Tight quantum depth lower bound for solving systems of linear equations

Since Harrow et al. [A. W. Harrow, A. Hassidim, and S. Lloyd, Phys. Rev. Lett. 103, 150502 (2009)] showed that a system of linear equations with $N$ variables and condition number $\kappa$ can be solved on a quantum computer in $poly\left[log\left(N\right),\kappa \right]$ 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 et al. [P. C. S. Costa, D. An, Y. R. Sanders, Y. Su, R. Babbush, and D. W. Berry, PRX Quantum 3, 040303 (2022)], with optimal query complexity $\Theta \left(\kappa \right)$. An important question that is 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 $poly\left[log\left(N\right),\kappa \right]$ has a lower bound of $\Omega \left(\kappa \right)$ on the depth of queries, which is tight up to a constant factor.