QAC0 Contains TC0 (with Many Copies of the Input)

2026-01-06

arXiv.org

AI Summary

This work investigates the computational power of constant-depth quantum circuits, denoted $\mathsf{QAC}^0$, and their advantage over classical constant-depth circuits $\mathsf{AC}^0$. By introducing multiple copies of the input and leveraging amplitude amplification, the authors transform approximate quantum constructions into exact implementations. This approach yields the first unconditional separation $\mathsf{QAC}^0 \not\subseteq \mathsf{AC}^0[p]$ for any prime $p$, and establishes the inclusion $\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$. The results not only clarify the superiority of $\mathsf{QAC}^0$ in computing nontrivial Boolean functions but also provide novel techniques for designing constant-depth quantum circuits.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

$\mathsf{QAC}^0$ is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical $\mathsf{AC}^0$ lower bounds to $\mathsf{QAC}^0$ have failed. We give one possible explanation of this: $\mathsf{QAC}^0$ circuits are significantly more powerful than their classical counterparts. We show the unconditional separation $\mathsf{QAC}^0\not\subset\mathsf{AC}^0[p]$ for decision problems, which also resolves for the first time whether $\mathsf{AC}^0$ could be more powerful than $\mathsf{QAC}^0$. Moreover, we prove that $\mathsf{QAC}^0$ circuits can compute a wide range of Boolean functions if given multiple copies of the input: $\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$. Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.

Problem

• QAC^0
• AC^0
• TC^0
• quantum circuits
• Boolean functions

Innovation

• QAC^0
• quantum circuits
• amplitude amplification
• circuit complexity
• TC^0