Optimal scaling quantum linear systems solver via discrete adiabatic theorem

TL;DR

This paper introduces a discrete adiabatic theorem and uses it to develop a quantum linear systems solver with optimal linear complexity in the condition number, improving efficiency and simplicity over previous methods.

Contribution

The paper proves a discrete adiabatic theorem and applies it to create an asymptotically optimal quantum algorithm for solving linear systems.

Findings

Algorithm complexity is $ ext{O}(\kappa ext{log}(1/\epsilon))$

The method achieves linear complexity in the condition number $\kappa$

The approach simplifies implementation and determines constant factors for gate count analysis.

Abstract

Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number $\kappa$ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of $\log(\kappa)$. Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions. We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in $\kappa$, matching a known lower bound on the complexity. Our $\mathcal{O}(\kappa\log(1/\epsilon))$ complexity is also optimal in terms of the combined scaling in $\kappa$ and the precision $\epsilon$. Compared to existing suboptimal methods, our algorithm is simpler and easier to implement. Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.