  Quantum algorithms for simulating quantum systems provide a clear and
provable advantage over classical algorithms in fault-tolerant settings. There
is also interest in quantum algorithms and their implementation in Noisy
Intermediate Scale Quantum (NISQ) settings. In these settings, various noise
sources and errors must be accounted for when executing any experiments.
Recently, NISQ devices have been verified as versatile testbeds for simulating
open quantum systems and have been used to simulate simple quantum channels.
Our goal is to solve the more complicated problem of simulating convex mixtures
of single qubit Pauli channels on NISQ devices. We consider two specific cases:
mixtures of Markovian channels that result in a non-Markovian channel (M+M=nM)
and mixtures of non-Markovian channels that result in a Markovian channel
(nM+nM=M). For the first case, we consider mixtures of Markovian single-qubit
Pauli channels; for the second case, we consider mixtures of Non-Markovian
single-qubit depolarising channels, which is a special case of the single-qubit
Pauli channel. We show that efficient circuits, which account for the topology
of currently available devices and current levels of decoherence, can be
constructed by heuristic approaches that reduce the number of CNOT gates used
in our circuit. We also present a strategy for regularising the process matrix
so that the process tomography yields a completely positive and
trace-preserving (CPTP) channel.
