Improved upper bounds on the stabilizer rank of magic states

• URL: http://arxiv.org/abs/2106.07740v2
• Date: Wed, 15 Dec 2021 14:29:18 GMT
• Title: Improved upper bounds on the stabilizer rank of magic states
• Authors: Hammam Qassim, Hakop Pashayan, David Gosset
• Abstract summary: improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|Trangle=sqrt2-1(|0rangle+eipi/4|1rangle)$ in the limit of large $m$. We obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $textpoly(n,m) 2m/2$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|T\rangle=\sqrt{2}^{-1}(|0\rangle+e^{i\pi/4}|1\rangle)$ in the limit of large $m$. In particular, we show that $|T\rangle^{\otimes m}$ can be exactly expressed as a superposition of at most $O(2^{\alpha m})$ stabilizer states, where $\alpha\leq 0.3963$, improving on the best previously known bound $\alpha \leq 0.463$. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an $n$-qubit Clifford + T circuit $U$ with $m$ uses of the T gate to within a given inverse polynomial relative error using a runtime $\mathrm{poly}(n,m)2^{\alpha m}$. We also provide improved upper bounds on the stabilizer rank of symmetric product states $|\psi\rangle^{\otimes m}$ more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $\text{poly}(n,m) 2^{m/2}$. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

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