[论文解读] Quantum Topological Data Analysis with Linear Depth and Exponential Speedup

Shashanka Ubaru, Ismail Yunus Akhalwaya+3 more|arXiv (Cornell University)|Aug 5, 2021

Topological and Geometric Data Analysis参考文献 55被引用 15

一句话总结

本文提出了一种新型量子算法用于拓扑数据分析，实现了仅需线性电路深度的指数级加速，使其可在近期含噪声中等规模量子（NISQ）设备上实现。通过用高效的泡利分解、量子拒绝采样和随机切比雪夫估计替代量子相位估计算法与格罗弗搜索，该方法在密集单纯复形的贝蒂数估计中实现了实际的量子优势。

ABSTRACT

Quantum computing offers the potential of exponential speedups for certain classical computations. Over the last decade, many quantum machine learning (QML) algorithms have been proposed as candidates for such exponential improvements. However, two issues unravel the hope of exponential speedup for some of these QML algorithms: the data-loading problem and, more recently, the stunning dequantization results of Tang et al. A third issue, namely the fault-tolerance requirements of most QML algorithms, has further hindered their practical realization. The quantum topological data analysis (QTDA) algorithm of Lloyd, Garnerone and Zanardi was one of the first QML algorithms that convincingly offered an expected exponential speedup. From the outset, it did not suffer from the data-loading problem. A recent result has also shown that the generalized problem solved by this algorithm is likely classically intractable, and would therefore be immune to any dequantization efforts. However, the QTDA algorithm of Lloyd et~al. has a time complexity of $O(n^4/(ε^2 δ))$ (where $n$ is the number of data points, $ε$ is the error tolerance, and $δ$ is the smallest nonzero eigenvalue of the restricted Laplacian) and requires fault-tolerant quantum computing, which has not yet been achieved. In this paper, we completely overhaul the QTDA algorithm to achieve an improved exponential speedup and depth complexity of $O(n\log(1/(δε)))$. Our approach includes three key innovations: (a) an efficient realization of the combinatorial Laplacian as a sum of Pauli operators; (b) a quantum rejection sampling approach to restrict the superposition to the simplices in the complex; and (c) a stochastic rank estimation method to estimate the Betti numbers. We present a theoretical error analysis, and the circuit and computational time and depth complexities for Betti number estimation.

研究动机与目标

• 解决先前量子拓扑数据分析（QTDA）算法在容错性和高电路深度方面的局限性。
• 在不依赖数据加载或相位估计算法的前提下，实现对贝蒂数估计的指数级加速。
• 开发一种可在近期量子硬件（NISQ设备）上实现的算法，电路深度为O(n log(1/(δε)))。
• 提供一个理论基础坚实、经过误差分析的方法，用随机秩估计替代量子相位估计。
• 通过在真实量子计算机上实现该算法，证明其实际可行性。

提出的方法

• 高效地将组合拉普拉斯算子分解为泡利算子之和，以实现低深度的量子电路实现。
• 使用量子拒绝采样和投影技术，将叠加态限制在特定阶数的单纯形上，替代先前工作中使用的格罗弗搜索。
• 采用随机切比雪夫方法进行秩估计，替代资源密集型的量子相位估计算法。
• 应用受控递增技术，将单纯形态与计数寄存器 entangle，以实现高效的态制备。
• 引入基于阈值的近似方案，在谱间隙δ较小时估计贝蒂数，过滤掉噪声较大的小特征值。
• 进行理论误差分析，将贝蒂数估计误差控制在ε容差范围内。

实验结果

研究问题

• RQ1 是否存在一种量子算法，可在仅需线性深度量子电路的前提下，实现对贝蒂数估计的指数级加速？
• RQ2 是否可能在QTDA中消除对容错量子计算和量子相位估计的需求？
• RQ3 该算法是否可实现在当前NISQ设备上，从而实现近期量子优势？
• RQ4 如何仅使用浅层量子电路，高精度地估计谱特征（如贝蒂数）？
• RQ5 能否通过过滤掉小特征值，使算法对噪声和小谱间隙具有鲁棒性？

主要发现

• 所提出的算法实现了O(n log(1/(δε)))的电路深度，使其可在近期NISQ设备上实现。
• 在广泛接受的假设下，该算法在密集单纯复形且谱间隙较大的情况下，相对于经典方法实现了指数级加速。
• 随机切比雪夫方法使得无需量子相位估计即可实现高精度的贝蒂数估计，显著降低了资源开销。
• 该算法已在真实量子计算机上成功实现并执行，验证了理论预测。
• 通过仅估计高于阈值δ的特征值，该方法对小谱间隙具有鲁棒性，有效过滤了噪声。
• 该方法的应用范围不仅限于QTDA，还可扩展至谱密度估计、数值秩近似以及其他量子机器学习中的线性代数问题。