求解线性方程组的紧致量子深度下界
Tight Quantum Depth Lower Bound for Solving Systems of Linear Equations
AI总结：
本文针对线性方程组量子求解问题，证明了时间复杂度为poly(log(N),κ)的量子算法的查询深度下界为Ω(κ)，明确了并行量子计算的优化极限，该下界在常数因子下紧致。
摘要：
自Harrow、Hassidim和Lloyd（2009年）证明含$N$个变量、条件数为$κ$的线性方程组可在量子计算机上以$\operatorname{poly}(\log(N), κ)$时间求解，且速度比任何经典算法呈指数级提升以来，该算法的改进与应用已得到广泛研究。目前该问题的最优量子算法由Costa、An、Sanders、Su、Babbush和Berry（2022年）提出，具有最优查询复杂度$Θ(κ)$。遗留的一个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，任何时间复杂度为$\operatorname{poly}(\log(N), κ)$的线性方程组求解量子算法，其查询深度下界为$Ω(κ)$，该下界在常数因子意义下是紧致的。
英文摘要：
Since Harrow, Hassidim, and Lloyd (2009) showed that a system of linear equations with $N$ variables and condition number $κ$ can be solved on a quantum computer in $\operatorname{poly}(\log(N), κ)$ 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 $Θ(κ)$. 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 $\operatorname{poly}(\log(N), κ)$ has a lower bound of $Ω(κ)$ on the depth of queries, which is tight up to a constant factor.
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