Self-concordant Schrödinger operators: spectral gaps and optimization without condition numbers

TL;DR

A novel quantum interior point method is constructed that applies to arbitrary self-concordant barriers and shows no condition-number dependence and is combined with techniques from semiclassical analysis, convex optimization, and quantum annealing.

Abstract

Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schrödinger operators has a long history of study due to its physical relevance, while in quantum computing spectral gaps are an important proxy for efficiency, such as in the quantum adiabatic algorithm. Motivated by convex optimization, we study Schrödinger operators associated with self-concordant barriers over convex domains and prove non-asymptotic lower bounds on the spectral gap for this class of operators. Significantly, we find that the spectral gap does not display any condition-number dependence when the usual Laplacian is replaced by the Laplace--Beltrami operator, which uses second-order information of the barrier and hence can take the curvature of the barrier into account. As an algorithmic application, we construct a novel quantum interior point method that applies to arbitrary self-concordant barriers and shows no condition-number dependence. To achieve this we combine techniques from semiclassical analysis, convex optimization, and quantum annealing.

Figures (1)

• Figure 1: The bump function $j$ from \ref{['lemma:j and barj properties']}.

Theorems & Definitions (67)

• Theorem 1.1: Spectral gap of $\mathcal{H}^E$, informal
• Theorem 1.2: Spectral gap of $\mathcal{H}^R$, informal
• Definition 2.1: Self-concordance
• Lemma 2.2: Renegar01
• Definition 2.3: Barrier
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.0: Spectral gap of $-\frac{1}{2}\Delta + \semi^2 f$