$\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input)
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| Format: | Preprint |
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2026
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| _version_ | 1866908750259945472 |
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| author | Grier, Daniel Morris, Jackson Wu, Kewen |
| author_facet | Grier, Daniel Morris, Jackson Wu, Kewen |
| contents | $\mathsf{QAC}^0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03243 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input) Grier, Daniel Morris, Jackson Wu, Kewen Computational Complexity Quantum Physics $\mathsf{QAC}^0$ 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. |
| title | $\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input) |
| topic | Computational Complexity Quantum Physics |
| url | https://arxiv.org/abs/2601.03243 |