Real-Time Krylov Theory for Quantum Computing Algorithms

TL;DR

This paper advances quantum algorithms by developing a generalized Krylov subspace method using real-time evolution, providing theoretical error bounds and demonstrating applications in electronic structure calculations.

Contribution

It introduces a generalized Krylov scheme for quantum algorithms, extending the VQPE method with theoretical analysis and practical insights into eigenstate extraction.

Findings

Fast spectral convergence justified by error bounds

Suppression of high-energy eigenstates via real-time diagonalization

Effective application to strongly correlated electronic systems

Abstract

Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.