Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm

TL;DR

This paper presents a quantum linear system solver that combines time-optimal adiabatic quantum computing and QAOA, achieving near-optimal runtime complexity and broad applicability to different matrix types.

Contribution

It introduces a method that attains near-optimal runtime for solving quantum linear systems using adiabatic quantum computing and QAOA, applicable to non-Hermitian matrices.

Findings

Achieves $ ilde{O}( ext{poly}( ext{log}(rac{ ext{cond}}{ ext{error}})))$ runtime complexity.

QAOA with optimal control matches the time-optimal AQC complexity.

Numerical results show QAOA outperforms other methods in runtime efficiency.

Abstract

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with $\mathcal{O}(\kappa~\text{poly}(\log(\kappa/\epsilon)))$ runtime, where $\kappa$ is the condition number, and $\epsilon$ is the target accuracy. This is near optimal with respect to both $\kappa$ and $\epsilon$. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.