Variational Quantum Linear Solver --- Carlos Bravo-Prieto

2020 APS March Meeting
Title: Variational Quantum Linear Solver
Speaker: Carlos Bravo-Prieto (@charl_bp on Twitter)

Abstract:
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare 'x' such that 'Ax' is proportional to 'b'. VQLS assumes that 'A' is given as a linear combination of unitaries, and we provide an efficient procedure for obtaining this decomposition when 'A' is sparse. Furthermore, we derive an operationally meaningful termination condition for VQLS that allows one to verify that a desired solution precision 'epsilon' is achieved. Specifically, we prove that 'C' is greater or equal than 'epsilon^2 / kappa^2', where 'C' is the VQLS cost function and 'kappa' is the condition number of 'A'. We present efficient quantum circuits to estimate 'C', while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of 1024x1024. Finally, we numerically solve non-trivial problems of size up to 2^50x2^50. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in 'epsilon', 'kappa', and the system size 'N'.