The diagonalization of quantum field Hamiltonians

TL;DR

The paper tackles the challenge of computing low-energy spectra in large or infinite-dimensional quantum Hamiltonians by proposing a diagonalization-first hybrid approach that uses Monte Carlo for the remaining basis states. The core contribution is the quasi-sparse eigenvector (QSE) diagonalization, an iterative, basis-agnostic algorithm that builds an optimal subspace to capture the most important basis components of low-energy eigenvectors, even for non-orthogonal bases and non-Hermitian or infinite-dimensional Hamiltonians. The method is validated on three tests—a random sparse real symmetric matrix, a random sparse complex non-Hermitian matrix, and the infinite-dimensional φ^4 theory in 1+1 dimensions—demonstrating convergence, stability, and the role of quasi-sparsity. Together, these results establish QSE as a robust, scalable foundation for the subsequent stochastic error correction stage and the overall diagonalization/Monte Carlo strategy, with potential to mitigate sign problems and enable efficient spectral computations in challenging quantum systems.

Abstract

We introduce a new diagonalization method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques.

Figures (8)

• Figure 1: Distribution of basis components for an eigenvector where the spacing between consecutive levels is $\Delta E=0.13x_{\text{rms}}$.
• Figure 2: Distribution of basis components for an eigenvector where $\Delta E=0.041x_{\text{rms}}$.
• Figure 3: Distribution of basis components for an eigenvector where $\Delta E=0.024x_{\text{rms}}$.
• Figure 4: Comparison of the four lowest exact energies $E_{i}$ and QSE results $E_{i}^{\text{QSE}}$ as functions of iteration number.
• Figure 5: Inner products between the normalized exact eigenvectors $\left| v_{i}\right\rangle$ and the QSE results $\left| v_{i}^{\text{QSE}}\right\rangle$ as functions of iteration number.