Trimmed Sampling Algorithm for the Noisy Generalized Eigenvalue Problem

Abstract

Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose entries are inner products of the basis states, and the process is unfortunately susceptible to even small errors. The problem is especially bad when matrix elements are evaluated using stochastic methods and have significant error bars. In this work, we introduce the trimmed sampling algorithm in order to solve this problem. Using the framework of Bayesian inference, we sample prior probability distributions determined by uncertainty estimates of the various matrix elements and likelihood functions composed of physics-informed constraints. The result is a probability distribution for the eigenvectors and observables which automatically comes with a reliable estimate of the error and performs far better than standard regularization methods. The method should have immediate use for a wide range of applications involving classical and quantum computing calculations of large quantum systems.

arXiv commentFinal version to appear in Physical Review Research. 6 pages and 3 figures (main), 5 pages and 8 pages (supplemental)

References in corpus (22)

• A variational eigenvalue solver on a quantum processor
• Precise solution of few-body problems with stochastic variational method on correlated Gaussian basis
• Cloud Quantum Computing of an Atomic Nucleus
• Excited and exotic charmonium spectroscopy from lattice QCD
• Lattice simulations for few- and many-body systems
• The Flavor Structure of the Excited Baryon Spectra from Lattice QCD
• Eigenvector continuation with subspace learning
• Global sensitivity analysis of bulk properties of an atomic nucleus
• Eigenvector Continuation as an Efficient and Accurate Emulator for Uncertainty Quantification
• Get on the BAND Wagon: A Bayesian Framework for Quantifying Model Uncertainties in Nuclear Dynamics
• Novel shape evolution in Sn isotopes from magic numbers 50 to 82
• Rodeo Algorithm for Quantum Computing
• Efficient emulators for scattering using eigenvector continuation
• Improved many-body expansions from eigenvector continuation
• Model reduction methods for nuclear emulators
• Convergence of Eigenvector Continuation
• Training and Projecting: A Reduced Basis Method Emulator for Many-Body Physics
• Illustration of universal relations for trapped four-fermion system with arbitrary s-wave scattering length
• GCM study of hexadecapole correlations in superdeformed ${}^{194}$Hg
• Wigner SU(4) symmetry, clustering, and the spectrum of ${}^{12}$C
• Bayesian probability updates using Sampling/Importance Resampling: Applications in nuclear theory
• Adiabatic projection method with Euclidean time subspace projection

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• Eigenvector Continuation and Projection-Based Emulators
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• Non-Hermitian quantum mechanics approach for extracting and emulating continuum physics based on bound-state-like calculations: Detailed description
• Non-Hermitian Quantum Mechanics Approach for Extracting and Emulating Continuum Physics Based on Bound-State-Like Calculations