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arXiv 2406.04846quant-ph
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无需使用 Solovay-Kitaev 定理的高效容错单量子比特门逼近与通用量子计算

Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem

H. F. Chau

备注：
The claim in this manuscript is incorrect due to a mistake in Eq. (6). The reason is that expression in Eq. (6) is derived in real arithmetic. It is incompatible with the modulo 2 arithmetic in the state ket
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AI总结：

本文针对 Nielsen 和 Chuang 提出的单量子比特门高效逼近挑战，给出一种不依赖 Solovay-Kitaev 定理的容错递归构造方法，将所需门数从 O(log^c[1/ε]) 改进为 O(log[1/ε] loglog[1/ε] logloglog[1/ε] ⋯)。

摘要：

任意精度的容错（FT）通用量子计算可以利用 Clifford 门 Z、S、CNOT 以及非 Clifford 门 T 来实现。此外，Kuperberg 最近对 Solovay-Kitaev 定理的改进表明，要以精度 $ε> 0$ 逼近任意单量子比特门，需要 $\text{O}(\log^c[1/ε])$ 个量子门，其中 $c > 1.44042$。能否做得更好？这是 Nielsen 和 Chuang 在其量子计算教材中提出的问题。具体而言，他们发布了一项挑战：从有限门集合中选取 $Ω(\log[1/ε])$ 个门，以容错或其他方式高效逼近单量子比特门。本文对该问题给出了部分回答，证明可以使用 $\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)$ 个容错门来实现，这些门选自一个依赖于 $ε$ 取值的有限集合。关键思想是以容错方式通过递归构造任意相位门的逼近，使其达到任意给定精度 $ε> 0$。该方法实现直接、易于理解，并且有趣的是不涉及 Solovay-Kitaev 定理。

英文摘要：

Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of $ε> 0$ requires $\text{O}(\log^c[1/ε])$ quantum gates with $c > 1.44042$. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using $Ω(\log[1/ε])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/ε] \log\log[1/ε] \log\log\log[1/ε] \cdots)$ FT gates chosen from a finite set depending on the value of $ε$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $ε> 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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