Double-bracket flow quantum algorithm for diagonalization

Abstract

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 Głazek-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 offdiagonal 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 doublebracket 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.Studying quantum many-body systems is an area where quantum computing may lead to practical advances outside the scope of what we can compute numerically. We often gain physics understanding by analyzing eigenstates and eigenvalues of quantum models so quantum algorithms for diagonalization could be key for achieving new insights. While looking into the past does not always inform accurately how to proceed into the future, it seems that three characteristics of numerical algorithms may continue being relevant. Firstly, the best-performing diagonalization algorithms use iterative schemes [1]. Secondly, they involve operations which come naturally to a classical architecture [2]. Finally, the diagonalization is the fixed-point of the iteration [3].Typical numerical diagonalization iterations involve powers of the input matrix [1]. This is an issue for quantum computers which support composing unitary evolutions governed by prescribed Hermitian matrices more naturally than multiplying or adding