[Paper Review] Computational Bottlenecks of Quantum Annealing

Sergey Knysh|arXiv (Cornell University)|Jun 29, 2015

Quantum Computing Algorithms and Architecture36 references3 citations

TL;DR

This paper identifies O(log N) computational bottlenecks in quantum annealing for large spin glass systems, showing that exponentially small energy gaps arise not only at the critical point but also within the spin glass phase due to avoided level crossings. Using exact non-perturbative methods on the two-pattern Gaussian Hopfield model, it demonstrates that these bottlenecks lead to time complexity scaling as a fractional power of N, offering a milder exponential slowdown than previously feared.

ABSTRACT

A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller $N$, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.

Motivation & Objective

• To understand the origin of computational bottlenecks in quantum annealing beyond the critical point.
• To rigorously analyze the role of avoided level crossings in the spin glass phase for large systems.
• To determine whether time complexity remains polynomial or becomes exponential in the presence of multiple small gaps.
• To assess the validity of perturbative arguments about small gaps in spin glasses using non-perturbative methods.
• To provide a theoretical foundation for the observed performance scaling in quantum annealing hardware.

Proposed method

• Uses exact, non-perturbative methods to analyze the two-pattern Gaussian Hopfield model.
• Models the transverse field evolution as a stochastic process governed by a Fokker-Planck equation with a time-dependent potential.
• Applies a scaling ansatz to reduce the Fokker-Planck equation to an ordinary differential equation (ODE) with eigenvalue problem.
• Derives the universal asymptotic solution using the Airy function, capturing the statistics of the lowest energy level near the global minimum.
• Introduces a dimensionless time variable τ and constructs a Markov process in (μ, ν, ϑ) space to simulate the dynamics of the system.
• Performs numerical simulations by rescaling the potential and extending τ to gather sufficient statistics, with results available upon request.

Experimental results

Research questions

• RQ1 What causes the energy gap to become exponentially small in the spin glass phase of quantum annealing?
• RQ2 How many such bottlenecks exist in large systems, and what is their scaling with system size N?
• RQ3 Why do avoided crossings in the spin glass phase lead to a time complexity that is only a fractional power of N rather than exponential?
• RQ4 How does the non-perturbative behavior of energy level statistics differ from perturbative predictions in spin glasses?
• RQ5 Can the mechanism of tunneling bottlenecks be rigorously described using stochastic processes and extreme value statistics?

Key findings

• The number of computational bottlenecks in the spin glass phase scales as O(log N), not as a single critical point.
• The energy gap at these bottlenecks scales as a stretched exponential, leading to time complexity that grows as a fractional power of N.
• The dominant contribution to the gap comes from the smallest eigenvalue α = 1/4 in the ODE solution, consistent with known results from extreme value statistics.
• The system exhibits a cascade of tunneling events at geometrically spaced values of the transverse field Γ, indicating a hierarchical structure in the energy landscape.
• The time complexity is milder than previously feared, with exponential slowdown only in a fractional power of N, not full exponential scaling.
• The analysis reveals that bottlenecks become easier as Γ → 0, contrary to the expectation that tunneling is suppressed in the classical limit.