Solving Large-Scale Linear Systems of Equations by a Quantum Hybrid Algorithm

Funding Information: The authors thank Alexey A. Melnikov for reviewing the manuscript and providing valuable comments. M.R.P. and G.S.P. acknowledge support from the Academy of Finland through the Finnish Center of Excellence in Quantum Technology QTF (projects 312296, 336810, 312295, 336813). The work of A.G. at Argonne was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE‐AC05‐00OR22725. The work of V.M.V., M.R.P., A.I.P., A.A.M., and G.B.L. was supported by Terra Quantum AG. Publisher Copyright: © 2022 The Authors. Annalen der Physik published by Wiley-VCH GmbH.Today's intermediate-scale quantum computers, although imperfect, already perform computational tasks that are manifestly beyond the capabilities of modern classical supercomputers. However, so far, quantum-enabled large-scale solutions have been realized only for limited set of problems. Here a hybrid algorithm based on phase estimation and classical optimization of the circuit width and depth is employed for solving a specific class of large linear systems of equations ubiquitous to many areas of science and engineering. A classification of linear systems based on the entanglement properties of the associated phase-estimation unitary operation is introduced, enabling a highly efficient search for solutions that is facilitated by a straightforward matrix-to-circuit map. A 217-dimensional problem is implemented on several IBM quantum computer superconducting quantum processors, a record-breaking result for a linear system solved by a quantum computer. Demonstrated realisation sets a clear benchmark in the quest for the future quantum speedup in the linear systems of equations solution.Peer reviewe