# Simulating quantum circuits by adiabatic computation: improved spectral gap bounds

## TL;DR

This paper improves the lower bounds on the spectral gap for adiabatic quantum computing, demonstrating that the gap decreases at most polynomially with the number of steps, which supports the computational equivalence with circuit models.

## Contribution

The authors provide two simplified proofs establishing an inverse polynomial lower bound on the spectral gap, improving previous estimates and suggesting broader applicability of these methods.

## Key findings

- Spectral gap lower bound is approximately π^2 / [8(L+1)^2] for large L
- Two proof techniques: eigenstate ansatz and Weyl's theorem-based approach
- Results support polynomial spectral gap decay in adiabatic quantum computing

## Abstract

Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L \gg 1$ is $\min_s\Delta \gtrsim \pi^2 / [8(L+1)^2]$, an improvement over previous estimates. Our first method is a direct approach based on an eigenstate ansatz, while the the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.