Variational Quantum Linear Solver: A Hybrid Algorithm for Linear Systems
Carlos Bravo-Prieto, Ryan LaRose, Marco Cerezo, Yiğit Subaşı, Łukasz Cincio, Patrick J. Coles · arXiv (Cornell University) · 2019
Solving linear systems of equations is central to many engineering and scientific fields. Several quantum algorithms have been proposed for linear systems, where the goal is to prepare $|x\rangle$ such that $A|x\rangle \propto |b\rangle$. While these algorithms are promising, the time horizon for their implementation is long due to the required quantum circuit depth. In this work, we propose a variational hybrid quantum-classical algorithm for solving linear systems, with the aim of reducing the circuit depth and doing much of the computation classically. We propose a cost function based on the overlap between $|b\rangle$ and $A|x\rangle$, and we derive an operational meaning for this cost in terms of the solution precision $\epsilon$. We also introduce a quantum circuit to estimate this cost, while showing that this cost cannot be efficiently estimated classically. Using Rigetti's quantum computer, we successfully implement our algorithm up to a problem size of $32 \times 32$. Furthermore, we numerically find that the complexity of our algorithm scales efficiently in both $1/\epsilon$ and $\kappa$, with $\kappa$ the condition number of $A$. Our algorithm provides a heuristic for quantum linear systems that could make this application more near term.