Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing

TL;DR

This paper introduces two simple, Hamiltonian-based quantum algorithms for solving linear systems, achieving efficient complexity and avoiding complex procedures like phase estimation, highlighting the potential of adiabatic-inspired quantum computing.

Contribution

The paper presents novel quantum algorithms based on evolution randomization that are simple, implementable, and nearly optimal in condition number dependence, distinct from existing methods.

Findings

Algorithms achieve $O( ext{kappa}^2 ext{log}( ext{kappa})/ extepsilon)$ and $O( ext{kappa} ext{log}( ext{kappa})/ extepsilon)$ complexity.

Algorithms do not require phase estimation or large ancillas.

Methods demonstrate exponential quantum speedup under certain assumptions.

Abstract

We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state $\vert x \rangle$ that is proportional to the solution of the system of linear equations $A \vec{x}=\vec{b}$. The time complexities of our algorithms are $O(\kappa^2 \log(\kappa)/\epsilon)$ and $O(\kappa \log(\kappa)/\epsilon)$, where $\kappa$ is the condition number of $A$ and $\epsilon$ is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of $A$, the projector onto the initial state $\vert b \rangle$, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of $\kappa$. Like previous methods, our techniques yield an exponential quantum speedup under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.