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Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions

Dylan Herman, Guneykan Ozgul, Anuj Apte, Junhyung Lyle Kim, Anupam Prakash, Jiayu Shen, Shouvanik Chakrabarti

TL;DR

The paper develops a novel, mechanism-based view for quantum advantage in global optimization of nonconvex functions by establishing a spectral-equivalence between Schrödinger operators and Langevin diffusion. Using the real-space adiabatic algorithm (RsAA), it shows polynomial-time optimization for broad function families, notably block-separable functions, via rigorous semiclassical and hypercontractivity analyses, while classical Langevin-type methods remain exponential in many cases. A key insight is that a unique global minimum yields robust quantum gaps (ground-state perspective) whereas the WKB/Langevin viewpoint can create degeneracies that slow classical dynamics, enabling a separation that does not rely on barrier tunneling. The work extends to non-separable functions through perturbation and intrinsic hypercontractivity, and it provides detailed runtime bounds, spectral-gap results, and comprehensive numerical benchmarking against off-the-shelf optimizers, establishing a theoretical foundation for quantum advantage in continuous optimization with broad implications for quantum-classical algorithm design. The results suggest new avenues connecting quantum algorithms, stochastic processes, and semiclassical analysis for nonconvex optimization in high dimensions.

Abstract

We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.

Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions

TL;DR

The paper develops a novel, mechanism-based view for quantum advantage in global optimization of nonconvex functions by establishing a spectral-equivalence between Schrödinger operators and Langevin diffusion. Using the real-space adiabatic algorithm (RsAA), it shows polynomial-time optimization for broad function families, notably block-separable functions, via rigorous semiclassical and hypercontractivity analyses, while classical Langevin-type methods remain exponential in many cases. A key insight is that a unique global minimum yields robust quantum gaps (ground-state perspective) whereas the WKB/Langevin viewpoint can create degeneracies that slow classical dynamics, enabling a separation that does not rely on barrier tunneling. The work extends to non-separable functions through perturbation and intrinsic hypercontractivity, and it provides detailed runtime bounds, spectral-gap results, and comprehensive numerical benchmarking against off-the-shelf optimizers, establishing a theoretical foundation for quantum advantage in continuous optimization with broad implications for quantum-classical algorithm design. The results suggest new avenues connecting quantum algorithms, stochastic processes, and semiclassical analysis for nonconvex optimization in high dimensions.

Abstract

We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.

Paper Structure

This paper contains 53 sections, 45 theorems, 262 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting up the Problem
  4. Problem Definition:
  5. Fixing the Quantum Algorithm:
  6. Classical Algorithms for Comparison:
  7. Overview of Results
  8. Connecting Schrödinger Operators to Classical Diffusion
  9. Mechanism: The Impact of Unique Global Minima
  10. Application 1: Block Coordinate-Separable Functions
  11. Structure-aware Classical Algorithms:
  12. Application 2: Beyond Coordinate-Separability
  13. Related Work
  14. Mechanism: Unique Global Minimum
  15. Connecting Schrödinger Operators to Diffusion
  16. ...and 38 more sections

Key Result

Theorem 1.3

Suppose $f : \mathcal{X}\rightarrow \mathbb{R}$ is $G$-Lipschitz function with a bound $\Lambda \geq \lVert f \rVert_{\infty}$ and has support in a compact domain $\mathcal{X}$ with $x_0 + [-2R, 2R]^d \subseteq \mathcal{X}$, and that for $\lambda \in [0, \lambda_{\max}]$, the spectral gap of $H(\lam for any fixed $y \in \mathcal{X}$, where $\Phi_{\lambda_{\max}}$ is the ground state of $H(\lambda_

Figures (13)

  • Figure 1: Correspondence between Quantum Dynamics (RsAA) and Langevin Diffusions (SGD)
  • Figure 2: Visualization of the unique/multiple global minima separation for the potential$f(x) = x^4 - (x - 1/32)^2 + 0.296$. Quantumly, as $\lambda$ increases, the ground state potential concentrates near the global minimum $x^{\star}$ around $-0.7$. Classically, WKB effective potential eventually contains multiple global minima, and it cannot distinguish between $x^{\star}$ and the local minimum of $f(x)$ around $0.7$.
  • Figure 3: Comparison of the spectral gaps of the original Hamiltonian with potential $f$ and the Witten Laplacian with the effective WKB potential.
  • Figure 4: Visualization of the unique/multiple global minima separation for the Levy function. Similarly to Figure \ref{['fig:degeneracy-separation-2d']}, the ground state potential concentrates near the global minimum, whereas WKB effective potential has multiple global minimizers. The exact form is: $f(x, y) = \sin^2(\pi w_1) + (w_1 - 1)^2 \left[ 1 + 10 \sin^2(\pi w_1 + 1) \right] + (w_2 - 1)^2 \left[ 1 + \sin^2(2\pi w_2) \right]$ where $w_1 = 1 + \frac{x - 1}{4}$ and $w_2 = 1 + \frac{y - 1}{4}$.
  • Figure 5: Visualization of the unique/multiple global minima separation for the separable version of Rastrigin function:$f_{\text{sep}}(x, y) = \left[ x^2 - 10 \cos(2\pi x) \right] + \left[ y^2 - 10 \cos(2\pi y) \right] + 20$.
  • ...and 8 more figures

Theorems & Definitions (79)

  • Definition 1.1: $\epsilon_f$-accurate binary oracle
  • Definition 1.2: Real-space Adiabatic Algorithm (RsAA)
  • Theorem 1.3: Theorem \ref{['thm:adiabatic_simulation']} Informal
  • Theorem 1.4: Theorem 2 shi2023learning
  • Definition 1.5: Ground State Potential
  • Proposition 1.5
  • Proposition 1.5
  • Definition 2.1: Markov Process
  • Definition 2.2: Confining Function
  • Definition 2.3: Dirichlet Form
  • ...and 69 more