A Semidefinite Programming algorithm for the Quantum Mechanical Bootstrap

TL;DR

This paper introduces a semidefinite programming algorithm within the quantum bootstrap framework to accurately compute eigenvalues of Schrödinger operators, leveraging linearized constraints and positivity conditions.

Contribution

The authors develop a novel SDP-based method that linearizes bootstrap constraints for eigenvalue estimation in quantum mechanics, providing high-precision bounds.

Findings

Achieved sharp bounds on eigenenergies for 1-D polynomial potentials

Demonstrated high-precision results using the SDP algorithm

Validated the method's effectiveness for arbitrary confining potentials

Abstract

We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasability problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in 1-D.