Perturbative expansions for simulating noisy Clifford circuits

Import with using QuantumClifford.Experimental.NoisyCircuits.

This module enables the simulation of noisy Clifford circuits through a perturbative expansion in the noise parameter (assuming the noise is small). Instead of simulating many Monte Carlo trajectories, only the leading order trajectories are exhaustively enumerated and simulated.

Here is an example of a purification circuit (the same circuit seen in the Monte Carlo example)

good_bell_state = S"XX
                    ZZ"
canonicalize_rref!(good_bell_state)
initial_state = MixedDestabilizer(good_bell_state⊗good_bell_state)

g1 = sCNOT(1,3) # CNOT between qubit 1 and qubit 3 (both with Alice)
g2 = sCNOT(2,4) # CNOT between qubit 2 and qubit 4 (both with Bob)
m = BellMeasurement([sMX(3),sMX(4)]) # Bell measurement on qubit 3 and 4
v = VerifyOp(good_bell_state,[1,2]) # Verify that qubit 1 and 2 indeed form a good Bell pair
epsilon = 0.01 # The error rate
n = NoiseOpAll(UnbiasedUncorrelatedNoise(epsilon))

# This circuit performs a depolarization at rate `epsilon` to all qubits,
# then bilater CNOT operations
# then a Bell measurement
# followed by checking whether the final result indeed corresponds to the correct Bell pair.
circuit = [n,g1,g2,m,v]

petrajectories(initial_state, circuit)

Dict{CircuitStatus, Float64} with 3 entries:
  true_success:CircuitStatus(1)  => 0.967065
  false_success:CircuitStatus(2) => 0.019406
  failure:CircuitStatus(3)       => 0.0129373

For more examples, see the notebook comparing the Monte Carlo and Perturbative method or this tutorial on entanglement purification.

Symbolic expansions

The perturbative expansion method works with symbolic variables as well. One can use any of the symbolic libraries available in Julia and simply plug symbolic parameters in lieu of numeric parameters. A detailed example is available as a Jupyter notebook.

Interface for custom operations

If you want to create a custom gate type (e.g. calling it Operation), you need to definite the following methods.

applyop_branches!(s::T, g::Operation; max_order=1)::Vector{Tuple{T,Symbol,Real,Int}} where T is a tableaux type like Stabilizer or a Register. The Symbol is the status of the operation, the Real is the probability for that branch, and the Int is the order of that branch.

There is also applynoise_branches! which is convenient for use in NoisyGate, but you can also just make up your own noise operator simply by implementing applyop_branches! for it.

You can also consult the list of implemented operators.