[论文解读] Composability of global phase invariant distance and its application to approximation error management

Priyanka Mukhopadhyay|arXiv (Cornell University)|Jun 13, 2021

Quantum Computing Algorithms and Architecture参考文献 61被引用 5

一句话总结

本文建立了量子线路近似中全局相位不变距离的可组合性，证明该度量下的误差界比算子范数下的更紧致。研究表明，将近似误差均匀分布在Rz门上可最小化T计数，从而在近似量子傅里叶变换等量子线路中实现更低的资源开销。

ABSTRACT

Many quantum algorithms can be written as a composition of unitaries, some of which can be exactly synthesized by a universal fault-tolerant gate set, while others can be approximately synthesized. A quantum compiler synthesizes each approximately synthesizable unitary up to some approximation error, such that the error of the overall unitary remains bounded by a certain amount. In this paper we consider the case when the errors are measured in the global phase invariant distance. Apart from deriving a relation between this distance and the Frobenius norm, we show that this distance composes. If a unitary is written as a composition (product and tensor product) of other unitaries, we derive bounds on the error of the overall unitary as a function of the errors of the composed unitaries. Our bound is better than the sum-of-error bound (Bernstein,Vazirani,1997), derived for the operator norm. This indicates that synthesizing a circuit using global phase invariant distance maybe done with less number of resources. Next we consider the following problem. Suppose we are given a decomposition of a unitary. The task is to distribute the errors in each component such that the T-count is optimized. Specifically, we consider those decompositions where $R_z( heta)$ gates are the only approximately synthesizable component. We prove analytically that for both the operator norm and global phase invariant distance, the error should be distributed equally among these components (given some approximations). The optimal number of T-gates obtained by using the global phase invariant distance is less. Furthermore, we show that in case of approximate Quantum Fourier Transform, the error obtained by pruning rotation gates is less when measured in this distance.

研究动机与目标

• 建立量子线路近似中全局相位不变距离的可组合性。
• 推导出在该度量下，组合酉算符的误差界相比算子范数下的误差和界更为紧致。
• 通过在Rz门间高效分配近似误差，优化量子线路中的T计数。
• 证明使用全局相位不变距离可降低近似量子算法（如QFT和QPE）的资源开销。

提出的方法

• 推导全局相位不变距离与Frobenius范数之间的关系。
• 证明全局相位不变距离在酉算符乘法和张量积下具有可组合性。
• 建立整体误差的解析界，作为各分量误差的函数。
• 将这些界应用于Rz门近似情形，证明误差均分可最小化T计数。
• 使用模拟退火和解析近似方法验证误差分配策略。
• 将结果应用于近似QFT和QPE，比较在全局相位不变距离与算子范数下的性能表现。

实验结果

研究问题

• RQ1 全局相位不变距离在酉算符复合（乘法与张量积）下是否可组合？
• RQ2 与算子范数下的误差和界相比，能否在该度量下为组合酉算符导出更紧致的误差界？
• RQ3 在近似线路中，是否Rz门间的误差均分策略能最优地最小化T计数？
• RQ4 在全局相位不变距离下测量误差，是否能降低近似QFT和QPE的资源开销？
• RQ5 该度量是否能在时间演化或其他量子算法中带来实际的门数减少优势？

主要发现

• 全局相位不变距离在酉算符乘法和张量积下具有可组合性，支持误差传播分析。
• 在全局相位不变距离下，整体误差界比算子范数下的误差和界更紧致。
• 对于Rz门近似，误差均分策略可最小化T计数，且该策略可实现更低的资源开销。
• 在近似QFT中，当使用全局相位不变距离测量误差时，剪枝旋转门可导致更低的算法误差，相比算子范数。
• 对于Rz(α)的QPE，使用全局相位不变距离测量误差时，所需T门数量更少。
• 结果表明，使用全局相位不变距离可降低量子线路中的T计数，尤其在QFT和QPE等模块化算法中更为显著。