Limitations of probabilistic error cancellation for open dynamics beyond sampling overhead

TL;DR

This paper investigates the fundamental limitations of probabilistic error cancellation in quantum simulations of open dynamics, revealing that Trotter-like errors arise from commutation relations and are unavoidable with stepwise mitigation methods.

Contribution

It provides a detailed analysis of errors in stepwise probabilistic error cancellation, highlighting their dependence on commutation relations and demonstrating fundamental limitations.

Findings

Errors depend on commutating relations between superoperators.

Stepwise mitigation introduces Trotter-like errors.

Limitations motivate development of continuous error cancellation methods.

Abstract

Quantum simulation of dynamics is an important goal in the NISQ era, within which quantum error mitigation may be a viable path towards modifying or eliminating the effects of noise. Most studies on quantum error mitigation have been focused on the resource cost due to its exponential scaling in the circuit depth. Methods such as probabilistic error cancellation rely on discretizing the evolution into finite time steps and applying the mitigation layer after each time step, modifying only the noise part without any Hamiltonian-dependence. This may lead to Trotter-like errors in the simulation results even if the error mitigation is implemented ideally, which means that the number of samples is taken as infinite. Here we analyze the aforementioned errors which have been largely neglected before. We show that, they are determined by the commutating relations between the superoperators of the unitary part, the device noise part and the noise part of the open dynamics to be simulated. We include both digital quantum simulation and analog quantum simulation setups, and consider defining the ideal error mitigation map both by exactly inverting the noise channel and by approximating it to the first order in the time step. We take single-qubit toy models to numerically demonstrate our findings. Our results illustrate fundamental limitations of applying probabilistic error cancellation in a stepwise manner to continuous dynamics, thus motivating the investigations of truly time-continuous error cancellation methods.