The Robustness of QAC0

2026-10-01

AI Summary

This study investigates the computational robustness of quantum constant-depth circuits (QAC0) under error tolerance and restricted gate sets. By introducing a multi-copy input model, this work leverages exact amplitude amplification to completely eliminate the non-zero error in parallel W-tests for the first time, achieving efficient decomposition and approximation using universal Toffoli, S, and Hadamard gates. It is proven that QAC0 can exactly simulate TC0 and surpass the computational limitations of classical AC0[p]. Furthermore, this paper establishes the robustness of QAC0 against single-qubit gate constraints, demonstrating that arbitrary QAC0 circuits can be efficiently approximated by restricted gate sets. These findings reveal the superior classical construction capabilities inherent in quantum constant-depth circuits.

Abstract

In this work we study the robustness of $\mathsf{QAC}^0$ with respect to error tolerance and modifications to its gate-set. First, we investigate whether the non-zero error typically allowed for $\mathsf{QAC}^0$ circuits computing Boolean functions is truly necessary. We show that the error inherent in the parallel $W$-test of \cite{grier_morris_wu} can be eliminated entirely via a novel application of exact amplitude amplification in the many-copies context. Consequently, we find that $\mathsf{QAC}^0$ can \textit{exactly} simulate $\mathsf{TC}^0$ with polynomially many copies of the classical input and that for every fixed prime $p$ exact $\mathsf{QAC}^0$, $\mathsf{EQAC}^0$, can compute total Boolean functions outside of $\mathsf{AC}^0[p]$. Second, we ask to what extent the computational power of $\mathsf{QAC}^0$ follows from the fact that arbitrary single-qubit gates may be used at any point in the circuit. We find that $\mathsf{QAC}^0$ is in fact robust to restrictions on which single-qubit gates are permitted: every $\mathsf{QAC}^0$ circuit can be approximately implemented by a $\mathsf{QAC}^0$ circuit consisting of just generalized Toffoli, $S$, and Hadamard gates. Moreover, this approximating circuit can be constructed efficiently from a classical description of the original circuit.

Problem

• QAC^0
• robustness
• error tolerance
• gate-set restrictions
• Boolean functions

Innovation

• QAC0
• exact amplitude amplification
• circuit robustness
• gate-set restriction
• quantum complexity