorf, A Ruthven, et al. (2019) -# "Large scale modular quantum computer architecture with atomic memory and photonic interconnects", -# `Phys. Rev. A 89 022317 -# `__. -# (`arXiv `__) -# -# .. [#QCCD2002] -# -# D. Kielpinski, C. Monroe, and D. Wineland. (2002) -# "Architecture for a large-scale ion-trap quantum computer", -# `Nature 417, 709–711 (2002). -# `__. -# -# .. [#Amini2010] -# -# J. Amini, H. Uys, J. Wesenberg, et al. (2010) -# "Toward scalable ion traps for quantum information processing", -# `New J. Phys 12 033031 -# `__. -# (`arXiv `__) -# -# -# .. [#Pino2021] -# -# J. Pino, J. Dreiling, J, C, Figgatt, et al. (2021) -# "Demonstration of the trapped-ion quantum CCD computer architecture". -# `Nature 592, 209–213 -# `__. -# (`arXiv `__) -# -# .. [#Blumel2021] -# -# R. Blumel, N. Grzesiak, N. Nguyen, et al. (2021) -# "Efficient Stabilized Two-Qubit Gates on a Trapped-Ion Quantum Computer" -# `Phys. Rev. Lett. 126, 220503 -# `__. -# (`arXiv `__) +# .. [#Saleem2020] # -# .. [#Niffenegger2020] +# Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization", +# `arXiv preprint arXiv:2010.06660 `__. # -# R. Niffenegger, J. Stuart, C.Sorace-Agaskar, et al. (2020) -# "Integrated multi-wavelength control of an ion qubit" -# `Nature volume 586, pages538–542 -# `__. -# (`arXiv `__) From 9729175c0909aa55c2f182de6e2fabaa68abe776 Mon Sep 17 00:00:00 2001 From: pangara Date: Fri, 14 Jan 2022 22:19:28 -0800 Subject: [PATCH 03/28] add demo --- demonstrations/dqva_mis/independent-sets.png | Bin 0 -> 113218 bytes demonstrations/dqva_mis/mixer-unitary.png | Bin 0 -> 63942 bytes demonstrations/dqva_mis/partial-mixers.png | Bin 0 -> 160156 bytes demonstrations/tutorial_dqva_mis.py | 798 ++++++++++++++++--- test-results/sphinx-gallery/junit.xml | 1 + 5 files changed, 695 insertions(+), 104 deletions(-) create mode 100644 demonstrations/dqva_mis/independent-sets.png create mode 100644 demonstrations/dqva_mis/mixer-unitary.png create mode 100644 demonstrations/dqva_mis/partial-mixers.png create mode 100644 test-results/sphinx-gallery/junit.xml diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png new file 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assessment of trapped ion quantum computers :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png .. related:: - tutorial_pasqal Quantum computation with neutral atoms + tutorial_qaoa_intro Introduction to QAOA + tutorial_qaoa_maxcut QAOA for MaxCut -*Author: * - -DQVA ansatz intro +*Author: Priya Angara* +This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) ` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) ` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. .. container:: alert alert-block alert-info Alert block """ -############################################################################## -# -# QAOA -# ~~~~~~~~~~~~~~~~~~ -# -# Something about QAOA -# -# .. figure:: ../demonstrations/trapped_ions/confining.png +###################################################################### +# QAOA, QAO-Ansatz, and DQVA +# -------------------------- +# +# The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid +# quantum-classical algorithm by `Farhi et +# al. `__ is a significant meta-heuristic +# for finding solutions to combinatorial optimization problems. The +# quantum part involves alternating between a *cost Hamiltonian*, +# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evalulate the +# objective function and a classical optimization loop updates the ansatz +# parameters. You can learn more about this in `PennyLane's tutorial on +# QAOA `__. +# +# For solving constrained combinatorial optimization problems such as MIS +# using QAOA, constraints can be imposed in two ways: 1. By adding a +# penalty term to the problem's objective function (the Lagrange +# multiplier approach) 2. By constructing the variational ansatz in a way +# such that constraints are satisfied at all times +# +# Hadfield et al., extend the QAOA framework and introduce the **Quantum +# Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more +# general families of operators thereby supporting potentially more useful +# sets of states by focusing on unitaries rather than Hamiltonians. For +# example, a mixing operator :math:`U_M (\beta)` is decomposed into a +# sequence of partial mixers :math:`U_{M, x}` which may not correspond to +# time evolution under a fixed mixer Hamiltonian :math:`H_M`. +# +# .. figure:: ../demonstrations/dqva_mis/independent_sets.png # :align: center -# :width: 70% +# :width: 100% # # .. # -# Image caption -# -# What are the limitations for the general QAOA? +# Independent sets of a graph +# +# The QAO-Ansatz guarantees that the output never leaves the set of +# feasible states (in contrast, the penalty term optimization requires an +# additional pruning step since the output contains both feasible and +# infeasible solutions). However, the QAO-Ansatz requires more complex +# quantum circuitry, which in the case of the MIS, manifests itself as +# Multi-Controlled Toffoli gates that require high connectivity between +# qubits. This makes the QAO-Ansatz inadequate for large graphs on current +# near-term quantum computers. +# +# This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that +# maximizes the performance of the QAO-Ansatz by: +# +# 1. *Warm starting the optimization* by starting with an initial state that is a feasible state +# or a superposition of feasible states. A feasible state can be found by +# using a classical approximate polynomial time algorithm. +# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* +# 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries +# +# Now that we have the basics in place, let's implement the QAO-Ansatz and +# the DQVA! + +# + + +###################################################################### +# Formulating the Maximum Independent Set (MIS) problem +# ----------------------------------------------------- +# +# Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices +# and :math:`E` is the set of edges, an *independent set* is a subset of +# vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'` +# share an edge. A *Maximum Independet Set* (MIS) is an independent set of +# the largest possible size. In this demo, our objective is to find the +# MIS of a given graph. The following image shows some independent sets of +# a graph with 5 vertices: +# +# .. figure:: ../demonstrations/dqva_mis/independent_sets.png +# :align: center +# :width: 100% # +# .. # - - +# Independent sets of a graph +# +# Here, independent sets can be represented as bitstrings +# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown +# above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are +# independent sets of size 0, 1, and 2 respectively. The Minimum Vertex +# Cover, i.e, vertices that cover all edges, is the dual of MIS. +# +# We will start with the QAO-Ansatz and then re-use most components to +# formulate the DQVA. +# +# First, we start with the necessary imports. +# import pennylane as qml from pennylane import qaoa -import numpy as np -from matplotlib import pyplot as plt +from pennylane import numpy as np import networkx as nx +from matplotlib import pyplot as plt +import copy + -G = nx.Graph() -G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)]) +###################################################################### +# Next, we define the graph shown above with 5 nodes for which we would +# like to find the Maximum Independent Set. +# -nx.draw(G, with_labels=True) +edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)] +# edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)] +graph = nx.Graph(edges) +nx.draw(graph, with_labels=True) plt.show() + +###################################################################### +# We will also instantiate a device with the number of wires being one +# more than the number of nodes (the extra wire is an ancilla). +# + +wires = len(graph.nodes) + 1 +dev = qml.device("default.qubit", wires=range(wires)) + + +###################################################################### +# For MIS, our goal is to maximize the number of vertices that form an +# independent set, which is equivalent to maximizing the `Hamming +# weight `__ of the +# bitstring. Therefore, our objective function is the Hamming weight +# operator, +# +# .. math:: +# +# +# C_{obj} = H = \sum_{i \in V} b_i, +# +# where +# +# .. math:: +# +# +# b_i = \frac{1}{2} (1 - Z_i). +# +# Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th +# qubit. +# + + +###################################################################### +# We need a couple of helper functions here: ``hamming_weight``, which +# calculates the Hamming weight of a given bitstring and ``is_indset``, +# which verifies whether a given bitstring is a valid independent set. +# + def hamming_weight(bitstr): return sum([1 for bit in bitstr if bit == '1']) + +def is_indset(bitstr, G): + nodes = list(G.nodes) + ind_set = [] + for idx, bit in enumerate(bitstr): + if bit == '1': + cur_neighbors = list(G.neighbors(idx)) + for node in ind_set: + if node in cur_neighbors: + return False + else: + ind_set.append(idx) + return True + + +###################################################################### +# Next, we define a cost unitary that incorporates the objective function, +# :math:`H` and is parameterized by :math:`\gamma`: +# +# .. math:: +# +# +# U_C(\gamma) = \prod_i e^{i \gamma H}. +# + +def cost_layer(gamma): + for qb in graph.nodes: + qml.RZ(2*gamma, wires=qb) + + +###################################################################### +# The mixer unitary for the QAO-Ansatz is defined as $ +# U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i +# e^{i:raw-latex:`\beta `M_i}$ +# +# where +# +# .. math:: +# +# +# M_i = X_i \tilde{B}, +# +# and +# +# .. math:: +# +# +# \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}. +# +# Here, where :math:`v_j` are the neighbors and :math:`l` is the number of +# neighbors for the :math:`i`-th node and +# +# .. math:: +# +# +# \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}. +# +# The mixer unitary can also be written as a product of :math:`N` partial +# mixers :math:`V_i`: +# +# .. math:: +# +# +# U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N (I + (e^{-i\beta X_i} - I)\tilde{B}) $ +# +# The partial mixers may not commute with each other, i.e., +# :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers +# may have different outputs. Therefore, the variational ansatz is defined +# up to a permutation: +# +# .. math:: +# +# +# U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta)) +# +# where P is the permutation's function of labels from :math:`1` to +# :math:`N`. +# +# These partial mixers are implemented using parameterized +# controlled-:math:`X` rotations sandwiched between two `multi-controlled +# Toffoli +# gates `__. +# The number of qubits we require is one more than the number of vertices. +# This extra qubit is used as an ancilliary qubit to serve as the target +# for the multi-controlled Toffoli gates. +# +# .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png +# :align: center +# :width: 100% +# +# .. +# +# Formulation of the mixer unitary +# + +def mixer_layer(beta, ancilla, mixer_order): -def mixer_layer(alpha, init_state, G, mixer_order): - """ - Apply the mixer unitary U_M(alpha) to circ - - Input - ----- - alpha : list[float] - The angle values of the parametrized gates - init_state : str - The current initial state for the ansatz, bits which are "1" are hit - with an X-gate at the beginning of the circuit and their partial mixers - are turned off. Bitstring is little-endian ordered. - G : NetworkX Graph - The graph we want to solve MIS on - mixer_order : list[int] - The order that the partial mixers should be applied in. For a list - such as [1,2,0,3] qubit 1's mixer is applied, then qubit 2's, and so on - """ - - # Apply partial mixers V_i(alpha_i) + # Permute the order of mixing unitaries if mixer_order is None: - mixer_order = list(G.nodes) - print('Mixer order:', mixer_order) + mixer_order = list(graph.nodes) + + for qubit in list(mixer_order): + neighbors = list(graph.neighbors(qubit)) + + # Apply the multi-controlled Toffoli, targetting the ancilla qubit + ctrl_qubits = [i for i in neighbors] + qml.MultiControlledX(control_wires = ctrl_qubits, + wires = [ancilla], + control_values ='0'*len(neighbors)) + + qml.CRX(2*beta, wires=[ancilla, qubit]) + + # Uncompute the ancilla + qml.MultiControlledX(control_wires = ctrl_qubits, + wires = [ancilla], + control_values ='0'*len(neighbors)) + + +###################################################################### +# We are now ready to build the circuit consisting of alternating cost and +# mixer layers. A list of ``params`` includes parameters for the mixer +# layer (even elements) and cost layer (odd elements). +# - # Pad the given alpha parameters to account for the zeroed angles +def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): + nq = len(graph.nodes) + # Initialize + if init_state is None: + init_state = '0'*nq + + else: + for qb, bit in enumerate(reversed(init_state)): + if bit == '1': + qml.PauliX(wires=qb) + assert (len(params) == 2*P),"Incorrect number of parameters!" + + betas = [a for i, a in enumerate(params) if i % 2 == 0] + gammas = [a for i, a in enumerate(params) if i % 2 == 1] + + for beta, gamma in zip(betas, gammas): + ancilla = nq # Use the last qubit as an ancilla + mixer_layer(beta, ancilla, mixer_order) + cost_layer(gamma) + + + +###################################################################### +# Now, let's build the circuit that finds the probabilities of measuring +# each bitstring. +# + +@qml.qnode(dev) +def probability_circuit(P, params=[], init_state=None, mixer_order=None): + qaoa_ansatz(P, params, init_state, mixer_order) + return qml.probs(wires=range(wires-1)) + + +###################################################################### +# Finally, we run the quantum-classical loop for ``m`` different mixer +# permutations. The expectation value is also calculated in the cost +# function ``f`` by running the probability circuit and calculating the +# Hamming weight (which is defined as the cost). In each iteration, an +# ansatz is constructed based on the mixer order, with optimal parameters +# from classical minimization of the cost function (Gradient Descent) +# +# .. math:: +# +# +# \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. +# + +def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): + # Select an ordering for the partial mixers + if mixer_order == None: + cur_permutation = np.random.permutation(list(graph.nodes)).tolist() + else: + cur_permutation = mixer_order + + # Define the function to be optimized + # Note that we are returning -cost since this is a minimization + def f(params): + probs = probability_circuit(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation) + avg_cost = 0 + for sample in range(0, len(probs)): + + x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))] + # Cost function is Hamming weight + avg_cost += probs[sample] * sum(x) + + return -avg_cost + + # Begin outer optimization loop + best_indset = init_state + best_init_state = init_state + cur_init_state = init_state + best_params = None + best_perm = copy.copy(cur_permutation) + + # Randomly permute the order of mixer unitaries m times + for mixer_round in range (1, m+1): + + new_hamming_weight = hamming_weight(cur_init_state) + inner_round = 1 + + while (inner_round < 2): + print('Start round {}.{}, Initial state = {}'.format(mixer_round, inner_round, cur_init_state)) + + # Begin inner variational loop + num_params = 2 * P + print('\tNum params =', num_params) + + init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params) + print('\tCurrent Mixer Order:', cur_permutation) + + # Optimize parameters + optimizer = qml.GradientDescentOptimizer(stepsize=0.5) + cur_params = init_params.copy() + + for i in range(70): + cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) + + opt_params = cur_params.copy() + + print('\tOptimal cost:', opt_cost) + + # Obtain probabilites + probs = probability_circuit(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation) + top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1] + # print (top_counts) + + best_hamming_weight = hamming_weight(best_indset) + better_strs = [] + + print(top_counts[:cutoff]) + for bitstr in top_counts[:cutoff]: + + this_hamming = hamming_weight(bitstr) + if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: + better_strs.append((bitstr, this_hamming)) + + better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) + + # If no improvement was made, break and go to next mixer round + if len(better_strs) == 0: + print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset) + break + + # Otherwise, save the new bitstring and repeat + best_indset, new_hamming_weight = better_strs[0] + best_init_state = cur_init_state + best_params = opt_params.copy() + best_perm = copy.copy(cur_permutation) + cur_init_state = best_indset + print('\tFound new independent set: {}, Hamming weight = {}'.format( + best_indset, new_hamming_weight)) + + inner_round = inner_round+1 + # Choose a new permutation of the mixer unitaries + cur_permutation = np.random.permutation(list(graph.nodes)).tolist() + + + print('\tRETURNING, best hamming weight:', new_hamming_weight) + return best_indset, best_params, best_init_state, best_perm + + +###################################################################### +# Let's run this to find the MIS! +# + +base_str = '0'*len(graph.nodes) + +out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1) +print('Init string: {}, Best MIS: {}'.format(''.join(base_str), out[0])) +print() + + +###################################################################### +# Starting with an all zero initial string will give us an independent +# set, but this may not be the maximum. Let's try to run with with a few +# different initial states. And also increase the cutoff value. ``cutoff`` +# indicates the number of bitstrings (sorted by the highest probabilities) +# that we consider that may improve the Hamming weight. +# + +base_str = '0'*len(graph.nodes) +for i in range(len(graph.nodes)): + init_str = list(base_str) + init_str[i] = '1' + out = solve_mis_qaoa(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=2) + print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0])) + print() + + +###################################################################### +# DQVA +# ---- +# +###################################################################### +# We will now formulate the MIS using the DQVA Ansatz. The cost function +# is the same as the QAO-Ansatz (the Hamming weight operator). +# +# In the DQVA, the way mixers are defined is slightly different from the +# QAO-Ansatz and are allowed to be independent. +# +# .. math:: +# +# +# U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N)) +# +# where :math:`k = 1, 2 \cdots p`. +# +# In the DQVA, whenever the :math:`j`-th bit of the initial state is one, +# the corresponding parameter :math:`\alpha_k^j` is set to 0. For example, +# if the initial state is :math:`\vert 01101 \rangle`, then +# +# .. math:: +# +# +# U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1) I_2 I_3 V_4^k(\alpha_k^4) I_5). +# +# Therefore, we are dynamically turning off parameters thereby improving +# the utilization of quantum resources. +# + +def mixer_dqva(alpha, ancilla, init_state, mixer_order): + + # Permute the order of mixing unitaries + if mixer_order is None: + mixer_order = list(graph.nodes) + pad_alpha = [None]*len(init_state) next_alpha = 0 + for qubit in mixer_order: - bit = list(reversed(init_state))[qubit] + bit = list(init_state)[qubit] if bit == '1' or next_alpha >= len(alpha): continue else: pad_alpha[qubit] = alpha[next_alpha] next_alpha += 1 - - print('init_state: {}, alpha: {}, pad_alpha: {}'.format(init_state, - alpha, pad_alpha)) - + for qubit in mixer_order: - if pad_alpha[qubit] == None or not G.has_node(qubit): + + if pad_alpha[qubit] == None or not graph.has_node(qubit): # Turn off mixers for qubits which are already 1 continue - - neighbors = list(G.neighbors(qubit)) - anc_idx = 4 - - qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors)) - - # apply an X rotation controlled by the state of the ancilla qubit - qml.CRX(2*pad_alpha[qubit], wires = [anc_idx, qubit]) + + neighbors = list(graph.neighbors(qubit)) - qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors)) - -def cost_layer(gamma, G): - """ - Apply a parameterized Z-rotation to every qubit - """ - for qb in G.nodes: - qml.RZ(2*gamma, wires=qb) - -def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None): - nq = len(G.nodes) - print(nq) - - # Step 1: Jump Start - # Run an efficient classical approximation algorithm to warm-start the optimization + # Apply the multi-controlled Toffoli, targetting the ancilla qubit + ctrl_qubits = [i for i in neighbors] + qml.MultiControlledX(control_wires = ctrl_qubits, + wires = [ancilla], + control_values ='0'*len(neighbors)) + + qml.CRX(2*pad_alpha[qubit], wires=[ancilla, qubit]) + + # Uncompute the ancilla + qml.MultiControlledX(control_wires = ctrl_qubits, + wires = [ancilla], + control_values ='0'*len(neighbors)) + + +###################################################################### +# The structure of the cost and mixer unitaries is similar QAO-Ansatz, +# however, we now alternate between :math:`p` applications of the mixing +# unitary :math:`U_M^k(\alpha_k)` and :math:`p` applications of the cost +# unitary :math:`U_C^k(\gamma_k)`: +# +# .. math:: +# +# +# \vert \alpha, \gamma \rangle = U_C^p(\gamma_p)U_M^p(\alpha_p)\cdots U_C^1(\gamma_p)U_M^1(\alpha_1)\vert c_1 \rangle +# +# where :math:`\vert c_1 \rangle` is an initial state. +# + +def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): + nq = len(graph.nodes) + + # Step1: Initialize if init_state is None: init_state = '0'*nq - - # Step 2: Mixer Initialization - for qb, bit in enumerate(reversed(init_state)): - if bit == '1': - qml.PauliX(wires = qb) - - # parse the variational parameters - # The dqva ansatz dynamically turns off partial mixers for qubits in |1> - # and adds extra mixers to the end of the circuit - num_nonzero = nq - hamming_weight(init_state) - assert (len(params) == (nq + 1) * P), "Incorrect number of parameters!" + + else: + for qb, bit in enumerate(reversed(init_state)): + if bit == '1': + qml.PauliX(wires=qb) + + num_nonzero = nq - hamming_weight(init_state) + assert (len(params) == (nq + 1)*P),"Incorrect number of parameters!" alpha_list = [] gamma_list = [] last_idx = 0 @@ -144,30 +573,191 @@ def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None): alpha_list.append(cur_section[:-1]) gamma_list.append(cur_section[-1]) last_idx = (p+1)*chunk - - # Add the leftover parameters as extra mixers + alpha_list.append(params[last_idx:]) - - for i in range(len(alpha_list)): - print('alpha_{}: {}'.format(i, alpha_list[i])) - if i < len(gamma_list): - print('gamma_{}: {}'.format(i, gamma_list[i])) - - # Construct the dqva ansatz - #for alphas, gamma in zip(alpha_list, gamma_list): + for i in range(len(alpha_list)): + ancilla = nq alphas = alpha_list[i] - mixer_layer(alphas, init_state, G, mixer_order) - + mixer_dqva(alphas, ancilla, init_state, mixer_order) if i < len(gamma_list): gamma = gamma_list[i] - cost_layer(gamma, G) -############################################################################## -# Text in between code goes here + cost_layer(gamma) + + + +###################################################################### +# Let's also define the probability circuit for the DQVA ansatz +# + +@qml.qnode(dev) +def probability_dqva(P, params=[], init_state=None, mixer_order=None): + dqva_ansatz(P, params, init_state, mixer_order) + return qml.probs(wires= dev.wires[:-1]) + + +###################################################################### +# The quantum-classical loop for DQVA also maximizes +# +# .. math:: +# +# +# \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle. +# +# The dynamic ansatz update includes: 1. Optimization of parameters using +# Gradient Descent and finding the Hamming weight with new parameters 2. +# If the Hamming weight of the new state is larger than the initial state +# , the initial state gets updated to this new state 3. Based on this new +# state, partial mixers are updated (i.e., turned off for ones and turned +# on for zeros) 4. Steps 2 and 3 are repeated until the Hamming weight can +# no longer be improved 5. If no new Hamming weight is obtained, the +# partial mixers are randomized and steps 2 and 3 are repeated to check if +# a better Hamming weight is found. The number of randomizations is +# controlled via a hyperparameter. +# + +def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff=1): + + # Select an ordering for the partial mixers + if mixer_order == None: + cur_permutation = np.random.permutation(list(graph.nodes)).tolist() + else: + cur_permutation = mixer_order + def f(params): + + probs = probability_dqva(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation) + avg_cost = 0 + for sample in range(0, len(probs)): + + x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))] + # Cost function is Hamming weight + avg_cost += probs[sample] * sum(x) + + return -avg_cost + + # Begin outer optimization loop + best_indset = init_state + best_init_state = init_state + cur_init_state = init_state + best_params = None + best_perm = copy.copy(cur_permutation) + + # Randomly permute the order of mixer unitaries m times + for mixer_round in range (1, m+1): + inner_round = 1 + new_hamming_weight = hamming_weight(cur_init_state) + + while True: + print('Start round {}.{}, Initial state = {}'.format(mixer_round, + inner_round, cur_init_state)) + + # Begin inner variational loop + num_params = P * (len(graph.nodes()) + 1) + print('\tNum params =', num_params) + init_params = np.random.uniform(low=0.0, high=2*np.pi, size=num_params) + print('\tCurrent Mixer Order:', cur_permutation) + + # Optimize parameters + optimizer = qml.GradientDescentOptimizer(stepsize=0.5) + cur_params = init_params.copy() + + for i in range(70): + cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) + + opt_params = cur_params.copy() + + print('\tOptimal cost:', opt_cost) + + probs = probability_dqva(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation) + + + top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1] + + best_hamming_weight = hamming_weight(best_indset) + better_strs = [] + + for bitstr in top_counts[:1]: + this_hamming = hamming_weight(bitstr) + if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: + better_strs.append((bitstr, this_hamming)) + better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) + + + # If no improvement was made, break and go to next mixer round + if len(better_strs) == 0: + print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset) + break + + # Otherwise, save the new bitstring and repeat + best_indset, new_hamming_weight = better_strs[0] + best_init_state = cur_init_state + best_params = opt_params + best_perm = copy.copy(cur_permutation) + cur_init_state = best_indset + print('\tFound new independent set: {}, Hamming weight = {}'.format( + best_indset, new_hamming_weight)) + inner_round += 1 + + + # Choose a new permutation of the mixer unitaries + cur_permutation = np.random.permutation(list(graph.nodes)).tolist() + + print('\tRETURNING, best hamming weight:', new_hamming_weight) + return best_indset, best_params, best_init_state, best_perm + + +###################################################################### +# We also run the algorithm for a number of initial states (instead of +# just one) to check what is the best Maximum Independent Set we obtain. +# The initial states are a set of independent sets of size one (this is +# trivial - any bitstring with a single "1" is a valid independent set) +# + +base_str = '0'*len(graph.nodes) +all_init_strs = [] +for i in range(len(graph.nodes)): + init_str = list(base_str) + init_str[i] = '1' + out = solve_mis_dqva(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=1) + print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0])) + print() + + +###################################################################### +# And voila, we have found an independent set of size 3 which is the state +# ``10101``! +# + + +###################################################################### +# Conclusion +# ---------- +# +###################################################################### +# In this demo, we looked into solving a constrained combinatorial +# optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz +# defines mixer unitaries with partial mixers in such a way that we never +# leave the set of feasible states. However, this comes at a high cost of +# using multi-controlled Toffoli gates. To overcome this, we looked at the +# DQVA which turned partial mixers on and off, and randomized the partial +# mixer ordering. Furthermore, Saleem et al., recommend starting with a +# set of initial states that are independent sets found by any classical +# polynomial time approximation algorithm to "warm start" the +# initialization. Furthermore, they propose the first useful application +# of `circuit-cutting techniques `__ to +# solve MIS, which involves classical partitioning of a large graph into +# sub-graphs, finding a solution using DQVA on a subgraph, and then +# preparing a set of states to be passed as an input, essentially +# stitching together solutions. +# + + + +###################################################################### # # References # ---------- diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml new file mode 100644 index 0000000000..113d2a2b94 --- /dev/null +++ b/test-results/sphinx-gallery/junit.xml @@ -0,0 +1 @@ + \ No newline at end of file From c2ecb63fb8fa701555603c5e8d4140debbe34ad9 Mon Sep 17 00:00:00 2001 From: pangara Date: Fri, 14 Jan 2022 23:22:56 -0800 Subject: [PATCH 04/28] minor edits --- demonstrations/tutorial_dqva_mis.py | 513 +++++++++++++++------------- 1 file changed, 272 insertions(+), 241 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 9dba349f1b..d6e4a13e9b 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -23,23 +23,23 @@ ###################################################################### # QAOA, QAO-Ansatz, and DQVA # -------------------------- -# +# # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid # quantum-classical algorithm by `Farhi et # al. `__ is a significant meta-heuristic # for finding solutions to combinatorial optimization problems. The # quantum part involves alternating between a *cost Hamiltonian*, -# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evalulate the +# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evaluate the # objective function and a classical optimization loop updates the ansatz # parameters. You can learn more about this in `PennyLane's tutorial on # QAOA `__. -# +# # For solving constrained combinatorial optimization problems such as MIS -# using QAOA, constraints can be imposed in two ways: 1. By adding a -# penalty term to the problem's objective function (the Lagrange -# multiplier approach) 2. By constructing the variational ansatz in a way -# such that constraints are satisfied at all times -# +# using QAOA, constraints can be imposed in two ways: +# +# 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach) +# 2. By constructing the variational ansatz in a way such that constraints are satisfied at all times +# # Hadfield et al., extend the QAOA framework and introduce the **Quantum # Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more # general families of operators thereby supporting potentially more useful @@ -47,15 +47,15 @@ # example, a mixing operator :math:`U_M (\beta)` is decomposed into a # sequence of partial mixers :math:`U_{M, x}` which may not correspond to # time evolution under a fixed mixer Hamiltonian :math:`H_M`. -# -# .. figure:: ../demonstrations/dqva_mis/independent_sets.png +# +# .. figure:: ../demonstrations/dqva_mis/partial_mixers.png # :align: center # :width: 100% # # .. # # Independent sets of a graph -# +# # The QAO-Ansatz guarantees that the output never leaves the set of # feasible states (in contrast, the penalty term optimization requires an # additional pruning step since the output contains both feasible and @@ -64,34 +64,34 @@ # Multi-Controlled Toffoli gates that require high connectivity between # qubits. This makes the QAO-Ansatz inadequate for large graphs on current # near-term quantum computers. -# +# # This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that -# maximizes the performance of the QAO-Ansatz by: -# +# maximizes the performance of the QAO-Ansatz by: +# # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state # or a superposition of feasible states. A feasible state can be found by -# using a classical approximate polynomial time algorithm. -# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* +# using a classical approximate polynomial-time algorithm. +# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries -# +# # Now that we have the basics in place, let's implement the QAO-Ansatz and # the DQVA! -# +# ###################################################################### # Formulating the Maximum Independent Set (MIS) problem # ----------------------------------------------------- -# +# # Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices # and :math:`E` is the set of edges, an *independent set* is a subset of # vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'` -# share an edge. A *Maximum Independet Set* (MIS) is an independent set of +# share an edge. A *Maximum Independent Set* (MIS) is an independent set of # the largest possible size. In this demo, our objective is to find the # MIS of a given graph. The following image shows some independent sets of # a graph with 5 vertices: -# +# # .. figure:: ../demonstrations/dqva_mis/independent_sets.png # :align: center # :width: 100% @@ -99,18 +99,18 @@ # .. # # Independent sets of a graph -# +# # Here, independent sets can be represented as bitstrings # :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown # above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are # independent sets of size 0, 1, and 2 respectively. The Minimum Vertex # Cover, i.e, vertices that cover all edges, is the dual of MIS. -# +# # We will start with the QAO-Ansatz and then re-use most components to # formulate the DQVA. -# +# # First, we start with the necessary imports. -# +# import pennylane as qml from pennylane import qaoa @@ -123,7 +123,7 @@ ###################################################################### # Next, we define the graph shown above with 5 nodes for which we would # like to find the Maximum Independent Set. -# +# edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)] # edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)] @@ -135,7 +135,7 @@ ###################################################################### # We will also instantiate a device with the number of wires being one # more than the number of nodes (the extra wire is an ancilla). -# +# wires = len(graph.nodes) + 1 dev = qml.device("default.qubit", wires=range(wires)) @@ -147,38 +147,40 @@ # weight `__ of the # bitstring. Therefore, our objective function is the Hamming weight # operator, -# +# # .. math:: -# -# +# +# # C_{obj} = H = \sum_{i \in V} b_i, -# +# # where -# +# # .. math:: -# -# +# +# # b_i = \frac{1}{2} (1 - Z_i). -# +# # Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th # qubit. -# +# ###################################################################### # We need a couple of helper functions here: ``hamming_weight``, which # calculates the Hamming weight of a given bitstring and ``is_indset``, # which verifies whether a given bitstring is a valid independent set. -# +# + def hamming_weight(bitstr): - return sum([1 for bit in bitstr if bit == '1']) + return sum([1 for bit in bitstr if bit == "1"]) + def is_indset(bitstr, G): nodes = list(G.nodes) ind_set = [] for idx, bit in enumerate(bitstr): - if bit == '1': + if bit == "1": cur_neighbors = list(G.neighbors(idx)) for node in ind_set: if node in cur_neighbors: @@ -191,74 +193,75 @@ def is_indset(bitstr, G): ###################################################################### # Next, we define a cost unitary that incorporates the objective function, # :math:`H` and is parameterized by :math:`\gamma`: -# +# # .. math:: -# -# -# U_C(\gamma) = \prod_i e^{i \gamma H}. -# +# +# +# U_C(\gamma) = \prod_i e^{i \gamma H}. +# + def cost_layer(gamma): for qb in graph.nodes: - qml.RZ(2*gamma, wires=qb) + qml.RZ(2 * gamma, wires=qb) ###################################################################### # The mixer unitary for the QAO-Ansatz is defined as $ # U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i # e^{i:raw-latex:`\beta `M_i}$ -# +# # where -# +# # .. math:: -# -# +# +# # M_i = X_i \tilde{B}, -# +# # and -# +# # .. math:: # -# +# # \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}. -# +# # Here, where :math:`v_j` are the neighbors and :math:`l` is the number of # neighbors for the :math:`i`-th node and -# +# # .. math:: -# -# +# +# # \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}. -# +# # The mixer unitary can also be written as a product of :math:`N` partial # mixers :math:`V_i`: -# +# # .. math:: -# -# -# U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N (I + (e^{-i\beta X_i} - I)\tilde{B}) $ -# +# +# +# U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N (I + (e^{-i\beta X_i} - I)\tilde{B}) +# # The partial mixers may not commute with each other, i.e., # :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers # may have different outputs. Therefore, the variational ansatz is defined # up to a permutation: -# +# # .. math:: -# -# +# +# # U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta)) -# +# # where P is the permutation's function of labels from :math:`1` to # :math:`N`. -# +# # These partial mixers are implemented using parameterized # controlled-:math:`X` rotations sandwiched between two `multi-controlled # Toffoli # gates `__. # The number of qubits we require is one more than the number of vertices. -# This extra qubit is used as an ancilliary qubit to serve as the target +# This extra qubit is used as an ancillary qubit to serve as the target # for the multi-controlled Toffoli gates. -# +# # .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png # :align: center # :width: 100% @@ -266,68 +269,71 @@ def cost_layer(gamma): # .. # # Formulation of the mixer unitary -# +# +# In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates +# controlled by the neighbors of the :math:`i`'th vertex. def mixer_layer(beta, ancilla, mixer_order): - + # Permute the order of mixing unitaries if mixer_order is None: mixer_order = list(graph.nodes) - + for qubit in list(mixer_order): neighbors = list(graph.neighbors(qubit)) # Apply the multi-controlled Toffoli, targetting the ancilla qubit ctrl_qubits = [i for i in neighbors] - qml.MultiControlledX(control_wires = ctrl_qubits, - wires = [ancilla], - control_values ='0'*len(neighbors)) - - qml.CRX(2*beta, wires=[ancilla, qubit]) - + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) + + qml.CRX(2 * beta, wires=[ancilla, qubit]) + # Uncompute the ancilla - qml.MultiControlledX(control_wires = ctrl_qubits, - wires = [ancilla], - control_values ='0'*len(neighbors)) + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) ###################################################################### # We are now ready to build the circuit consisting of alternating cost and # mixer layers. A list of ``params`` includes parameters for the mixer # layer (even elements) and cost layer (odd elements). -# +# + def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): nq = len(graph.nodes) # Initialize if init_state is None: - init_state = '0'*nq - + init_state = "0" * nq + else: for qb, bit in enumerate(reversed(init_state)): - if bit == '1': + if bit == "1": qml.PauliX(wires=qb) - assert (len(params) == 2*P),"Incorrect number of parameters!" - - betas = [a for i, a in enumerate(params) if i % 2 == 0] + assert len(params) == 2 * P, "Incorrect number of parameters!" + + betas = [a for i, a in enumerate(params) if i % 2 == 0] gammas = [a for i, a in enumerate(params) if i % 2 == 1] - + for beta, gamma in zip(betas, gammas): - ancilla = nq # Use the last qubit as an ancilla + ancilla = nq # Use the last qubit as an ancilla mixer_layer(beta, ancilla, mixer_order) cost_layer(gamma) - ###################################################################### # Now, let's build the circuit that finds the probabilities of measuring # each bitstring. -# +# + @qml.qnode(dev) def probability_circuit(P, params=[], init_state=None, mixer_order=None): qaoa_ansatz(P, params, init_state, mixer_order) - return qml.probs(wires=range(wires-1)) + return qml.probs(wires=range(wires - 1)) ###################################################################### @@ -339,12 +345,13 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None): # from classical minimization of the cost function (Gradient Descent) # # .. math:: -# -# -# \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. -# +# +# +# \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. +# + -def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): +def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): # Select an ordering for the partial mixers if mixer_order == None: cur_permutation = np.random.permutation(list(graph.nodes)).tolist() @@ -354,70 +361,82 @@ def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cuto # Define the function to be optimized # Note that we are returning -cost since this is a minimization def f(params): - probs = probability_circuit(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation) + probs = probability_circuit( + P, params=params, init_state=cur_init_state, mixer_order=cur_permutation + ) avg_cost = 0 for sample in range(0, len(probs)): - + x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))] # Cost function is Hamming weight avg_cost += probs[sample] * sum(x) - + return -avg_cost - + # Begin outer optimization loop best_indset = init_state best_init_state = init_state cur_init_state = init_state best_params = None best_perm = copy.copy(cur_permutation) - + # Randomly permute the order of mixer unitaries m times - for mixer_round in range (1, m+1): - + for mixer_round in range(1, m + 1): + new_hamming_weight = hamming_weight(cur_init_state) inner_round = 1 - - while (inner_round < 2): - print('Start round {}.{}, Initial state = {}'.format(mixer_round, inner_round, cur_init_state)) + + while inner_round < 2: + print( + "Start round {}.{}, Initial state = {}".format( + mixer_round, inner_round, cur_init_state + ) + ) # Begin inner variational loop num_params = 2 * P - print('\tNum params =', num_params) + print("\tNum params =", num_params) init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params) - print('\tCurrent Mixer Order:', cur_permutation) + print("\tCurrent Mixer Order:", cur_permutation) # Optimize parameters optimizer = qml.GradientDescentOptimizer(stepsize=0.5) cur_params = init_params.copy() - + for i in range(70): cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) opt_params = cur_params.copy() - print('\tOptimal cost:', opt_cost) + print("\tOptimal cost:", opt_cost) # Obtain probabilites - probs = probability_circuit(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation) - top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1] + probs = probability_circuit( + P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation + ) + top_counts = list( + map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) + )[::-1] # print (top_counts) - + best_hamming_weight = hamming_weight(best_indset) better_strs = [] - + print(top_counts[:cutoff]) for bitstr in top_counts[:cutoff]: - + this_hamming = hamming_weight(bitstr) if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: better_strs.append((bitstr, this_hamming)) - + better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) # If no improvement was made, break and go to next mixer round if len(better_strs) == 0: - print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset) + print( + "\tNone of the measured bitstrings had higher Hamming weight than:", best_indset + ) break # Otherwise, save the new bitstring and repeat @@ -426,114 +445,117 @@ def f(params): best_params = opt_params.copy() best_perm = copy.copy(cur_permutation) cur_init_state = best_indset - print('\tFound new independent set: {}, Hamming weight = {}'.format( - best_indset, new_hamming_weight)) - - inner_round = inner_round+1 + print( + "\tFound new independent set: {}, Hamming weight = {}".format( + best_indset, new_hamming_weight + ) + ) + + inner_round = inner_round + 1 # Choose a new permutation of the mixer unitaries cur_permutation = np.random.permutation(list(graph.nodes)).tolist() - - print('\tRETURNING, best hamming weight:', new_hamming_weight) + print("\tRETURNING, best hamming weight:", new_hamming_weight) return best_indset, best_params, best_init_state, best_perm ###################################################################### # Let's run this to find the MIS! -# +# -base_str = '0'*len(graph.nodes) +base_str = "0" * len(graph.nodes) out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1) -print('Init string: {}, Best MIS: {}'.format(''.join(base_str), out[0])) +print("Init string: {}, Best MIS: {}".format("".join(base_str), out[0])) print() ###################################################################### -# Starting with an all zero initial string will give us an independent -# set, but this may not be the maximum. Let's try to run with with a few -# different initial states. And also increase the cutoff value. ``cutoff`` +# Starting with an all-zero initial string will give us an independent +# set, but this may not be the maximum. Now, we run this with a few +# different initial states and also increase the cutoff value. ``cutoff`` # indicates the number of bitstrings (sorted by the highest probabilities) # that we consider that may improve the Hamming weight. -# +# -base_str = '0'*len(graph.nodes) +base_str = "0" * len(graph.nodes) for i in range(len(graph.nodes)): init_str = list(base_str) - init_str[i] = '1' - out = solve_mis_qaoa(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=2) - print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0])) + init_str[i] = "1" + out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2) + print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0])) print() ###################################################################### # DQVA # ---- -# +# ###################################################################### # We will now formulate the MIS using the DQVA Ansatz. The cost function # is the same as the QAO-Ansatz (the Hamming weight operator). -# +# # In the DQVA, the way mixers are defined is slightly different from the # QAO-Ansatz and are allowed to be independent. -# +# # .. math:: -# -# +# +# # U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N)) -# +# # where :math:`k = 1, 2 \cdots p`. -# +# # In the DQVA, whenever the :math:`j`-th bit of the initial state is one, # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example, # if the initial state is :math:`\vert 01101 \rangle`, then -# +# # .. math:: -# -# +# +# # U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1) I_2 I_3 V_4^k(\alpha_k^4) I_5). -# +# # Therefore, we are dynamically turning off parameters thereby improving # the utilization of quantum resources. # + def mixer_dqva(alpha, ancilla, init_state, mixer_order): - + # Permute the order of mixing unitaries if mixer_order is None: mixer_order = list(graph.nodes) - - pad_alpha = [None]*len(init_state) + + pad_alpha = [None] * len(init_state) next_alpha = 0 - + for qubit in mixer_order: bit = list(init_state)[qubit] - if bit == '1' or next_alpha >= len(alpha): + if bit == "1" or next_alpha >= len(alpha): continue else: pad_alpha[qubit] = alpha[next_alpha] next_alpha += 1 - + for qubit in mixer_order: - + if pad_alpha[qubit] == None or not graph.has_node(qubit): # Turn off mixers for qubits which are already 1 continue - + neighbors = list(graph.neighbors(qubit)) - + # Apply the multi-controlled Toffoli, targetting the ancilla qubit ctrl_qubits = [i for i in neighbors] - qml.MultiControlledX(control_wires = ctrl_qubits, - wires = [ancilla], - control_values ='0'*len(neighbors)) - - qml.CRX(2*pad_alpha[qubit], wires=[ancilla, qubit]) - + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) + + qml.CRX(2 * pad_alpha[qubit], wires=[ancilla, qubit]) + # Uncompute the ancilla - qml.MultiControlledX(control_wires = ctrl_qubits, - wires = [ancilla], - control_values ='0'*len(neighbors)) + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) ###################################################################### @@ -541,41 +563,42 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order): # however, we now alternate between :math:`p` applications of the mixing # unitary :math:`U_M^k(\alpha_k)` and :math:`p` applications of the cost # unitary :math:`U_C^k(\gamma_k)`: -# +# # .. math:: -# -# +# +# # \vert \alpha, \gamma \rangle = U_C^p(\gamma_p)U_M^p(\alpha_p)\cdots U_C^1(\gamma_p)U_M^1(\alpha_1)\vert c_1 \rangle -# +# # where :math:`\vert c_1 \rangle` is an initial state. -# +# + def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): nq = len(graph.nodes) # Step1: Initialize if init_state is None: - init_state = '0'*nq - + init_state = "0" * nq + else: for qb, bit in enumerate(reversed(init_state)): - if bit == '1': + if bit == "1": qml.PauliX(wires=qb) - - num_nonzero = nq - hamming_weight(init_state) - assert (len(params) == (nq + 1)*P),"Incorrect number of parameters!" + + num_nonzero = nq - hamming_weight(init_state) + assert len(params) == (nq + 1) * P, "Incorrect number of parameters!" alpha_list = [] gamma_list = [] last_idx = 0 for p in range(P): chunk = num_nonzero + 1 - cur_section = params[p*chunk:(p+1)*chunk] + cur_section = params[p * chunk : (p + 1) * chunk] alpha_list.append(cur_section[:-1]) gamma_list.append(cur_section[-1]) - last_idx = (p+1)*chunk - + last_idx = (p + 1) * chunk + alpha_list.append(params[last_idx:]) - + for i in range(len(alpha_list)): ancilla = nq alphas = alpha_list[i] @@ -586,39 +609,38 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): cost_layer(gamma) - ###################################################################### # Let's also define the probability circuit for the DQVA ansatz -# +# + @qml.qnode(dev) def probability_dqva(P, params=[], init_state=None, mixer_order=None): dqva_ansatz(P, params, init_state, mixer_order) - return qml.probs(wires= dev.wires[:-1]) + return qml.probs(wires=dev.wires[:-1]) ###################################################################### # The quantum-classical loop for DQVA also maximizes -# +# # .. math:: -# -# -# \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle. -# -# The dynamic ansatz update includes: 1. Optimization of parameters using -# Gradient Descent and finding the Hamming weight with new parameters 2. -# If the Hamming weight of the new state is larger than the initial state -# , the initial state gets updated to this new state 3. Based on this new -# state, partial mixers are updated (i.e., turned off for ones and turned -# on for zeros) 4. Steps 2 and 3 are repeated until the Hamming weight can -# no longer be improved 5. If no new Hamming weight is obtained, the -# partial mixers are randomized and steps 2 and 3 are repeated to check if -# a better Hamming weight is found. The number of randomizations is -# controlled via a hyperparameter. -# - -def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff=1): - +# +# +# \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle. +# +# The dynamic ansatz update includes: +# +# 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters +# 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state +# 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros) +# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved +# 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if +# a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. +# + + +def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): + # Select an ordering for the partial mixers if mixer_order == None: cur_permutation = np.random.permutation(list(graph.nodes)).tolist() @@ -626,70 +648,78 @@ def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff cur_permutation = mixer_order def f(params): - - probs = probability_dqva(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation) + + probs = probability_dqva( + P, params=params, init_state=cur_init_state, mixer_order=cur_permutation + ) avg_cost = 0 for sample in range(0, len(probs)): - x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))] # Cost function is Hamming weight avg_cost += probs[sample] * sum(x) - + return -avg_cost - + # Begin outer optimization loop best_indset = init_state best_init_state = init_state cur_init_state = init_state best_params = None best_perm = copy.copy(cur_permutation) - + # Randomly permute the order of mixer unitaries m times - for mixer_round in range (1, m+1): + for mixer_round in range(1, m + 1): inner_round = 1 new_hamming_weight = hamming_weight(cur_init_state) - + while True: - print('Start round {}.{}, Initial state = {}'.format(mixer_round, - inner_round, cur_init_state)) - + print( + "Start round {}.{}, Initial state = {}".format( + mixer_round, inner_round, cur_init_state + ) + ) + # Begin inner variational loop num_params = P * (len(graph.nodes()) + 1) - print('\tNum params =', num_params) - init_params = np.random.uniform(low=0.0, high=2*np.pi, size=num_params) - print('\tCurrent Mixer Order:', cur_permutation) - + print("\tNum params =", num_params) + init_params = np.random.uniform(low=0.0, high=2 * np.pi, size=num_params) + print("\tCurrent Mixer Order:", cur_permutation) + # Optimize parameters optimizer = qml.GradientDescentOptimizer(stepsize=0.5) cur_params = init_params.copy() - + for i in range(70): cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) opt_params = cur_params.copy() - - print('\tOptimal cost:', opt_cost) - - probs = probability_dqva(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation) - - top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1] + print("\tOptimal cost:", opt_cost) + + probs = probability_dqva( + P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation + ) + + top_counts = list( + map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) + )[::-1] best_hamming_weight = hamming_weight(best_indset) better_strs = [] - + for bitstr in top_counts[:1]: this_hamming = hamming_weight(bitstr) if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: better_strs.append((bitstr, this_hamming)) better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) - # If no improvement was made, break and go to next mixer round if len(better_strs) == 0: - print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset) + print( + "\tNone of the measured bitstrings had higher Hamming weight than:", best_indset + ) break # Otherwise, save the new bitstring and repeat @@ -698,45 +728,47 @@ def f(params): best_params = opt_params best_perm = copy.copy(cur_permutation) cur_init_state = best_indset - print('\tFound new independent set: {}, Hamming weight = {}'.format( - best_indset, new_hamming_weight)) + print( + "\tFound new independent set: {}, Hamming weight = {}".format( + best_indset, new_hamming_weight + ) + ) inner_round += 1 - # Choose a new permutation of the mixer unitaries cur_permutation = np.random.permutation(list(graph.nodes)).tolist() - print('\tRETURNING, best hamming weight:', new_hamming_weight) + print("\tRETURNING, best hamming weight:", new_hamming_weight) return best_indset, best_params, best_init_state, best_perm ###################################################################### -# We also run the algorithm for a number of initial states (instead of +# We also run the algorithm for several initial states (instead of # just one) to check what is the best Maximum Independent Set we obtain. # The initial states are a set of independent sets of size one (this is # trivial - any bitstring with a single "1" is a valid independent set) -# +# -base_str = '0'*len(graph.nodes) +base_str = "0" * len(graph.nodes) all_init_strs = [] for i in range(len(graph.nodes)): init_str = list(base_str) - init_str[i] = '1' - out = solve_mis_dqva(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=1) - print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0])) + init_str[i] = "1" + out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1) + print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0])) print() ###################################################################### # And voila, we have found an independent set of size 3 which is the state # ``10101``! -# +# ###################################################################### # Conclusion # ---------- -# +# ###################################################################### # In this demo, we looked into solving a constrained combinatorial # optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz @@ -753,8 +785,7 @@ def f(params): # sub-graphs, finding a solution using DQVA on a subgraph, and then # preparing a set of states to be passed as an input, essentially # stitching together solutions. -# - +# ###################################################################### From 7220453aa61672fbc954d588f0df721e96ff4b6f Mon Sep 17 00:00:00 2001 From: pangara Date: Sat, 15 Jan 2022 11:27:37 -0800 Subject: [PATCH 05/28] fix errors --- demonstrations/tutorial_dqva_mis.py | 12 +++--------- test-results/sphinx-gallery/junit.xml | 2 +- 2 files changed, 4 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index d6e4a13e9b..4721da2a30 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -4,7 +4,7 @@ ============================================ .. meta:: - :property="og:description": Description and assessment of trapped ion quantum computers + :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png .. related:: @@ -14,9 +14,6 @@ *Author: Priya Angara* This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) ` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) ` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. -.. container:: alert alert-block alert-info - - Alert block """ @@ -488,10 +485,8 @@ def f(params): ###################################################################### -# DQVA -# ---- -# -###################################################################### +# Dynamic Quantum Variational Ansatz +# ---------------------------------- # We will now formulate the MIS using the DQVA Ansatz. The cost function # is the same as the QAO-Ansatz (the Hamming weight operator). # @@ -769,7 +764,6 @@ def f(params): # Conclusion # ---------- # -###################################################################### # In this demo, we looked into solving a constrained combinatorial # optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz # defines mixer unitaries with partial mixers in such a way that we never diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 113d2a2b94..7fd1edd634 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file From cafd6825092f29083a16282887347489fc9b5897 Mon Sep 17 00:00:00 2001 From: pangara Date: Sat, 15 Jan 2022 18:01:57 -0800 Subject: [PATCH 06/28] update toc --- demonstrations/tutorial_dqva_mis.py | 2 +- demos_optimization.rst | 7 +++++++ 2 files changed, 8 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 4721da2a30..aee60c882c 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -5,7 +5,7 @@ .. meta:: :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA - :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png + :property="og:image": https://pennylane.ai/qml/_images/mixer_unitary.png .. related:: tutorial_qaoa_intro Introduction to QAOA diff --git a/demos_optimization.rst b/demos_optimization.rst index 26704466c0..46908048de 100644 --- a/demos_optimization.rst +++ b/demos_optimization.rst @@ -139,6 +139,12 @@ in quantum neural networks. :description: :doc:`demos/tutorial_qaoa_maxcut` :tags: autograd +.. customgalleryitem:: + :tooltip: QAO-Ansatz and DQVA for MIS. + :figure: demonstrations/dqva_mis/mixer-unitary.png + :description: :doc:`demos/tutorial_dqva_mis` + :tags: autograd + :html:`
` @@ -166,3 +172,4 @@ in quantum neural networks. demos/qonn demos/tutorial_isingmodel_PyTorch demos/tutorial_qaoa_maxcut + demos/tutorial_dqva_mis \ No newline at end of file From 721673f1565594c1fdfa540b23824ec9190b35e8 Mon Sep 17 00:00:00 2001 From: pangara Date: Sun, 16 Jan 2022 09:31:45 -0800 Subject: [PATCH 07/28] fix latex issues --- demonstrations/tutorial_dqva_mis.py | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index aee60c882c..b653582592 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -204,9 +204,12 @@ def cost_layer(gamma): ###################################################################### -# The mixer unitary for the QAO-Ansatz is defined as $ -# U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i -# e^{i:raw-latex:`\beta `M_i}$ +# The mixer unitary for the QAO-Ansatz is defined as +# +# .. math:: +# +# +# U_M(\beta) = \prod\_i e^{i \beta M_i} # # where # @@ -787,8 +790,6 @@ def f(params): # References # ---------- # -# .. [#Saleem2020] -# -# Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization", +# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization", # `arXiv preprint arXiv:2010.06660 `__. # From 428cf66bb9906aca8c05b6896dd9cf01020f1111 Mon Sep 17 00:00:00 2001 From: pangara Date: Sun, 16 Jan 2022 10:17:27 -0800 Subject: [PATCH 08/28] fix figure --- demonstrations/tutorial_dqva_mis.py | 16 +++++++--------- 1 file changed, 7 insertions(+), 9 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index b653582592..13fa144094 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -13,7 +13,7 @@ *Author: Priya Angara* -This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) ` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) ` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. +This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) `__ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) `__ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. """ @@ -45,7 +45,7 @@ # sequence of partial mixers :math:`U_{M, x}` which may not correspond to # time evolution under a fixed mixer Hamiltonian :math:`H_M`. # -# .. figure:: ../demonstrations/dqva_mis/partial_mixers.png +# .. figure:: ../demonstrations/dqva_mis/partial-mixers.png # :align: center # :width: 100% # @@ -74,8 +74,6 @@ # Now that we have the basics in place, let's implement the QAO-Ansatz and # the DQVA! -# - ###################################################################### # Formulating the Maximum Independent Set (MIS) problem @@ -89,7 +87,7 @@ # MIS of a given graph. The following image shows some independent sets of # a graph with 5 vertices: # -# .. figure:: ../demonstrations/dqva_mis/independent_sets.png +# .. figure:: ../demonstrations/dqva_mis/independent-sets.png # :align: center # :width: 100% # @@ -262,7 +260,7 @@ def cost_layer(gamma): # This extra qubit is used as an ancillary qubit to serve as the target # for the multi-controlled Toffoli gates. # -# .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png +# .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png # :align: center # :width: 100% # @@ -490,7 +488,7 @@ def f(params): ###################################################################### # Dynamic Quantum Variational Ansatz # ---------------------------------- -# We will now formulate the MIS using the DQVA Ansatz. The cost function +# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]. The cost function # is the same as the QAO-Ansatz (the Hamming weight operator). # # In the DQVA, the way mixers are defined is slightly different from the @@ -635,7 +633,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None): # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. # - +# def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): @@ -790,6 +788,6 @@ def f(params): # References # ---------- # -# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization", +# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2020) "Approaches to Constrained Quantum Approximate Optimization", # `arXiv preprint arXiv:2010.06660 `__. # From 2429a16a556da9b4aa3ad5115785141623ad87ef Mon Sep 17 00:00:00 2001 From: pangara Date: Sun, 16 Jan 2022 10:31:30 -0800 Subject: [PATCH 09/28] fix images --- demonstrations/tutorial_dqva_mis.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 13fa144094..b6c6e13ca1 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -5,7 +5,7 @@ .. meta:: :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA - :property="og:image": https://pennylane.ai/qml/_images/mixer_unitary.png + :property="og:image": https://pennylane.ai/qml/_images/mixer-unitary.png .. related:: tutorial_qaoa_intro Introduction to QAOA From 3ce6a5605326e66649a228a99549e1d584ed8b0f Mon Sep 17 00:00:00 2001 From: pangara Date: Tue, 18 Jan 2022 10:04:51 -0800 Subject: [PATCH 10/28] fix figure size --- demonstrations/tutorial_dqva_mis.py | 4 ++-- test-results/sphinx-gallery/junit.xml | 2 +- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index b6c6e13ca1..bf4febacd5 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -47,7 +47,7 @@ # # .. figure:: ../demonstrations/dqva_mis/partial-mixers.png # :align: center -# :width: 100% +# :width: 90% # # .. # @@ -262,7 +262,7 @@ def cost_layer(gamma): # # .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png # :align: center -# :width: 100% +# :width: 80% # # .. # diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 7fd1edd634..2cc58ae1f5 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file From c9845cc9da9dbca6a46b52305ff183477851bfd7 Mon Sep 17 00:00:00 2001 From: pangara Date: Tue, 18 Jan 2022 10:42:11 -0800 Subject: [PATCH 11/28] typo --- demonstrations/tutorial_dqva_mis.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index bf4febacd5..9baf9e01d1 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -98,7 +98,7 @@ # Here, independent sets can be represented as bitstrings # :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown # above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are -# independent sets of size 0, 1, and 2 respectively. The Minimum Vertex +# independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex # Cover, i.e, vertices that cover all edges, is the dual of MIS. # # We will start with the QAO-Ansatz and then re-use most components to From 985644c40b2f1724ff3884cac5ede5dcd751a4dc Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Wed, 19 Jan 2022 13:43:09 -0800 Subject: [PATCH 12/28] Apply suggestions from code review Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com> --- demonstrations/tutorial_dqva_mis.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 9baf9e01d1..e1e82beaaa 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -31,7 +31,7 @@ # parameters. You can learn more about this in `PennyLane's tutorial on # QAOA `__. # -# For solving constrained combinatorial optimization problems such as MIS +# For solving constrained combinatorial optimization problems such as maximal independent set (MIS) # using QAOA, constraints can be imposed in two ways: # # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach) @@ -223,7 +223,7 @@ def cost_layer(gamma): # # \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}. # -# Here, where :math:`v_j` are the neighbors and :math:`l` is the number of +# Here, where :math:`v_j` are the neighbors, :math:`l` is the number of # neighbors for the :math:`i`-th node and # # .. math:: @@ -774,7 +774,7 @@ def f(params): # mixer ordering. Furthermore, Saleem et al., recommend starting with a # set of initial states that are independent sets found by any classical # polynomial time approximation algorithm to "warm start" the -# initialization. Furthermore, they propose the first useful application +# initialization. They also propose the first useful application # of `circuit-cutting techniques `__ to # solve MIS, which involves classical partitioning of a large graph into # sub-graphs, finding a solution using DQVA on a subgraph, and then From 0a75897956c34e52a521af1835d9781fa58886e7 Mon Sep 17 00:00:00 2001 From: pangara Date: Wed, 19 Jan 2022 13:43:28 -0800 Subject: [PATCH 13/28] minor edits --- demonstrations/tutorial_dqva_mis.py | 4 +++- test-results/sphinx-gallery/junit.xml | 2 +- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 9baf9e01d1..03dc29d7f0 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -340,7 +340,7 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None): # function ``f`` by running the probability circuit and calculating the # Hamming weight (which is defined as the cost). In each iteration, an # ansatz is constructed based on the mixer order, with optimal parameters -# from classical minimization of the cost function (Gradient Descent) +# from classical minimization of the cost function using gradient descent: # # .. math:: # @@ -413,6 +413,8 @@ def f(params): probs = probability_circuit( P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation ) + + # Sort bitstrings by decreasing probability top_counts = list( map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) )[::-1] diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 2cc58ae1f5..282bc3bbb5 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file From b04f721c20b87250d5f70d09f3ff153fce4ac40a Mon Sep 17 00:00:00 2001 From: pangara Date: Wed, 19 Jan 2022 13:50:26 -0800 Subject: [PATCH 14/28] add highlight to MIS 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to find the Maximum Independent Set. +# like to find the Maximum Indep endent Set. # edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)] -# edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)] graph = nx.Graph(edges) nx.draw(graph, with_labels=True) plt.show() From f360c16447419405746b5ec24b37f2b33b164017 Mon Sep 17 00:00:00 2001 From: pangara Date: Thu, 20 Jan 2022 14:36:01 -0800 Subject: [PATCH 16/28] change 1 -> I --- demonstrations/dqva_mis/independent-sets.png | Bin 560262 -> 557650 bytes demonstrations/tutorial_dqva_mis.py | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png index 7d8d0680a8ef418b7a56bb243369f5f6076b6e23..5a607bab1a24530b0ac6a428252c7ccea679a404 100644 GIT binary patch delta 408695 zcmZ5IdmvP4->1uFdsB9^l~&lUHeKjKBGlSKZpDm|8m8J(Gb@UrOyf*zSG!&4igK+m 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b/demonstrations/tutorial_dqva_mis.py index b56ddd402a..45f99429ac 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -152,7 +152,7 @@ # .. math:: # # -# b_i = \frac{1}{2} (1 - Z_i). +# b_i = \frac{1}{2} (I - Z_i). # # Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th # qubit. @@ -228,7 +228,7 @@ def cost_layer(gamma): # .. math:: # # -# \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}. +# \tilde{b}_{v_j} = \frac{I+Z_{v_j}}{2}. # # The mixer unitary can also be written as a product of :math:`N` partial # mixers :math:`V_i`: From 3ba0702ec3eb287f3c076558161621ca46b412f5 Mon Sep 17 00:00:00 2001 From: pangara Date: Thu, 20 Jan 2022 14:48:49 -0800 Subject: [PATCH 17/28] add note on bitstrings --- demonstrations/tutorial_dqva_mis.py | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 45f99429ac..2dada3f66e 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -96,8 +96,9 @@ # Independent sets of a graph with the MIS highlighted. # # Here, independent sets can be represented as bitstrings -# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown -# above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are +# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. This is convenient as the outputs of +# quantum computations are also bitstrings! +# In the graph shown above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are # independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex # Cover, i.e, vertices that cover all edges, is the dual of MIS. # From 049227ec052a12b33c276a46cac6cc98fd6b526a Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Sun, 23 Jan 2022 12:03:12 -0800 Subject: [PATCH 18/28] Apply suggestions from code review Co-authored-by: Angus Lowe Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com> --- demonstrations/tutorial_dqva_mis.py | 90 +++++++++++++++-------------- 1 file changed, 47 insertions(+), 43 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 2dada3f66e..cbd643a9d4 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -18,11 +18,11 @@ """ ###################################################################### -# QAOA, QAO-Ansatz, and DQVA +#QAO-Ansatz and DQVA # -------------------------- # # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid -# quantum-classical algorithm by `Farhi et +# quantum-classical algorithm due to `Farhi et # al. `__ is a significant meta-heuristic # for finding solutions to combinatorial optimization problems. The # quantum part involves alternating between a *cost Hamiltonian*, @@ -37,13 +37,14 @@ # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach) # 2. By constructing the variational ansatz in a way such that constraints are satisfied at all times # -# Hadfield et al., extend the QAOA framework and introduce the **Quantum +# Hadfield et al., extend QAOA by introducing the **Quantum # Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more -# general families of operators thereby supporting potentially more useful -# sets of states by focusing on unitaries rather than Hamiltonians. For -# example, a mixing operator :math:`U_M (\beta)` is decomposed into a -# sequence of partial mixers :math:`U_{M, x}` which may not correspond to -# time evolution under a fixed mixer Hamiltonian :math:`H_M`. +# general families of unitary operators. This has the potential to narrow the focus +# of the algorithm on a more useful set of states. For +# example, the mixing operator :math:`U_M (\beta)` can be a product +# of partial mixers :math:`U_{M, x}(\beta)` which may not commute. +# In contrast, the original proposal for QAOA used a mixing operator +# which was a simple product of single-qubit gates. # # .. figure:: ../demonstrations/dqva_mis/partial-mixers.png # :align: center @@ -53,13 +54,13 @@ # # Structure of a partial mixer # -# The QAO-Ansatz guarantees that the output never leaves the set of -# feasible states (in contrast, the penalty term optimization requires an -# additional pruning step since the output contains both feasible and -# infeasible solutions). However, the QAO-Ansatz requires more complex -# quantum circuitry, which in the case of the MIS, manifests itself as +# The QAO-Ansatz can be used to guarantee that the state of the circuit never leaves the set of +# feasible states. In contrast, the penalty term approach requires an +# additional pruning step since the output can correspond to an +# infeasible solution. However, the QAO-Ansatz requires more complicated +# quantum circuits. In the case of MIS, one must apply # Multi-Controlled Toffoli gates that require high connectivity between -# qubits. This makes the QAO-Ansatz inadequate for large graphs on current +# qubits, limiting the practicality of the QAO-Ansatz for large graphs on # near-term quantum computers. # # This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that @@ -118,7 +119,7 @@ ###################################################################### # Next, we define the graph shown above with 5 nodes for which we would -# like to find the Maximum Indep endent Set. +# like to find the Maximum Independent Set. # edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)] @@ -140,7 +141,7 @@ # For MIS, our goal is to maximize the number of vertices that form an # independent set, which is equivalent to maximizing the `Hamming # weight `__ of the -# bitstring. Therefore, our objective function is the Hamming weight +# bitstring. Therefore, our objective function is encoded by the Hamming weight # operator, # # .. math:: @@ -186,13 +187,13 @@ def is_indset(bitstr, G): ###################################################################### -# Next, we define a cost unitary that incorporates the objective function, +# Next, we define a cost unitary that incorporates the Hamming weight operator, # :math:`H` and is parameterized by :math:`\gamma`: # # .. math:: # # -# U_C(\gamma) = \prod_i e^{i \gamma H}. +# U_C(\gamma) = e^{i \gamma H}. # @@ -207,24 +208,24 @@ def cost_layer(gamma): # .. math:: # # -# U_M(\beta) = \prod\_i e^{i \beta M_i} +# U_M(\beta) = \prod_j e^{i \beta M_j} # -# where +# where the product is over each node :math:`j`, # # .. math:: # # -# M_i = X_i \tilde{B}, +# M_j = X_j \tilde{B}, # # and # # .. math:: # # -# \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}. +# \tilde{B} = \prod_{k=1}^{l} \tilde{b}_{v_k}. # -# Here, where :math:`v_j` are the neighbors, :math:`l` is the number of -# neighbors for the :math:`i`-th node and +# Here, :math:`v_k` are the neighbors, :math:`l` is the number of +# neighbors for the :math:`j`-th node and # # .. math:: # @@ -239,15 +240,14 @@ def cost_layer(gamma): # # U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N (I + (e^{-i\beta X_i} - I)\tilde{B}) # -# The partial mixers may not commute with each other, i.e., -# :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers -# may have different outputs. Therefore, the variational ansatz is defined +# As mentioned earlier, the partial mixers may not commute with each other, i.e., +# :math:`[V_i, V_j] \neq 0`. Therefore, the variational ansatz is defined # up to a permutation: # # .. math:: # # -# U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta)) +# U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_n(\beta)) # # where P is the permutation's function of labels from :math:`1` to # :math:`N`. @@ -271,8 +271,18 @@ def cost_layer(gamma): # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates # controlled by the neighbors of the :math:`i`'th vertex. -def mixer_layer(beta, ancilla, mixer_order): - +def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): + """ + Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations + + Args: + beta (List): A list of values for the mixing parameter + ancilla (int): The qubit to be used as the ancilla + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + + Returns: + None + """ # Permute the order of mixing unitaries if mixer_order is None: mixer_order = list(graph.nodes) @@ -311,7 +321,8 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): for qb, bit in enumerate(reversed(init_state)): if bit == "1": qml.PauliX(wires=qb) - assert len(params) == 2 * P, "Incorrect number of parameters!" + if len(params) == 2 * P: + raise ValueError("Incorrect number of parameters!") betas = [a for i, a in enumerate(params) if i % 2 == 0] gammas = [a for i, a in enumerate(params) if i % 2 == 1] @@ -372,9 +383,7 @@ def f(params): return -avg_cost # Begin outer optimization loop - best_indset = init_state - best_init_state = init_state - cur_init_state = init_state + best_indset = best_init_state = cur_init_state = init_state best_params = None best_perm = copy.copy(cur_permutation) @@ -386,9 +395,7 @@ def f(params): while inner_round < 2: print( - "Start round {}.{}, Initial state = {}".format( - mixer_round, inner_round, cur_init_state - ) + f"Start round {mixer_round}.{inner_round}, Initial state = {cur_init_state}" ) # Begin inner variational loop @@ -418,7 +425,6 @@ def f(params): top_counts = list( map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) )[::-1] - # print (top_counts) best_hamming_weight = hamming_weight(best_indset) better_strs = [] @@ -446,9 +452,7 @@ def f(params): best_perm = copy.copy(cur_permutation) cur_init_state = best_indset print( - "\tFound new independent set: {}, Hamming weight = {}".format( - best_indset, new_hamming_weight - ) + f"\tFound new independent set: {best_indset}, Hamming weight = {new_hamming_weight}" ) inner_round = inner_round + 1 @@ -466,7 +470,7 @@ def f(params): base_str = "0" * len(graph.nodes) out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1) -print("Init string: {}, Best MIS: {}".format("".join(base_str), out[0])) +print(f"Init string: {base_str}, Best MIS: {out[0]}") print() @@ -483,7 +487,7 @@ def f(params): init_str = list(base_str) init_str[i] = "1" out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2) - print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0])) + print(f"Init string: {init_str}, Best MIS: {out[0]}") print() From be7b85de833c549b97496edd5bbc2d0b7581fbb5 Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Wed, 26 Jan 2022 11:47:23 -0800 Subject: [PATCH 19/28] Update demonstrations/tutorial_dqva_mis.py --- demonstrations/tutorial_dqva_mis.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index cbd643a9d4..5039790699 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -271,7 +271,7 @@ def cost_layer(gamma): # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates # controlled by the neighbors of the :math:`i`'th vertex. -def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): +def mixer_layer(beta, ancilla, mixer_order): """ Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations From 43b919b563001c4481af47dbcef5280dafbcc652 Mon Sep 17 00:00:00 2001 From: pangara Date: Wed, 26 Jan 2022 11:49:12 -0800 Subject: [PATCH 20/28] minor edits --- demonstrations/tutorial_dqva_mis.py | 45 ++++++++++++++++----------- test-results/sphinx-gallery/junit.xml | 6 +++- 2 files changed, 31 insertions(+), 20 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index cbd643a9d4..bfd1785bd4 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -1,10 +1,10 @@ r""".. _dqva_mis: -Solving MIS with the QAO-Ansatz and the DQVA -============================================ +Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization +======================================================================== .. meta:: - :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA + :property="og:description": Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization :property="og:image": https://pennylane.ai/qml/_images/mixer-unitary.png .. related:: @@ -13,7 +13,7 @@ *Author: Priya Angara* -This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) `__ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) `__ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. +This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) `__ [#Saleem2020]_ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) `__ [#Hadfield2019]_ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. """ @@ -23,8 +23,7 @@ # # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid # quantum-classical algorithm due to `Farhi et -# al. `__ is a significant meta-heuristic -# for finding solutions to combinatorial optimization problems. The +# al. `__ [#Farhi2015]_ is an approach to solving combinatorial optimization problems using quantum computers. The # quantum part involves alternating between a *cost Hamiltonian*, # :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evaluate the # objective function and a classical optimization loop updates the ansatz @@ -272,17 +271,17 @@ def cost_layer(gamma): # controlled by the neighbors of the :math:`i`'th vertex. def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): - """ - Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations - - Args: - beta (List): A list of values for the mixing parameter - ancilla (int): The qubit to be used as the ancilla - mixer_order (Optional[List]=None): The desired permutation of the partial mixers - - Returns: - None - """ + """ + Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations + + Args: + beta (List): A list of values for the mixing parameter + ancilla (int): The qubit to be used as the ancilla + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + + Returns: + None + """ # Permute the order of mixing unitaries if mixer_order is None: mixer_order = list(graph.nodes) @@ -322,7 +321,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): if bit == "1": qml.PauliX(wires=qb) if len(params) == 2 * P: - raise ValueError("Incorrect number of parameters!") + raise ValueError("Incorrect number of parameters!") betas = [a for i, a in enumerate(params) if i % 2 == 0] gammas = [a for i, a in enumerate(params) if i % 2 == 1] @@ -781,7 +780,7 @@ def f(params): # set of initial states that are independent sets found by any classical # polynomial time approximation algorithm to "warm start" the # initialization. They also propose the first useful application -# of `circuit-cutting techniques `__ to +# of `circuit-cutting techniques `__ [#Saleem2021]_ to # solve MIS, which involves classical partitioning of a large graph into # sub-graphs, finding a solution using DQVA on a subgraph, and then # preparing a set of states to be passed as an input, essentially @@ -797,3 +796,11 @@ def f(params): # .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2020) "Approaches to Constrained Quantum Approximate Optimization", # `arXiv preprint arXiv:2010.06660 `__. # +# .. [#Farhi2014] E. Farhi, J. Goldstone, S. Gutmann. (2014) "A Quantum Approximate Optimization Algorithm", +# `arXiv preprint arXiv:1411.4028 `__. +# +# .. [#Hadfield2019] S. Hadfield, Z. Wang, B. O’Gorman, E. Rieffel, D. Venturelli, R. Biswas. (2019) "From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz", +# `arXiv preprint arXiv:1709.03489 `__. +# +# .. [#Saleem2021] Z. H. Saleem, T. Tomesh, M. A. Perlin, P. Gokhale M. Suchara. (2021) "Quantum Divide and Conquer for Combinatorial Optimization and Distributed Computing", +# `arXiv preprint arXiv:2107.07532 `__. diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 282bc3bbb5..05b2ff1dc1 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1,5 @@ - \ No newline at end of file +Traceback (most recent call last): + File "C:\Users\panga\Documents\PhD\Research\xanadu\qml\demonstrations\tutorial_dqva_mis.py", line 274, in <module> + def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): +NameError: name 'List' is not defined + \ No newline at end of file From 07a40131d3490e4d4774fd4338de643aac84260b Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Mon, 31 Jan 2022 09:58:16 -0800 Subject: [PATCH 21/28] Update demonstrations/tutorial_dqva_mis.py Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com> --- demonstrations/tutorial_dqva_mis.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 72d6b56fd1..3f1783d6c8 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -65,8 +65,8 @@ # maximizes the performance of the QAO-Ansatz by: # # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state -# or a superposition of feasible states. A feasible state can be found by -# using a classical approximate polynomial-time algorithm. +# or a superposition of feasible states. A feasible state can be found by +# using a classical approximate polynomial-time algorithm. # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries # From 8871c82fbcb8dbf3448415ac5ab804d1c9abac7d Mon Sep 17 00:00:00 2001 From: pangara Date: Mon, 31 Jan 2022 10:10:48 -0800 Subject: [PATCH 22/28] changes --- demonstrations/tutorial_dqva_mis.py | 2 +- test-results/sphinx-gallery/junit.xml | 6 +----- 2 files changed, 2 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 72d6b56fd1..80f82ab7c7 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -633,7 +633,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None): # 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters # 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros) -# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved +# 4. Steps 2 and 3 are repeated unti l the Hamming weight can no longer be improved # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. # diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 05b2ff1dc1..03ed374dee 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1,5 +1 @@ -Traceback (most recent call last): - File "C:\Users\panga\Documents\PhD\Research\xanadu\qml\demonstrations\tutorial_dqva_mis.py", line 274, in <module> - def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): -NameError: name 'List' is not defined - \ No newline at end of file + \ No newline at end of file From 06279a2b41a8cc5800eb09b7c01d76aeff664b93 Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Mon, 31 Jan 2022 12:34:13 -0800 Subject: [PATCH 23/28] Apply suggestions from code review Co-authored-by: Angus Lowe Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com> --- demonstrations/tutorial_dqva_mis.py | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 3f1783d6c8..daca247223 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -229,15 +229,15 @@ def cost_layer(gamma): # .. math:: # # -# \tilde{b}_{v_j} = \frac{I+Z_{v_j}}{2}. +# \tilde{b}_{v_k} = \frac{I + Z_{v_k}}{2}. # # The mixer unitary can also be written as a product of :math:`N` partial -# mixers :math:`V_i`: +# mixers :math:`V_j`: # # .. math:: # # -# U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N (I + (e^{-i\beta X_i} - I)\tilde{B}) +# U_M(\beta) = \prod_{j=1}^n V_j (\beta) =\prod_{j=1}^n (I + (e^{-i\beta X_j} - I)\tilde{B}) # # As mentioned earlier, the partial mixers may not commute with each other, i.e., # :math:`[V_i, V_j] \neq 0`. Therefore, the variational ansatz is defined @@ -345,11 +345,11 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None): ###################################################################### # Finally, we run the quantum-classical loop for ``m`` different mixer -# permutations. The expectation value is also calculated in the cost -# function ``f`` by running the probability circuit and calculating the -# Hamming weight (which is defined as the cost). In each iteration, an -# ansatz is constructed based on the mixer order, with optimal parameters -# from classical minimization of the cost function using gradient descent: +# permutations. To evaluate the cost function ``f``, the expectation value +# :math:`\langle C_{obj}\rangle is obtained by running the probability circuit +# and calculating the average Hamming weight. In each iteration, an +# ansatz is constructed based on the mixer order and the optimal parameters +# are determined by a classical minimization of the cost function using gradient descent. # # .. math:: # @@ -503,7 +503,7 @@ def f(params): # # U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N)) # -# where :math:`k = 1, 2 \cdots p`. +# where :math:`k = 1, 2, \dots, p`. # # In the DQVA, whenever the :math:`j`-th bit of the initial state is one, # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example, @@ -585,9 +585,9 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): qml.PauliX(wires=qb) num_nonzero = nq - hamming_weight(init_state) - assert len(params) == (nq + 1) * P, "Incorrect number of parameters!" - alpha_list = [] - gamma_list = [] + if len(params) == (nq + 1) * P: + raise ValueError("Incorrect number of parameters!") + alpha_list = gamma_list = [] last_idx = 0 for p in range(P): chunk = num_nonzero + 1 @@ -635,7 +635,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None): # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros) # 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if -# a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. +# a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. # # @@ -755,7 +755,7 @@ def f(params): init_str = list(base_str) init_str[i] = "1" out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1) - print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0])) + print(f"Init string: {init_str}, Best MIS: {out[0]}") print() From 32119b1c897f6ff0a44562a74ca19a0c64ae9f0d Mon Sep 17 00:00:00 2001 From: pangara Date: Mon, 31 Jan 2022 12:51:21 -0800 Subject: [PATCH 24/28] changes --- demonstrations/tutorial_dqva_mis.py | 37 ++++++++++++++++------------- 1 file changed, 20 insertions(+), 17 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 3c8d85c2f6..8ebeed9822 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -65,8 +65,8 @@ # maximizes the performance of the QAO-Ansatz by: # # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state -# or a superposition of feasible states. A feasible state can be found by -# using a classical approximate polynomial-time algorithm. +# or a superposition of feasible states. A feasible state can be found by +# using a classical approximate polynomial-time algorithm. # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries # @@ -95,11 +95,10 @@ # Independent sets of a graph with the MIS highlighted. # # Here, independent sets can be represented as bitstrings -# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. This is convenient as the outputs of +# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`, where the i'th bit being one represents inclusion of the ith vertex in the set. This is convenient as the outputs of # quantum computations are also bitstrings! # In the graph shown above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are -# independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex -# Cover, i.e, vertices that cover all edges, is the dual of MIS. +# independent sets of size 0, 1, 2, and 3 respectively. # # We will start with the QAO-Ansatz and then re-use most components to # formulate the DQVA. @@ -171,20 +170,24 @@ def hamming_weight(bitstr): return sum([1 for bit in bitstr if bit == "1"]) +# def is_indset(bitstr, G): +# nodes = list(G.nodes) +# ind_set = [] +# for idx, bit in enumerate(bitstr): +# if bit == "1": +# cur_neighbors = list(G.neighbors(idx)) +# for node in ind_set: +# if node in cur_neighbors: +# return False +# else: +# ind_set.append(idx) +# return True + def is_indset(bitstr, G): - nodes = list(G.nodes) - ind_set = [] - for idx, bit in enumerate(bitstr): - if bit == "1": - cur_neighbors = list(G.neighbors(idx)) - for node in ind_set: - if node in cur_neighbors: - return False - else: - ind_set.append(idx) + for edge in list(G.edges): + if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1": + return False return True - - ###################################################################### # Next, we define a cost unitary that incorporates the Hamming weight operator, # :math:`H` and is parameterized by :math:`\gamma`: From 91988e70697d0af42f7a9f28bb26e778db01e07f Mon Sep 17 00:00:00 2001 From: pangara Date: Thu, 3 Feb 2022 22:27:50 -0800 Subject: [PATCH 25/28] apply suggestions --- demonstrations/tutorial_dqva_mis.py | 258 ++++++++++++-------------- test-results/sphinx-gallery/junit.xml | 2 +- 2 files changed, 123 insertions(+), 137 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index b7da11065d..9bef9848b6 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -23,8 +23,8 @@ # # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid # quantum-classical technique due to `Farhi et -# al. `__ [#Farhi2015]_ is an approach to solving combinatorial optimization problems using quantum computers. The -# quantum part evaluates the objective function and involves alternating between a *cost Hamiltonian*, +# al. `__ [#Farhi2014]_ is an approach to solving combinatorial optimization problems using quantum computers. The +# quantum part evaluates the objective function and involves alternating between unitaries corresponding to a *cost Hamiltonian*, # :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`. A classical optimization loop updates the ansatz # parameters. You can learn more about this in `PennyLane's tutorial on # QAOA `__. @@ -65,8 +65,8 @@ # maximizes the performance of the QAO-Ansatz by: # # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state -# or a superposition of feasible states. A feasible state can be found by -# using a classical approximate polynomial-time algorithm. +# or a superposition of feasible states. A feasible state can be found by +# using a classical approximate polynomial-time algorithm. # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries # @@ -112,8 +112,7 @@ import networkx as nx from matplotlib import pyplot as plt import copy -from typing import List, Optional - +from typing import List, Optional, Tuple, Callable ###################################################################### # Next, we define the graph shown above with 5 nodes for which we would @@ -134,7 +133,6 @@ wires = len(graph.nodes) + 1 dev = qml.device("default.qubit", wires=range(wires)) - ###################################################################### # For MIS, our goal is to maximize the number of vertices that form an # independent set, which is equivalent to maximizing the `Hamming @@ -166,28 +164,17 @@ # -def hamming_weight(bitstr): +def hamming_weight(bitstr: str) -> int: return sum([1 for bit in bitstr if bit == "1"]) -# def is_indset(bitstr, G): -# nodes = list(G.nodes) -# ind_set = [] -# for idx, bit in enumerate(bitstr): -# if bit == "1": -# cur_neighbors = list(G.neighbors(idx)) -# for node in ind_set: -# if node in cur_neighbors: -# return False -# else: -# ind_set.append(idx) -# return True - -def is_indset(bitstr, G): +def is_indset(bitstr:str, G: nx.Graph) -> bool: for edge in list(G.edges): if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1": return False return True + + ###################################################################### # Next, we define a cost unitary that incorporates the Hamming weight operator, # :math:`H` and is parameterized by :math:`\gamma`: @@ -199,7 +186,7 @@ def is_indset(bitstr, G): # -def cost_layer(gamma): +def cost_layer(gamma: List) -> None: for qb in graph.nodes: qml.RZ(2 * gamma, wires=qb) @@ -264,16 +251,41 @@ def cost_layer(gamma): # # .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png # :align: center -# :width: 80% +# :width: 60% # # .. # # Formulation of the mixer unitary # # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates -# controlled by the neighbors of the :math:`i`'th vertex. +# controlled by the neighbors of the :math:`i`'th vertex. Let us define a +# helper function that applies the operations shown in the figure above. + + +def apply_multi_controlled_toffoli(ancilla: int, beta:List, qubit: int) -> None: + """Helper function to apply the multi-controlled Toffoli, targeting the ancilla qubit""" + + neighbors = list(graph.neighbors(qubit)) + + # Apply the multi-controlled Toffoli, targetting the ancilla qubit + ctrl_qubits = [i for i in neighbors] + + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) -def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): + qml.CRX(2 * beta, wires=[ancilla, qubit]) + + # Uncompute the ancilla + qml.MultiControlledX( + control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) + ) + +# We are now ready to build the mixer layer for the QAO-Ansatz by applying the +# multi-controlled Toffoli operations on each of the qubits in the graph. + + +def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> None: """ Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations @@ -290,31 +302,20 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None): mixer_order = list(graph.nodes) for qubit in list(mixer_order): - neighbors = list(graph.neighbors(qubit)) - - # Apply the multi-controlled Toffoli, targetting the ancilla qubit - ctrl_qubits = [i for i in neighbors] - qml.MultiControlledX( - control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) - ) - - qml.CRX(2 * beta, wires=[ancilla, qubit]) - - # Uncompute the ancilla - qml.MultiControlledX( - control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) - ) + apply_multi_controlled_toffoli(ancilla, beta, qubit) ###################################################################### # We are now ready to build the circuit consisting of alternating cost and # mixer layers. A list of ``params`` includes parameters for the mixer # layer (even elements) and cost layer (odd elements). -# - +# First let's define a helper function to initialize the ansatz. +# This function applies Pauli-X operations on qubits to +# prepare an initial state. -def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): +def initialize_ansatz(init_state: Optional[str]=None) -> None: + """ Helper function to initialize circuit ansatz""" nq = len(graph.nodes) - # Initialize + if init_state is None: init_state = "0" * nq @@ -322,6 +323,13 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): for qb, bit in enumerate(reversed(init_state)): if bit == "1": qml.PauliX(wires=qb) + + +def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None: + nq = len(graph.nodes) + + initialize_ansatz(init_state) + if len(params) != 2 * P: raise ValueError("Incorrect number of parameters!") @@ -334,6 +342,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): cost_layer(gamma) + ###################################################################### # Now, let's build the circuit that finds the probabilities of measuring # each bitstring. @@ -341,7 +350,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None): @qml.qnode(dev) -def probability_circuit(P, params=[], init_state=None, mixer_order=None): +def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray: qaoa_ansatz(P, params, init_state, mixer_order) return qml.probs(wires=range(wires - 1)) @@ -352,16 +361,50 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None): # :math:`\langle C_{obj}\rangle is obtained by running the probability circuit # and calculating the average Hamming weight. In each iteration, an # ansatz is constructed based on the mixer order and the optimal parameters -# are determined by a classical minimization of the cost function using gradient descent. -# -# .. math:: -# -# -# \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. +# are determined by a classical minimization of the cost function. # +# Let us construct the optimizer that uses gradient descent to minimize a +# cost function f. + +def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List, float]: + """ Helper function to optimize circuit parameters""" + + optimizer = qml.GradientDescentOptimizer(stepsize=0.5) + cur_params = init_params.copy() + + for i in range(70): + cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) + + return cur_params, opt_cost + +# We will also define a helper function that compares the independent sets +# generated by the solver with a currently-known best independent set. +# This is done by comparing the hamming weights of the bitstrings +# with the highest probabilities with the best independent set. -def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): + +def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List: + """ Helper function to calculate better independent sets""" + top_counts = list( + map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) + )[::-1] + + best_hamming_weight = hamming_weight(best_indset) + better_strs = [] + + print(top_counts[:cutoff]) + for bitstr in top_counts[:cutoff]: + + this_hamming = hamming_weight(bitstr) + if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: + better_strs.append((bitstr, this_hamming)) + + better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) + return better_strs + +def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, + threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: # Select an ordering for the partial mixers if mixer_order == None: cur_permutation = np.random.permutation(list(graph.nodes)).tolist() @@ -405,15 +448,9 @@ def f(params): init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params) print("\tCurrent Mixer Order:", cur_permutation) - + # Optimize parameters - optimizer = qml.GradientDescentOptimizer(stepsize=0.5) - cur_params = init_params.copy() - - for i in range(70): - cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) - - opt_params = cur_params.copy() + opt_params, opt_cost = optimize_params(f, init_params) print("\tOptimal cost:", opt_cost) @@ -423,21 +460,7 @@ def f(params): ) # Sort bitstrings by decreasing probability - top_counts = list( - map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) - )[::-1] - - best_hamming_weight = hamming_weight(best_indset) - better_strs = [] - - print(top_counts[:cutoff]) - for bitstr in top_counts[:cutoff]: - - this_hamming = hamming_weight(bitstr) - if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: - better_strs.append((bitstr, this_hamming)) - - better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) + better_strs = better_ind_sets(probs, best_indset, cutoff) # If no improvement was made, break and go to next mixer round if len(better_strs) == 0: @@ -464,6 +487,7 @@ def f(params): return best_indset, best_params, best_init_state, best_perm + ###################################################################### # Let's run this to find the MIS! # @@ -471,10 +495,9 @@ def f(params): base_str = "0" * len(graph.nodes) out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1) -print(f"Init string: {base_str}, Best MIS: {out[0]}") +print(f"Init string: {base_str}, Best MIS: {out}") print() - ###################################################################### # Starting with an all-zero initial string will give us an independent # set, but this may not be the maximum. Now, we run this with a few @@ -487,15 +510,15 @@ def f(params): for i in range(len(graph.nodes)): init_str = list(base_str) init_str[i] = "1" - out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2) - print(f"Init string: {init_str}, Best MIS: {out[0]}") + input_string = "".join(init_str) + out = solve_mis_qaoa(input_string, P=1, m=4, threshold=1e-5, cutoff=2) + print(f"Init string: {input_string}, Best MIS: {out[0]}") print() - ###################################################################### # Dynamic Quantum Variational Ansatz # ---------------------------------- -# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]. The cost function +# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]_. The cost function # is the same as the QAO-Ansatz (the Hamming weight operator). # # In the DQVA, the way mixers are defined is slightly different from the @@ -522,7 +545,7 @@ def f(params): # -def mixer_dqva(alpha, ancilla, init_state, mixer_order): +def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional[List]=None) -> None: # Permute the order of mixing unitaries if mixer_order is None: @@ -544,21 +567,7 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order): if pad_alpha[qubit] == None or not graph.has_node(qubit): # Turn off mixers for qubits which are already 1 continue - - neighbors = list(graph.neighbors(qubit)) - - # Apply the multi-controlled Toffoli, targetting the ancilla qubit - ctrl_qubits = [i for i in neighbors] - qml.MultiControlledX( - control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) - ) - - qml.CRX(2 * pad_alpha[qubit], wires=[ancilla, qubit]) - - # Uncompute the ancilla - qml.MultiControlledX( - control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) - ) + apply_multi_controlled_toffoli(ancilla, pad_alpha[qubit], qubit) ###################################################################### # The structure of the cost and mixer unitaries is similar QAO-Ansatz, @@ -575,22 +584,16 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order): # -def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): +def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None: nq = len(graph.nodes) - # Step1: Initialize - if init_state is None: - init_state = "0" * nq - - else: - for qb, bit in enumerate(reversed(init_state)): - if bit == "1": - qml.PauliX(wires=qb) + initialize_ansatz(init_state) num_nonzero = nq - hamming_weight(init_state) - if len(params) == (nq + 1) * P: + if len(params) != (nq + 1) * P: raise ValueError("Incorrect number of parameters!") - alpha_list = gamma_list = [] + alpha_list = [] + gamma_list = [] last_idx = 0 for p in range(P): chunk = num_nonzero + 1 @@ -612,13 +615,14 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None): + ###################################################################### # Let's also define the probability circuit for the DQVA ansatz # @qml.qnode(dev) -def probability_dqva(P, params=[], init_state=None, mixer_order=None): +def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray: dqva_ansatz(P, params, init_state, mixer_order) return qml.probs(wires=dev.wires[:-1]) @@ -636,13 +640,13 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None): # 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters # 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros) -# 4. Steps 2 and 3 are repeated unti l the Hamming weight can no longer be improved +# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. # # -def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1): +def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: # Select an ordering for the partial mixers if mixer_order == None: @@ -665,9 +669,7 @@ def f(params): return -avg_cost # Begin outer optimization loop - best_indset = init_state - best_init_state = init_state - cur_init_state = init_state + best_indset = best_init_state = cur_init_state = init_state best_params = None best_perm = copy.copy(cur_permutation) @@ -691,13 +693,7 @@ def f(params): print("\tCurrent Mixer Order:", cur_permutation) # Optimize parameters - optimizer = qml.GradientDescentOptimizer(stepsize=0.5) - cur_params = init_params.copy() - - for i in range(70): - cur_params, opt_cost = optimizer.step_and_cost(f, cur_params) - - opt_params = cur_params.copy() + opt_params, opt_cost = optimize_params(f, init_params) print("\tOptimal cost:", opt_cost) @@ -705,18 +701,8 @@ def f(params): P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation ) - top_counts = list( - map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)) - )[::-1] - - best_hamming_weight = hamming_weight(best_indset) - better_strs = [] - - for bitstr in top_counts[:1]: - this_hamming = hamming_weight(bitstr) - if is_indset(bitstr, graph) and this_hamming > best_hamming_weight: - better_strs.append((bitstr, this_hamming)) - better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) + # Sort bitstrings by decreasing probability + better_strs = better_ind_sets(probs, best_indset, cutoff) # If no improvement was made, break and go to next mixer round if len(better_strs) == 0: @@ -753,12 +739,12 @@ def f(params): # base_str = "0" * len(graph.nodes) -all_init_strs = [] for i in range(len(graph.nodes)): init_str = list(base_str) init_str[i] = "1" - out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1) - print(f"Init string: {init_str}, Best MIS: {out[0]}") + input_string = "".join(init_str) + out = solve_mis_dqva(input_string, P=1, m=4, threshold=1e-5, cutoff=1) + print(f"Init string: {input_string}, Best MIS: {out[0]}") print() diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 03ed374dee..24f7a4ec36 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file From 167d23ae6cb77384faac2d14be1ea99c11616ea8 Mon Sep 17 00:00:00 2001 From: pangara Date: Thu, 3 Feb 2022 22:29:25 -0800 Subject: [PATCH 26/28] apply suggestions --- demonstrations/tutorial_dqva_mis.py | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 9bef9848b6..7c18c47aa3 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -312,6 +312,7 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> N # This function applies Pauli-X operations on qubits to # prepare an initial state. + def initialize_ansatz(init_state: Optional[str]=None) -> None: """ Helper function to initialize circuit ansatz""" nq = len(graph.nodes) @@ -366,6 +367,7 @@ def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[ # Let us construct the optimizer that uses gradient descent to minimize a # cost function f. + def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List, float]: """ Helper function to optimize circuit parameters""" @@ -403,6 +405,7 @@ def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List: better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True) return better_strs + def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: # Select an ordering for the partial mixers @@ -644,7 +647,7 @@ def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter. # -# + def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: From 14b48e29a8fabe11025941f9769f2c1bcb8d51aa Mon Sep 17 00:00:00 2001 From: pangara Date: Fri, 4 Feb 2022 10:14:01 -0800 Subject: [PATCH 27/28] add docstrings --- demonstrations/tutorial_dqva_mis.py | 87 ++++++++++++++++++++++++++- test-results/sphinx-gallery/junit.xml | 2 +- 2 files changed, 85 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 7c18c47aa3..29940e8658 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -165,10 +165,13 @@ def hamming_weight(bitstr: str) -> int: + """Helper function to calculate the hamming weight of a bitstring""" return sum([1 for bit in bitstr if bit == "1"]) def is_indset(bitstr:str, G: nx.Graph) -> bool: + """Helper function that returns True when a bitstring is a valid independent set""" + for edge in list(G.edges): if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1": return False @@ -187,6 +190,15 @@ def is_indset(bitstr:str, G: nx.Graph) -> bool: def cost_layer(gamma: List) -> None: + """ + Builds the QAO-Ansatz cost layer for max independent set problem using PennyLane operations + + Args: + gamma (List): A list of values for the cost parameter + + Returns: + None + """ for qb in graph.nodes: qml.RZ(2 * gamma, wires=qb) @@ -314,7 +326,7 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> N def initialize_ansatz(init_state: Optional[str]=None) -> None: - """ Helper function to initialize circuit ansatz""" + """Helper function to initialize circuit ansatz""" nq = len(graph.nodes) if init_state is None: @@ -327,6 +339,18 @@ def initialize_ansatz(init_state: Optional[str]=None) -> None: def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None: + """ + Builds the QAO-Ansatz for max independent set problem + + Args: + P (int): Number of parameters + params (Optional[List]=[]): A list of 2*P parameters + init_state (Optional[str]=None): Bitstring representing the initial state + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + + Returns: + None + """ nq = len(graph.nodes) initialize_ansatz(init_state) @@ -352,6 +376,8 @@ def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=Non @qml.qnode(dev) def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray: + """Obtains probabilities for the QAO-Ansatz given a set of parameters, initial state, and the mixer order""" + qaoa_ansatz(P, params, init_state, mixer_order) return qml.probs(wires=range(wires - 1)) @@ -408,6 +434,22 @@ def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List: def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: + """ + Solver for max independent set problem using the QAO-Ansatz + + Args: + init_state (Optional[str]=None): Bitstring representing the initial state + P (int): Number of parameters + m (int): Number of mixer rounds + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + threshold (Optional[float]=1e-5): A threshold to remove low probability outcomes + cutoff (Optional[int]=1): number of "better" independent sets to consider + Returns: + str: Best independent set + np.ndarray: Best parameters + str: Best initial state + List: Best mixer order + """ # Select an ordering for the partial mixers if mixer_order == None: cur_permutation = np.random.permutation(list(graph.nodes)).tolist() @@ -549,7 +591,17 @@ def f(params): def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional[List]=None) -> None: - + """ + Builds the DQVA mixer layer for max independent set problem using PennyLane operations + + Args: + beta (List): A list of values for the mixing parameter + ancilla (int): The qubit to be used as the ancilla + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + + Returns: + None + """ # Permute the order of mixing unitaries if mixer_order is None: mixer_order = list(graph.nodes) @@ -588,6 +640,18 @@ def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None: + """ + Builds the DQVA for max independent set problem + + Args: + P (int): Number of parameters + params (Optional[List]=[]): a list of 2*P parameters + init_state (Optional[str]=None): bitstring representing the initial state + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + + Returns: + None + """ nq = len(graph.nodes) initialize_ansatz(init_state) @@ -626,6 +690,8 @@ def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=Non @qml.qnode(dev) def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray: + """Obtains probabilities for the DQVA given a set of parameters, initial state, and the mixer order""" + dqva_ansatz(P, params, init_state, mixer_order) return qml.probs(wires=dev.wires[:-1]) @@ -650,7 +716,22 @@ def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]: - + """ + Solver for max independent set problem using the DQVA + + Args: + init_state (Optional[str]=None): Bitstring representing the initial state + P (int): Number of parameters + m (int): Number of mixer rounds + mixer_order (Optional[List]=None): The desired permutation of the partial mixers + threshold (Optional[float]=1e-5): A threshold to remove low probability outcomes + cutoff (Optional[int]=1): number of "better" independent sets to consider + Returns: + str: Best independent set + np.ndarray: Best parameters + str: Best initial state + List: Best mixer order + """ # Select an ordering for the partial mixers if mixer_order == None: cur_permutation = np.random.permutation(list(graph.nodes)).tolist() diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml index 24f7a4ec36..c3f661e6b4 100644 --- a/test-results/sphinx-gallery/junit.xml +++ b/test-results/sphinx-gallery/junit.xml @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file From c3e4f7fb4b306c7c432ef7833068863bc8af9300 Mon Sep 17 00:00:00 2001 From: Priya Angara Date: Tue, 8 Feb 2022 09:25:12 -0800 Subject: [PATCH 28/28] Apply suggestions from code review Co-authored-by: Olivia Di Matteo <2068515+glassnotes@users.noreply.github.com> --- demonstrations/tutorial_dqva_mis.py | 23 +++++++++++++---------- 1 file changed, 13 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py index 29940e8658..340685dced 100644 --- a/demonstrations/tutorial_dqva_mis.py +++ b/demonstrations/tutorial_dqva_mis.py @@ -18,7 +18,7 @@ """ ###################################################################### -#QAO-Ansatz and DQVA +# QAO-Ansatz and DQVA # -------------------------- # # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid @@ -29,7 +29,7 @@ # parameters. You can learn more about this in `PennyLane's tutorial on # QAOA `__. # -# For solving constrained combinatorial optimization problems such as maximal independent set (MIS) +# For solving constrained combinatorial optimization problems such as maximum independent set (MIS) # using QAOA, constraints can be imposed in two ways: # # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach) @@ -57,7 +57,7 @@ # additional pruning step since the output can correspond to an # infeasible solution. However, the QAO-Ansatz requires more complicated # quantum circuits. In the case of MIS, one must apply -# Multi-Controlled Toffoli gates that require high connectivity between +# multi-controlled Toffoli gates that require high connectivity between # qubits, limiting the practicality of the QAO-Ansatz for large graphs on # near-term quantum computers. # @@ -78,7 +78,7 @@ # Formulating the Maximum Independent Set (MIS) problem # ----------------------------------------------------- # -# Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices +# Given a graph :math:`G = (V, E)` where :math:`V` is the set of vertices # and :math:`E` is the set of edges, an *independent set* is a subset of # vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'` # share an edge. A *Maximum Independent Set* (MIS) is an independent set of @@ -159,7 +159,7 @@ ###################################################################### # We need a couple of helper functions here: ``hamming_weight``, which -# calculates the Hamming weight of a given bitstring and ``is_indset``, +# calculates the Hamming weight of a given bitstring, and ``is_indset``, # which verifies whether a given bitstring is a valid independent set. # @@ -180,7 +180,7 @@ def is_indset(bitstr:str, G: nx.Graph) -> bool: ###################################################################### # Next, we define a cost unitary that incorporates the Hamming weight operator, -# :math:`H` and is parameterized by :math:`\gamma`: +# :math:`H`, and is parameterized by :math:`\gamma`: # # .. math:: # @@ -269,7 +269,7 @@ def cost_layer(gamma: List) -> None: # # Formulation of the mixer unitary # -# In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates +# In the figure above, an :math:`X` rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates # controlled by the neighbors of the :math:`i`'th vertex. Let us define a # helper function that applies the operations shown in the figure above. @@ -293,6 +293,7 @@ def apply_multi_controlled_toffoli(ancilla: int, beta:List, qubit: int) -> None: control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors) ) +###################################################################### # We are now ready to build the mixer layer for the QAO-Ansatz by applying the # multi-controlled Toffoli operations on each of the qubits in the graph. @@ -385,7 +386,7 @@ def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[ ###################################################################### # Finally, we run the quantum-classical loop for ``m`` different mixer # permutations. To evaluate the cost function ``f``, the expectation value -# :math:`\langle C_{obj}\rangle is obtained by running the probability circuit +# :math:`\langle C_{obj}\rangle` is obtained by running the probability circuit # and calculating the average Hamming weight. In each iteration, an # ansatz is constructed based on the mixer order and the optimal parameters # are determined by a classical minimization of the cost function. @@ -406,9 +407,11 @@ def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List return cur_params, opt_cost + +###################################################################### # We will also define a helper function that compares the independent sets # generated by the solver with a currently-known best independent set. -# This is done by comparing the hamming weights of the bitstrings +# This is done by comparing the Hamming weights of the bitstrings # with the highest probabilities with the best independent set. @@ -576,7 +579,7 @@ def f(params): # # where :math:`k = 1, 2, \dots, p`. # -# In the DQVA, whenever the :math:`j`-th bit of the initial state is one, +# In the DQVA, whenever the :math:`j`-th bit of the initial state is 1, # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example, # if the initial state is :math:`\vert 01101 \rangle`, then #

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