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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.09931 |
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| _version_ | 1866912642484928512 |
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| author | Karamchedu, Chaitanya Fox, Matthew Gottesman, Daniel |
| author_facet | Karamchedu, Chaitanya Fox, Matthew Gottesman, Daniel |
| contents | Say a collection of $n$-qu$d$it gates $Γ$ is eventually universal if and only if there exists $N_0 \geq n$ such that for all $N \geq N_0$, one can approximate any $N$-qu$d$it unitary to arbitrary precision by a circuit over $Γ$. In this work, we improve the best known upper bound on the smallest $N_0$ with the above property. Our new bound is roughly $d^4n$, where $d$ is the local dimension (the `$d$' in qu$d$it), whereas the previous bound was roughly $d^8n$. For qubits ($d = 2$), our result implies that if an $n$-qubit gate set is eventually universal, then it will exhibit universality when acting on a $16n$ qubit system, as opposed to the previous bound of a $256n$ qubit system. In other words, if adding just $15n$ ancillary qubits to a quantum system (as opposed to the previous bound of $255 n$ ancillary qubits) does not boost a gate set to universality, then no number of ancillary qubits ever will. Our proof relies on the invariants of finite linear groups as well as a classification result for all finite groups that are unitary $2$-designs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on Eventually Universal Quantum Gate Sets Karamchedu, Chaitanya Fox, Matthew Gottesman, Daniel Quantum Physics Computational Complexity Say a collection of $n$-qu$d$it gates $Γ$ is eventually universal if and only if there exists $N_0 \geq n$ such that for all $N \geq N_0$, one can approximate any $N$-qu$d$it unitary to arbitrary precision by a circuit over $Γ$. In this work, we improve the best known upper bound on the smallest $N_0$ with the above property. Our new bound is roughly $d^4n$, where $d$ is the local dimension (the `$d$' in qu$d$it), whereas the previous bound was roughly $d^8n$. For qubits ($d = 2$), our result implies that if an $n$-qubit gate set is eventually universal, then it will exhibit universality when acting on a $16n$ qubit system, as opposed to the previous bound of a $256n$ qubit system. In other words, if adding just $15n$ ancillary qubits to a quantum system (as opposed to the previous bound of $255 n$ ancillary qubits) does not boost a gate set to universality, then no number of ancillary qubits ever will. Our proof relies on the invariants of finite linear groups as well as a classification result for all finite groups that are unitary $2$-designs. |
| title | Bounds on Eventually Universal Quantum Gate Sets |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2510.09931 |