Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, +3 more
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 such that . We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision is achieved. Specifically, we prove that , where is the VQLS cost function and is the condition number of . We present efficient quantum circuits to estimate , 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 . Finally, we numerically solve non-trivial problems of size up to . For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in , , and the system size .
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