[论文解读] Resource analysis of the quantum linear system algorithm

Artur Scherer, Benoît Valiron+4 more|arXiv (Cornell University)|May 25, 2015

Quantum Computing Algorithms and Architecture被引用 4

一句话总结

本文对量子线性系统算法（QLSA）进行了详细的逻辑资源分析，利用Quipper和人工方法估算N=332,020,680时的门数、量子比特需求、电路宽度和深度。结果表明，即使不考虑量子预言机成本，QLSA的电路深度也高达$10^{25}$量级，而包含预言机时更增至$10^{29}$，凸显预言机资源消耗巨大，且要实现实用性，必须将资源消耗降低多个数量级。

ABSTRACT

We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described generalization [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the standard set {X, Y, Z, H, S, T, CNOT}. To perform these estimates, we used an approach that combines manual analysis with automated estimates generated via the Quipper quantum programming language and compiler. Our estimates pertain to the example problem size N=332,020,680 beyond which, according to a crude big-O complexity comparison, QLSA is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy 0.01 requires an approximate circuit width 340 and circuit depth of order $10^{25}$ if oracle costs are excluded, and a circuit width and depth of order $10^8$ and $10^{29}$, respectively, if oracle costs are included, indicating that the commonly ignored oracle resources are considerable. In addition to providing detailed logical resource estimates, it is also the purpose of this paper to demonstrate explicitly how these impressively large numbers arise with an actual circuit implementation of a quantum algorithm. While our estimates may prove to be conservative as more efficient advanced quantum-computation techniques are developed, they nevertheless provide a valid baseline for research targeting a reduction of the resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.

研究动机与目标

• 在容错量子电路模型中，为量子线性系统算法（QLSA）提供精确的逻辑资源估算。
• 量化预言机资源成本对整体资源扩展的影响，这些成本在以往分析中常被忽略。
• 通过展示QLSA超越经典算法所需资源规模，为未来研究建立基准。
• 通过使用Quipper和人工分析的具象电路实现，阐明此类巨大资源需求的来源。

提出的方法

• 使用Quipper量子编程语言和编译器，自动生成QLSA电路的资源估算。
• 将自动化的Quipper估算结果与人工分析相结合，以优化门数和电路参数。
• 聚焦于标准容错门集：{X, Y, Z, H, S, T, CNOT}，并报告各类门的计数结果。
• 针对问题规模N=332,020,680，分析电路宽度、深度、总门操作数以及量子比特/辅助量子比特需求。
• 区分排除和包含预言机成本的估算，以隔离其对资源开销的贡献。
• 通过大O时间复杂度比较，证明所选问题规模为QLSA可能超越经典求解器的临界阈值。

实验结果

研究问题

• RQ1 对于大规模问题规模，实现QLSA所需的精确逻辑资源需求（量子比特、门数、深度、宽度）是什么？
• RQ2 预言机资源成本如何影响QLSA的整体资源扩展？为何这些成本常被忽略？
• RQ3 所需电路深度和宽度在多大程度上使QLSA在当前容错量子硬件上变得不切实际？
• RQ4 具象电路实现能否揭示该算法巨大资源需求的根源？
• RQ5 为实现QLSA在经典线性系统求解器上的实际量子优势，资源需降低至何种数量级？

主要发现

• 对于问题规模N=332,020,680，当不包含预言机成本时，QLSA的电路宽度约为340个逻辑量子比特。
• 不考虑预言机成本时，电路深度估算为$10^{25}$次操作，表明其时间资源需求极高。
• 当包含预言机成本时，电路宽度增至约$10^8$量级，深度则增至$10^{29}$，表明预言机主导了整体资源开销。
• 研究揭示预言机资源并非可忽略，显著增加了整体复杂性，挑战了以往分析中的假设。
• 结果提供了一个具体基准，表明QLSA的资源需求目前远超容错量子硬件的能力，必须降低多个数量级才具可行性。
• 分析确认，即使采用最优门合成，QLSA仍因深度和宽度需求达到天文量级，对近期容错量子计算机而言仍不切实际。