Pauli path integral

Main Reference:

• A polynomial-time classical algorithm for noisy quantum circuits

• quantum mean value

• A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling

• non-unital …

Talk:

• STOC 2023 by Liu

• Simons by Liu

• QIP by Aharonov

…. todo

PauliPropagation.jl is a Julia package for Pauli propagation simulation of quantum circuits and quantum systems.

MSRudolph/PauliPropagation.jl

Trotter circuit

using PauliPropagation

## number of qubits
nqubits = 32

## define the observable
# here I...IZI...I
observable = PauliString(nqubits, :Z, 16)

## define the circuit
# the number of layers
nlayers = 32

# bricklayertopology is also the default if you don't provide any
topology = bricklayertopology(nqubits; periodic=true)

# a circuit containing RX and RZZ Pauli gates on the topology
# derived from the Trotterization of a transverse field Ising Hamiltonian
circuit = tfitrottercircuit(nqubits, nlayers; topology=topology)

# time step
dt = 0.1
# count the number of parameters
nparams = countparameters(circuit)
# define the parameter vector
parameters = ones(nparams) * dt

## the truncations
# maximum Pauli weight
max_weight = 6
# minimal coefficient magnitude
min_abs_coeff = 1e-4

## propagate through the circuit with our best (and currently only propagation method)
pauli_sum = propagate(circuit, observable, parameters; max_weight=max_weight, min_abs_coeff=min_abs_coeff)

## overlap with the initial state
overlapwithzero(pauli_sum)
# yields 0.154596728241...

0.15459672824142454