Randomized adiabatic quantum linear solver algorithm with optimal complexity scaling and detailed running costs
To address the suboptimal complexity $O(kappa cdot ext{poly}(1/varepsilon))$ of existing quantum linear solvers with respect to condition number $kappa$ and precision $varepsilon$, this paper proposes the first randomized adiabatic quantum algorithm achieving optimal complexity $O(kappa log(1/varepsilon))$. Methodologically, it replaces costly Hamiltonian simulation with a low-overhead random-walk operator, thereby eliminating classical preprocessing bottlenecks; integrates Poissonization-based sampling, block-encoding techniques, optimized circuit design, and a novel fused filtering scheme to enhance hardware compatibility. Theoretically, we derive a tight closed-form upper bound on runtime: for Hermitian matrices, only approximately $867kappa$ block-encoding queries are required at $varepsilon = 10^{-10}$—yielding an exponential improvement in precision dependence over prior art and constant-factor hardware acceleration.