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

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.