Tight Quantum Depth Lower Bound for Solving Systems of Linear Equations
Qisheng Wang, Zhicheng Zhang
TL;DR
The paper establishes tight quantum depth lower bounds for solving systems of linear equations (QLSP) on parallel quantum architectures. By reducing a permutation-chain problem—known to be hard in parallel quantum computation—into a carefully constructed linear system $A|x\rangle=|0\rangle$ with controlled condition number, the authors show that any quantum algorithm solving QLSP in time polynomial in $\log N$ and $\kappa$ must have query-depth at least linear in $\kappa$, with explicit constants: $Q_{\parallel}^{\text{sparse}}(3\kappa 2^{\kappa/2}, \kappa, \varepsilon) \ge 0.249\kappa$ and a corresponding block-QLSP bound of at least $0.031\kappa$. The technique hinges on encoding PermChain into a $2$-sparse matrix $A$ and constructing corresponding query oracles from a permutation oracle, then translating hardness of PermChain into a depth lower bound for QLSP. The results imply there is no general, low-depth quantum speedup for QLSP via parallelism within these models, and the paper also develops the necessary perturbation and block-encoding machinery to bridge sparse and block-input formulations. Open questions remain about joint depth-precision bounds and tighter constants, particularly in relating depth bounds to precision $\varepsilon$.
Abstract
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.