Digital simulation of convex mixtures of Markovian and non-Markovian single qubit Pauli channels on NISQ devices

Abstract

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.

Key points

• This work simulates the convex mixtures of single qubit Markovian and non-Markovian quantum channels on NISQ devices provided by the IMBQE.

• The circuits used to implement the channels take into account the topolgy of the quantum device used as well as the number of CNOT gates used.

• We present a strategy for regularising the process matrix to ensure the quantum process tomography yields a CPTP channel. Something that is not correctly implemented in Qiskit.

• A method is outlined for finding mixtures of non-Markovian depolarising channels that yield a Markovian depolarising channel. It is also shown that, one cannot convexly mix two Markovian depolarising channels that leads to a non-Markovian depolarising channel.