[论文解读] Quantum-inspired classical algorithm for molecular vibronic spectra

Changhun Oh, Youngrong Lim+3 more|arXiv (Cornell University)|Feb 3, 2022

Quantum Computing Algorithms and Architecture参考文献 54被引用 4

一句话总结

本文提出了一种受量子启发的经典算法，利用傅里叶变换和稀疏快速傅里叶变换高效计算分子振动-电子光谱，表明通常被认为具有量子优势的 Fock 态和高斯玻色采样问题，均可通过经典方法高精度求解。关键结果是，这些特定的振动-电子光谱问题并不提供量子优势，但更一般的某些问题可能仍是经典计算困难的，因而适合实现量子加速。

ABSTRACT

We have recently seen the first plausible claims for quantum advantage using sampling problems such as random circuit sampling and Gaussian boson sampling. The obvious next step is to channel the potential quantum advantage to solving practical applications rather than proof-of-principle experiments. Recently, a quantum simulator, specifically a Gaussian boson sampler, has been proposed to generate molecular vibronic spectra efficiently, which is an essential property of molecules and an important tool for analyzing chemical components and studying molecular structures. Computing molecular vibronic spectra has been a challenging task, and its best-known classical algorithm scales combinatorially in the system size. Thus, it is a candidate of tasks for which quantum devices provide computational advantages. In this work, we propose a quantum-inspired classical algorithm for molecular vibronic spectra for harmonic potential. We first show that the molecular vibronic spectra problem corresponding to Fock-state boson sampling can be efficiently solved using a classical algorithm as accurately as running a boson sampler. In particular, we generalize Gurvits's algorithm to approximate Fourier components of the spectra of Fock-state boson sampling and prove using Parseval's relation that the error of the spectra can be suppressed as long as that of the Fourier components are small. We also show that the molecular vibronic spectra problems of Gaussian boson sampling, which corresponds to the actual molecular vibronic spectra problem in chemistry, can be exactly solved even without Gurvits-type algorithms. Consequently, we demonstrate that those problems are not candidates of quantum advantage. We then provide a more general molecular vibronic spectra problem, which is also chemically well-motivated, for which we might be able to take advantage of a boson sampler.

研究动机与目标

• 研究被提议作为量子优势候选的分子振动-电子光谱计算，是否能通过经典算法高效求解。
• 确定对应于 Fock 态和高斯玻色采样问题的振动-电子光谱问题的计算复杂度。
• 开发一种与玻色采样精度相当的经典算法。
• 识别是否存在一种化学上合理的振动-电子光谱问题，其经典计算仍困难，因而适合作为量子优势的候选。

提出的方法

• 将 Gurlvits 的算法推广，以在可控误差下近似 Fock 态玻色采样光谱的傅里叶分量。
• 利用 Parseval 恒等式，从傅里叶分量的误差界 bounds 光谱的最终误差。
• 应用稀疏快速傅里叶变换，当光谱通道数超过多项式规模时，高效计算光谱。
• 证明对于高斯玻色采样，傅里叶分量可使用正 P 表象精确且高效地计算。
• 利用广义 P 表象的结构，实现对期望值的高效采样与计算。
• 使用随机哈希和稀疏 FFT 中的迭代峰值检测，处理超多项式规模的权重向量。

实验结果

研究问题

• RQ1 对应于 Fock 态玻色采样问题的分子振动-电子光谱，能否以与玻色采样器相同的精度被经典近似？
• RQ2 分子振动-电子光谱的高斯玻色采样问题，能否被经典方法精确或高效地求解？
• RQ3 所提出的经典算法是否实现了与量子玻色采样器相同的误差界？
• RQ4 是否存在即使在该方法提出后仍保持经典困难的化学上合理的振动-电子光谱问题？
• RQ5 傅里叶变换与稀疏 FFT 的框架能否用于高效计算具有加法误差保证的光谱？

主要发现

• 借助 Parseval 恒等式，Fock 态玻色采样问题的分子振动-电子光谱可被经典近似，其误差由傅里叶分量估计误差所界。
• 若傅里叶分量的估计误差在 ε 以内，则该算法可实现小于 ε 的光谱误差，确保与玻色采样器相当的精度。
• 对于高斯玻色采样，傅里叶分量可使用正 P 表象精确且高效地计算，使整个问题可被经典求解。
• 稀疏快速傅里叶变换即使在光谱通道数随系统尺寸超多项式增长时，也能实现高效计算。
• 该方法无法高效求解更一般的振动-电子光谱问题，表明此类问题可能仍为量子优势的候选。
• 本文结论为：基于 Fock 态和高斯玻色采样的振动-电子光谱问题不提供量子优势，但更广泛的问题类别可能仍存在。