Double-bracket flow quantum algorithm for diagonalization

A quantum algorithm for preparing eigenstates of quantum systems is presented which makes use of only forward and backward evolutions under a prescribed Hamiltonian and phase flips. It is based on the Glazek-Wilson-Wegner flow method from condensed-matter physics or more generally double-bracket flows considered in dynamical systems. The phase flips are used to implement a dephasing of off-diagonal interaction terms and evolution reversal is employed for the quantum computer to approximate the group commutator needed for unitary propagation under the double-bracket generator of the diagonalizing flow. The presented algorithm is recursive and involves no qubit overheads. Its efficacy for near-term quantum devices is discussed using numerical examples. In particular, variational double-bracket flow generators, optimized flow step durations and heuristics for pinching via efficient unitary mixing approximations are considered. More broadly, this work opens a pathway for constructing purposeful quantum algorithms based on double-bracket flows also for tasks different from diagonalization and thus enlarges the quantum computing toolkit geared towards practical physics problems.