Near-term quantum algorithms for linear systems of equations

Solving linear systems of equations is an essential component in science and technology, including in many machine learning algorithms. Existing quantum algorithms have demonstrated large speedup in solving linear systems, but the required quantum resources are not available on near-term quantum devices. In this work, we study potential near-term quantum algorithms for linear systems of equations. We investigate the use of variational algorithms for solving $Ax = b$ and analyze the optimization landscape for variational algorithms. We found that a wide range of variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, still suffer from a fundamentally different plateau problem. To circumvent this issue, we design a potentially near-term adaptive alternating algorithm based on a core idea: the classical combination of variational quantum states. We have conducted numerical experiments solving linear systems as large as $2^{300} \times 2^{300}$ by considering special systems that can be simulated efficiently on a classical computer. These experiments demonstrate the algorithm's ability to scale to system sizes within reach in near-term quantum devices of about $100$-$300$ qubits.