通过梯度下降视角改进量子线性系统求解器
Improving quantum linear system solvers via a gradient descent perspective
摘要：
求解线性方程组是量子计算中最重要的基础之一，它有可能在许多不同的领域提供实用的量子优势，包括优化，模拟和机器学习。在这项工作中，我们从凸优化的角度重新审视量子线性系统求解器，特别是梯度下降型算法。这导致了运行时间中相当大的常数因子改进（或者相反，在相同的运行时间/电路深度下，误差小了几个数量级）。更准确地说，我们首先展示Childs、Kothari和Somma的渐进最优量子线性系统求解器如何与凸函数$\|A\vec x - \vec b\|_2^2$上的梯度下降算法相关：他们的线性系统求解器是基于阶数- $(t-1)$多项式的切比雪夫基的截断（在$A$中），它将初始解$\vec{x}_1 := \vec{b}$映射到基本梯度下降算法中的第$t$个解$\vec{x}_t$。然后，而不是从基本的梯度下降算法开始，我们使用的最佳Chebyshev迭代方法（它可以被看作是一个加速梯度下降算法），并表明，这导致量子求解器相当大的改进。
英文摘要：
Solving systems of linear equations is one of the most important primitives in quantum computing that has the potential to provide a practical quantum advantage in many different areas, including in optimization, simulation, and machine learning. In this work, we revisit quantum linear system solvers from the perspective of convex optimization, and in particular gradient descent-type algorithms. This leads to a considerable constant-factor improvement in the runtime (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth). More precisely, we first show how the asymptotically optimal quantum linear system solver of Childs, Kothari, and Somma is related to the gradient descent algorithm on the convex function $\|A\vec x - \vec b\|_2^2$: their linear system solver is based on a truncation in the Chebyshev basis of the degree-$(t-1)$ polynomial (in $A$) that maps the initial solution $\vec{x}_1 := \vec{b}$ to the $t$-th iterate $\vec{x}_t$ in the basic gradient descent algorithm. Then, instead of starting from the basic gradient descent algorithm, we use the optimal Chebyshev iteration method (which can be viewed as an accelerated gradient descent algorithm) and show that this leads to considerable improvements in the quantum solver.
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