Log-concavity and tunneling: adiabatic quantum optimization for convex functions (with a spike)

2026-06-22

AI Summary

This work investigates the acceleration mechanism of quantum tunneling in adiabatic quantum optimization for spike-shaped convex potentials. By introducing the discrete log-concavity of the ground state into the analysis of one-dimensional discrete Schrödinger operators, the study establishes a theoretical framework applicable to nonsmooth potentials. It extends the classical Brascamp–Lieb inequality from the continuous to the discrete setting and generalizes the Hidden Weighted Subset (HWS) problem from linear to quadratic potentials. Combining spectral theory, the adiabatic theorem, and perturbation analysis, the authors derive a more general lower bound on the spectral gap, demonstrating that quantum tunneling retains its computational advantage even in the presence of spike-shaped convex barriers. This provides new theoretical support for quantum optimization in scenarios involving non-analytic potential landscapes.

Abstract

Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes

Problem

• quantum tunneling
• adiabatic quantum optimization
• convex functions
• log-concavity
• spike

Innovation

• log-concavity
• adiabatic quantum optimization
• quantum tunneling
• spectral gap
• convex potentials