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

2026-03-06

AI Summary

This work investigates whether polynomial-size quantum circuits followed by arbitrary sparse classical post-processing (SCP) can be efficiently simulated classically. By leveraging Fourier analysis, characterizations of Boolean function sparsity, constructions of commuting gate circuits, and probabilistic simulation techniques, the authors establish—for the first time—a necessary and sufficient condition for the classical simulability of quantum circuits under arbitrary SCP, thereby generalizing Van den Nest’s result. As applications, they demonstrate that the quantum parts of IQP, Clifford Magic, and Simon’s algorithm, although potentially hard to simulate in isolation, become efficiently classically simulable when followed by any SCP. Furthermore, they show that constant-depth quantum circuits under certain communication gate constraints are also efficiently simulable, opening new avenues for classical simulation of quantum computations.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

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.

Problem

• classical simulability
• quantum circuits
• sparse classical post-processing
• Fourier spectrum
• constant-depth circuits

Innovation

• classical simulability
• sparse classical post-processing
• Fourier sparsity
• constant-depth quantum circuits
• commuting quantum circuits