Real-Time Krylov Theory for Quantum Computing Algorithms

Yizhi Shen1,2,3, Katherine Klymko4, James Sud1,5, David B. Williams-Young6, Wibe A. de Jong6, and Norm M. Tubman1

1NASA Ames Research Center, Moffett Field, CA 94035, USA
2KBR, 601 Jefferson St., Houston, TX 77002
3Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
5USRA Research Institute for Advanced Computer Science, Mountain View, CA 94043, USA
6Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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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.

Near-term approaches exploiting quantum dynamics have shown great promise in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In this work, we make a timely step forward in understanding their properties by exploring and analyzing a generalized Krylov subspace scheme where the subspace is constructed from a parametrized real-time evolution. We establish an error bound on the spectral convergence, and derive how the overlap with high energy eigenstates becomes rapidly suppressed. In addition, we investigate various algorithmic implementations and consider performance when stochasticity is added to the target Hamiltonian. To demonstrate the practical edge of such real-time methods, we discuss fundamental problems in quantum computation such as electronic structure predictions on strongly correlated materials, highlighting the encouraging efficacy.

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