Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

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