发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出两种无逆量子编译算法，将编译指数分别降至约2.138和2.988，首次超越信息论下界3，证明无逆编译存在显著效率提升空间。
AI 中文摘要
Solovay-Kitaev 算法指出，使用任何通用门集，都可以在 polylog(1/ε) 时间内高效地将酉算子编译到精度 ε。虽然 Solovay 和 Kitaev 的初始工作处理的是封闭逆的门集，但最近 Bouland 和 Giurgicǎ-Tiron 表明，可以高效地编译没有逆的任意门集。然而，他们的编译算法的指数很高。例如，一个量子比特的无逆指数超过 8.62，而根据 Oszmaniec、Sawicki 和 Horodecki 的结果，从信息论角度看，在任何维度下无逆指数为 3 是可能的。这引出了一个自然的问题：能否改进 Bouland 和 Giurgicǎ-Tiron 算法中的指数？是否存在无逆指数低于 3 的情况，从而超越现有的信息论论证？为了回答这个问题，我们首先表明，无逆编译问题的一个特例——即由量子比特的非理性旋转补充的 Pauli 门——允许一个指数至多约为 2.138 的高效编译算法。我们的算法融合了 Kuperberg 以及 Sardharwalla、Cubitt、Harrow 和 Linden 先前算法的思想，构建了一个高效的“Pauli 高尔夫”编译程序。然后，我们将这些思想扩展到量子比特的一般无逆情况，给出一个指数至多为 2.988 的编译算法。这一结果使用了一种新颖的近似“瞄准”酉校正形式来提高编译效率。我们的结果给出了两个超越信息论论证的量子编译算法实例，并表明无逆编译存在许多效率提升的可能性。
英文摘要
The Solovay-Kitaev algorithm states that it is possible to efficiently compile unitaries to accuracy $ε$ in $\text{polylog}(1/ε)$ time using any universal gate set. While Solovay and Kitaev's initial work addressed inverse-closed gate sets, recently Bouland and Giurgicǎ-Tiron showed it is possible to efficiently compile arbitrary gate sets without inverses. However, the exponent of their compilation algorithm is high. For example the inverse-free exponent for a qubit is over $8.62$, while information theoretically it is known that an inverse-free exponent of $3$ is possible in any dimension by a result of Oszmaniec, Sawicki, and Horodecki. This leads to a natural question: can one improve the exponent in Bouland and Giurgicǎ-Tiron's algorithm? And are there any cases where the inverse-free exponent is below 3, thus surpassing existing information-theoretic arguments? To answer this question, we first show that a special case of the inverse-free compiling problem -- namely Pauli gates supplemented by an irrational rotation of a qubit - admits a highly efficient compilation algorithm with an exponent at most $\approx 2.138$. Our algorithm fuses ideas from prior algorithms of Kuperberg and Sardharwalla, Cubitt, Harrow and Linden to construct an efficient ``Pauli golf'' compilation routine. We then extend these ideas to the general inverse-free case for a qubit, where we give a compiling algorithm with exponent at most $2.988$. This result uses a novel form of approximate ``aiming'' of unitary corrections to improve the compilation efficiency. Our results give two examples of quantum compiling algorithms surpassing information-theoretic arguments, and demonstrate that there are many efficiency gains possible for inverse-free compilation.
