QuantumBayesian.jl

Julia package for simulating quantum evolution efficiently, including the quantum Bayesian approach to stochastic measurement update and continuous readout. The package handles tensor products between arbitrary d-dimensional Hilbert space factors (including partial trace). Note: now requires julia v0.6+.

The package uses an efficient (sparse) matrix backend, with minimal structural overhead and a lightweight interface. By using proper completely positive maps to simulate time evolution, the package avoids common pitfalls when simulating (stochastic) differential equations for quantum systems. This technique guarantees that a physically sensible state is produced during the evolution, with time-step size affecting only solution precision, and not solution integrity.

See the notebooks directory for more detailed usage examples.

Simple Qubit Examples

using QuantumBayesian
using PyPlot

q = qubit()

QuantumBayesian.QFactor: Qubit
Dims  : 2
Ops   : "d", "y", "x", "u", "z", "i"

Ω  = 2*π;        # Rabi frequency
τ = 3.0;         # Measurement collapse timescale
Γ = 1/(2*τ);     # Measurement dephasing rate
T = (0.0, 6*τ);  # Time duration of simulation;
dt = 1e-2;       # Simulation timestep (coarse to show method precision)

# Initial condition
init = ground(q)

# Time-dependent Hamiltonian
f(t) = 2*exp(-(t-3*τ)^2/2)/sqrt(2π)
H(t) = f(t)*(Ω/2)*q("y");

# Measurement dephasing
DM = [sqrt(Γ/2)*q("z")];
# Stochastic monitoring (quantum-limited efficiency)
SM = [(q("z"), τ, 1.0)]

# Bloch coordinate expectation values 
fs = collect(ρ -> real(expect(ρ, q(l))) for l in ["x","y","z"])

# Lindblad pulse trajectory (plotting code omitted)
t = trajectory(lind(dt, H, clist=DM), init, T, fs..., dt=dt)

INFO: Trajectory: steps = 1799, points = 1000, values = 3
INFO: Time elapsed: 0.086969712 s, Steps per second: 20685.362278766657

# Stochastic pulse trajectory (plotting code omitted)
t = trajectory(meas(dt, H, mclist=SM), init, T, fs..., dt=dt)

INFO: Trajectory: steps = 1799, points = 1000, values = 3
INFO: Readout: values = 1
INFO: Time elapsed: 0.118673362 s, Steps per second: 15159.257053828136

# Stochastic pulse ensemble (plotting code omitted)
# No parallelization of ensemble creation
t = ensemble(2500, meas(dt, H, mclist=SM), init, T, fs..., dt=dt)
# Mean plotted with Std-Dev shaded behind it

INFO: Trajectories: 2500, steps each: 1799, points each: 1000, values each = 3
INFO: Readouts: values each = 1
INFO: Time elapsed: 262.68186513 s, Steps: 2500000, Steps per second: 9517.21581070228

# Stochastic CW Z-Y measurement (plotting code omitted)
SM = [(q("z"), τ, 0.4), (q("y"), τ, 0.4)]
t = trajectory(meas(dt, H, mclist=SM), init, T, fs..., dt=dt)

INFO: Trajectory: steps = 899, points = 899, values = 3
INFO: Readout: values = 2
INFO: Time elapsed: 0.081660735 s, Steps per second: 11008.962875487221

# Stochastic CW Z-Y measurement ensemble (plotting code omitted)
# Parallelization enabled with 4 additional processes
SM = [(q("z"), τ, 0.4), (q("y"), τ, 0.4)]
t = ensemble(2000, meas(dt, H, mclist=SM), init, T, fs..., dt=dt)
# Mean plotted with Std-Dev shaded behind it

INFO: Trajectories: 2000, steps each: 899, points each: 899, values each = 3
INFO: Readouts: values each = 2
INFO: Time elapsed: 86.065250033 s, Steps: 1798000, Steps per second: 20891.126201464504

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant Number 1915015. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.