Block Lanczos method for excited states on a quantum computer

TL;DR

This paper extends the quantum Lanczos recursion method to a block version for better resolution of multiple excitations and degeneracies, analyzing its complexity, error scaling, and potential for non-Hermitian operators.

Contribution

The paper introduces a block Lanczos routine for quantum computers to improve excited state calculations and discusses its complexity and error properties.

Findings

Block Lanczos improves degeneracy resolution.

Error in ground state energy scales linearly with Lanczos coefficient uncertainty.

Incremental operator addition keeps complexity manageable.

Abstract

The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is in principle capable of obtaining excitations, the extension to a block Lanczos routine can resolve degeneracies with better precision and only costs $\mathcal{O}(d^2)$ for $d$ excitations on top of the previously introduced quantum Lanczos recursion method. The formal complexity in applying all operators to the system at once with oblivious amplitude amplification is exponential, but this cost can be kept small to obtain the ground state by incrementally adding operators. The error of the ground state energy based on the accuracy of the Lanczos coefficients is investigated and the error of the ground state energy. It is demonstrated to scale linearly with the uncertainty of the Lanczos coefficients. Extension to non-Hermitian operators is also discussed.