A shortcut to an optimal quantum linear system solver

Abstract

Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(\kappa \log(1/\varepsilon))$, where $\kappa$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))\kappa \ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(\kappa))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(\kappa \log\log(\kappa)\log\log\log(\kappa)+\kappa\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(\kappa)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(\kappa\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56\kappa+1.05\kappa \ln(1/\varepsilon)+o(\kappa)$ for the complexity of our optimal QLSS.