$\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)

• URL: http://arxiv.org/abs/2601.03243v1
• Date: Tue, 06 Jan 2026 18:40:44 GMT
• Title: $\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)
• Authors: Daniel Grier, Jackson Morris, Kewen Wu,
• Abstract summary: $mathsfQAC0$ is the class of constant-depth quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates.<n>We show that $mathsfQAC0$ circuits are significantly more powerful than their classical counterparts.
• Score: 0.9023122463034333
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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