Real-Time Krylov Theory for Quantum Computing Algorithms

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.

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• Sampling Error Analysis in Quantum Krylov Subspace Diagonalization
• Measurement-efficient quantum Krylov subspace diagonalisation
• Solving lattice gauge theories using the quantum Krylov algorithm and qubitization
• Quantum subspace expansion approach for simulating dynamical response functions of Kitaev spin liquids
• Estimating Eigenenergies from Quantum Dynamics: A Unified Noise-Resilient Measurement-Driven Approach
• Classical Benchmarks for Variational Quantum Eigensolver Simulations of the Hubbard Model
• Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor
• Rovibrational energy levels of H${}_2$O by quantum computing
• Cheaper and more noise-resilient quantum state preparation using eigenvector continuation
• Exploring fixed points and eigenstates of quantum systems with reinforcement learning
• Transversal architecture for megaquop-scale quantum simulation with neutral atoms
• Variational Quantum Subspace Construction via Symmetry-Preserving Cost Functions