Higher-order product formulas for double bracket iteration quantum algorithms

Abstract

This work aims to study the effects of adding higher-order terms into the group commutator formula in the previously studied double-bracket iterations (DBI) quantum algorithm. In particular, the effects of adding a six-gate third-order product formula, S3, is studied and compared to the previously studied DBI algorithm for second-order S2 group commutator terms. In the original study, double-bracket iterations for constructing diagonalizing quantum circuits were implemented. The method involves interleaving evolutions generated by the input Hamiltonian with variational choices of diagonal evolutions during implementation on a quantum computer. To address near-term implementation challenges, the proposal includes optimizations for diagonal evolution generators and recursion step durations. Numerical examples demonstrate that even with a limited number of recursion steps, double-bracket iterations possess sufficient expressive power to approximate eigenstates of relevant quantum models. Importantly, this method overcomes train ability limitations associated with brute force optimization of unstructured circuits and presents a more feasible implementation compared to quantum phase estimation. The study not only paves the way for practical near-term quantum computing experiments but also expands the quantum computing toolkit by introducing purposeful quantum algorithms based on double-bracket flows.