🤖 AI Summary
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.
📝 Abstract
Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. [Subac{s}i et al., Phys. Rev. Lett. (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number $kappa$ of the linear system and the target error $epsilon$. Here we go beyond these results in several ways. Firstly, using filtering [Lin et al., Quantum (2019)] and Poissonization techniques [Cunningham et al., arXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling $O(kappa log(1/epsilon))$ - an exponential improvement in $epsilon$, and a shaving of a $log kappa$ scaling factor in $kappa$. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation - which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is $867 kappa$ at $epsilon=10^{-10}$ for Hermitian matrices.