Near-term quantum algorithms for linear systems of equations

TL;DR

This paper introduces near-term quantum algorithms for solving large linear systems, leveraging variational methods and classical combination of quantum states, with promising scalability demonstrated through classical simulations.

Contribution

The paper proposes a novel near-term quantum algorithm based on classical combination of variational quantum states and provides theoretical guarantees and scalability analysis.

Findings

Algorithms can solve systems up to 2^300 size

Scalability demonstrated for 100-300 qubits

Supports heuristic and gradient-based improvements

Abstract

Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum devices. In this work, we study near-term quantum algorithms for linear systems of equations of the form $Ax = b$. We investigate the use of variational algorithms and analyze their optimization landscapes. There exist types of linear systems for which variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, suffer from a different plateau problem. To circumvent this issue, we design near-term algorithms based on a core idea: the classical combination of variational quantum states (CQS). We exhibit several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called Ansatz tree. The CQS approach and the Ansatz tree also admit the systematic application of heuristic approaches, including a gradient-based search. We have conducted numerical experiments solving linear systems as large as $2^{300} \times 2^{300}$ by considering cases where we can simulate the quantum algorithm efficiently on a classical computer. These experiments demonstrate the algorithms' ability to scale to system sizes within reach in near-term quantum devices of about $100$-$300$ qubits.