Partitioned Quantum Cellular Automata (PQCA)

A quantum cellular automaton iteratively applies some update circuit to some initial state. A partitioned quantum cellular automaton (PQCA) derives its update circuit by partitioning a lattice of qubits into cells, and then applying the same circuit to each cell. The full update circuit is created by composing several such partitioned updates. There is a review of Quantum Cellular Automata by Terry Farrelly, published in Quantum, and available at doi:10.22331/q-2020-11-30-368.

This python module allows for the easy creation and execution of partitioned quantum cellular automata. To create an automaton you will need:

• A starting state (list of 0s and 1s)
• Update Frames (see pqca.update_frame)
• A simulator / quantum computer (see pqca.backend)

An Update Frame combines a tessellation with a circuit to be applied to each cell in that tessellation. A tessellation just partitions a list of qubits into cells. For example pqca.tessellation.one_dimensional(10,2) partitions 10 qubits into 5 cells, each of size 2. The Update Frame would then need to be a circuit on 2 qubits. For more complicated tessellations you can use, e.g. pqca.tessellation.n_dimensional([4,2,4],[2,2,2]) which partitions 32 qubits as though they were arranged in a lattice of shape 4 x 2 x 4, with each cell of size 2 x 2 x 2. The Update Frame would then need to be a circuit on 8 qubits.

One can then call next(automaton) which will advance the internal state of the automaton and return the new state.

Making music with PQCA

Check the tutorials for examples on using the pqca package and learn how to render music from the 1-D and 2-D quantum CA. We show how to save MIDI and Music XML (score) files, which can be uploaded to music editors (e.g., Sibelius and MuseScore).

Installation

Install via pip from the command line with the command:

pip install pqca

Example

Here is an example that creates two update frames, both applying a simple CX gate, but with offset tessellations.

# Create circuit
cx_circuit = qiskit.QuantumCircuit(2)
cx_circuit.cx(0, 1)

# Create tessellation
tes = pqca.tessellation.one_dimensional(10, 2)

# Create update frames
update_1 = pqca.UpdateFrame(tes, cx_circuit)
update_2 = pqca.UpdateFrame(tes.shifted_by(1), cx_circuit)


# Create initial state
initial_state = [1]*10

# Specify a backend; `pqca.backend.qiskit()` returns IBM's Aer simulator by default
# See backend.py for more details and instructions on coding your own backend
backend = pqca.backend.qiskit()

# Create the automaton
automaton = pqca.Automaton(initial_state, [update_1, update_2], backend)

# The automaton can be called like any other iterator
# The following line advances the internal state, and returns the new state
next(automaton)

Documentation

Detailed documentation can be found at readthedocs.io as well as in the docstrings of the python files themselves.

Licensing

The source code is available under the MIT licence and can be found on Hector Miller-Bakewell's github.

Acknowledgements

This package was created as part of the QuTune Project.