Real-Time Krylov Theory for Quantum Computing Algorithms

Abstract

Quantum computers provide alternative avenues to access ground and excited state properties of systems 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 using 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, applicable 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 methods, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

Quantum eigenvector continuation for chemistry applications

“…Within the context of these molecules, we investigate the use of EC and details of its implementation, particularly the special considerations that are unique to the ab initio setting. We evaluate the problems entirely on a classical simulator, however, the method pertains to quantum computation in the same sense that all subspace methods do [26][27][28]. It is relatively expensive to find the ground state of a model at a single parameter point [29] due to a combination of a large variational search space [30,31] barren plateaus [32][33][34], and deep circuits that are not amenable to today's hardware, etc.…”