Introduction to stochastic error correction methods

TL;DR

The paper tackles truncation errors in subspace diagonalization for large or infinite-dimensional quantum Hamiltonians by introducing stochastic error correction (SEC), a Monte Carlo–augmented diagonalization framework. It presents two SEC variants: a series method that builds an $E^{-1}$ expansion around a good subspace eigenvector and a non-series stochastic Lanczos approach that uses matrix diffusion Monte Carlo to evaluate Krylov- and cross-subspace matrix elements. The authors demonstrate SEC on three non-perturbative problems—$\phi_{2+1}^4$ theory, compact $U(1)$ in $2+1$ dimensions, and the 2D Hubbard model—obtaining results in agreement with established literature, accessing excited states with fixed quantum numbers, and mitigating certain basis-size and sign-problem challenges. Collectively, SEC provides a flexible, scalable strategy for tackling challenging quantum many-body problems beyond the reach of traditional diagonalization or Monte Carlo methods.

Abstract

We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in the initial diagonalization. The method is part of a new approach to computational quantum physics which combines both diagonalization and Monte Carlo techniques.

Figures (1)

• Figure 1: Energy eigenvalues as functions of $\frac{g}{4!}$ as calculated by QSE diagonalization with first-order error corrections.