Multi-qubit Toffoli with exponentially fewer T gates
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908582090375168 |
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| author | Gosset, David Kothari, Robin Zhang, Chenyi |
| author_facet | Gosset, David Kothari, Robin Zhang, Chenyi |
| contents | Prior work of Beverland et al. has shown that any exact Clifford+$T$ implementation of the $n$-qubit Toffoli gate must use at least $n$ $T$ gates. Here we show how to get away with exponentially fewer $T$ gates, at the cost of incurring a tiny $1/\mathrm{poly}(n)$ error that can be neglected in most practical situations. More precisely, the $n$-qubit Toffoli gate can be implemented to within error $ε$ in the diamond distance by a randomly chosen Clifford+$T$ circuit with at most $O(\log(1/ε))$ $T$ gates. We also give a matching $Ω(\log(1/ε))$ lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use $Ω(n)$ $T$ gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the $T$-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-qubit Toffoli with exponentially fewer T gates Gosset, David Kothari, Robin Zhang, Chenyi Quantum Physics Prior work of Beverland et al. has shown that any exact Clifford+$T$ implementation of the $n$-qubit Toffoli gate must use at least $n$ $T$ gates. Here we show how to get away with exponentially fewer $T$ gates, at the cost of incurring a tiny $1/\mathrm{poly}(n)$ error that can be neglected in most practical situations. More precisely, the $n$-qubit Toffoli gate can be implemented to within error $ε$ in the diamond distance by a randomly chosen Clifford+$T$ circuit with at most $O(\log(1/ε))$ $T$ gates. We also give a matching $Ω(\log(1/ε))$ lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use $Ω(n)$ $T$ gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the $T$-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue. |
| title | Multi-qubit Toffoli with exponentially fewer T gates |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2510.07223 |