Classical simulability of quantum circuits followed by sparse classical post-processing

TL;DR

This paper characterizes when polynomial-size quantum circuits with sparse classical post-processing are classically simulable, extending previous results and analyzing the complexity for constant-depth circuits with such post-processing.

Contribution

It provides a necessary and sufficient condition for classical simulability of quantum circuits with sparse post-processing, and analyzes the complexity for constant-depth circuits with this post-processing.

Findings

Quantum circuits with sparse post-processing are classically simulable under certain conditions.

Constant-depth quantum circuits with sparse post-processing are simulable with access to commuting quantum circuits.

The paper extends previous results to broader classes of quantum circuits and post-processing functions.

Abstract

We study the classical simulability of a polynomial-size quantum circuit $C_n$ on $n$ qubits followed by sparse classical post-processing (SCP) on $m$ bits, where $m \leq n \leq {\rm poly}(m)$. The SCP is described by a non-zero Boolean function $f_m$ that is classically computable in polynomial time and is sparse, i.e., has a peaked Fourier spectrum. First, we provide a necessary and sufficient condition on $C_n$ such that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable. This characterization extends the result of Van den Nest and implies that various quantum circuits followed by SCP are classically simulable. Examples include IQP circuits, Clifford Magic circuits, and the quantum part of Simon's algorithm, even though these circuits alone are hard to simulate classically. Then, we consider the case where $C_n$ has constant depth $d$. While it is unlikely that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable, we show that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on $n+1$ qubits. Each such circuit consists of at most deg($f_m$) commuting gates and each commuting gate acts on at most $2^d+1$ qubits, where deg($f_m$) is the Fourier degree of $f_m$. This provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.