张量格式线性系统问题的高效量子算法
An Efficient Quantum Algorithm for Linear System Problem in Tensor Format
AI总结：
本文针对低秩张量和形式的线性系统提出一种基于绝热启发的量子线性系统算法，严格证明其总复杂度在维度上为多重对数级别，解决了经典Krylov方法总收敛代价无保证的问题。
摘要：
求解线性系统是许多算法的基础。近年来，量子线性系统算法（QLSAs）引起了广泛关注，因为它们在问题维度上比经典算法以指数级更快的速度收敛到解。然而，这些QLSAs中所假设的预言机的低复杂度电路实现，构成了在实际求解线性系统中实现量子加速的主要瓶颈。在本工作中，我们专注于将QLSAs应用于表示为低秩张量和的线性系统，这类系统出现在离散化偏微分方程的求解中。先前的工作使用改进的Krylov子空间方法来求解此类线性系统，其每次迭代复杂度为维度的多重对数级别，但对总收敛代价没有保证。我们基于最近受绝热启发的QLSA的进展提出了一种量子算法，并对其实现的电路深度进行了详细分析。我们严格证明，我们实现的总复杂度在维度上是多重对数级别的，这与经典启发式方法的每次迭代复杂度相当。
英文摘要：
Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.
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