Stretched Exponential Scaling of Parity-Restricted Energy Gaps in a Random Transverse-Field Ising Model

TL;DR

This paper proves that in a one-dimensional random transverse-field Ising model, the energy gap in the parity-restricted subspace scales as a stretched exponential, impacting the efficiency of quantum annealing.

Contribution

It generalizes previous results by showing stretched exponential scaling of energy gaps in a broader class of spin glass models with i.i.d. randomness.

Findings

Energy gap follows stretched exponential scaling in 1D models

Implications for quantum annealing efficiency

Generalization to models with continuous or biased randomness

Abstract

The success of a quantum annealing algorithm requires a polynomial scaling of the energy gap. Recently it was shown that a two-dimensional transverse-field Ising model on a square lattice with nearest-neighbor $\pm J$ random coupling has a polynomial energy gap in the symmetric subspace of the parity operator [Nature 631, 749-754 (2024)], indicating the efficient preparation of its ground states by quantum annealing. However, it is not clear if this result can be generalized to other spin glass models with continuous or biased randomness. Here we prove that under general independent and identical distributions (i.i.d.) of the exchange energies, the energy gap of a one-dimensional random transverse-field Ising model follows a stretched exponential scaling even in the parity-restricted subspace. We discuss the implication of this result to quantum annealing problems.