Since Harrow, Hassidim, and Lloyd (2009) showed that a system of linear equations with $N$ variables and condition number $\kappa$ can be solved on a quantum computer in $\operatorname{poly}(\log(N), \kappa)$ time, exponentially faster than any classical algorithms, its improvements and applications have been extensively investigated. The state-of-the-art quantum algorithm for this problem is due to Costa, An, Sanders, Su, Babbush, and Berry (2022), with optimal query complexity $\Theta(\kappa)$. An important question left is whether parallelism can bring further optimization. In this paper, we study the limitation of parallel quantum computing on this problem. We show that any quantum algorithm for solving systems of linear equations with time complexity $\operatorname{poly}(\log(N), \kappa)$ has a lower bound of $\Omega(\kappa)$ on the depth of queries, which is tight up to a constant factor.

翻译：自 Harrow、Hassidim 和 Lloyd（2009年）证明具有 $N$ 个变量和条件数 $\kappa$ 的线性方程组可在量子计算机上以 $\operatorname{poly}(\log(N), \kappa)$ 的时间求解，其速度相比任何经典算法均呈指数级提升以来，其改进与应用得到了广泛研究。针对该问题的最先进量子算法由 Costa、An、Sanders、Su、Babbush 和 Berry（2022年）提出，其查询复杂度达到最优值 $\Theta(\kappa)$。一个遗留的重要问题是并行性是否能带来进一步优化。本文研究了并行量子计算在该问题上的局限性。我们证明，任何求解线性方程组且时间复杂度为 $\operatorname{poly}(\log(N), \kappa)$ 的量子算法，其查询深度均存在 $\Omega(\kappa)$ 的下界，该下界在常数因子范围内是紧致的。