Bounds on Eventually Universal Quantum Gate Sets

• URL: http://arxiv.org/abs/2510.09931v1
• Date: Sat, 11 Oct 2025 00:05:41 GMT
• Title: Bounds on Eventually Universal Quantum Gate Sets
• Authors: Chaitanya Karamchedu, Matthew Fox, Daniel Gottesman,
• Abstract summary: For qubits, 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.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Say a collection of $n$-qu$d$it gates $\Gamma$ 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 $\Gamma$. 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.

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