Trust Region Bayesian Optimization of Annealing Schedules on a Quantum Annealer

TL;DR

This work addresses the challenge of designing effective annealing schedules for quantum annealers under hardware noise and resource constraints. It introduces TuRBO, a trust-region Bayesian optimization framework that jointly tunes the total annealing time $T$ and a Fourier-based schedule $s(t)$, with a Gaussian process surrogate and an EI acquisition within adaptive trust regions. The method is hardware-aware, incorporating post-processing, readout budgets, and feasibility checks, and is validated on traveling salesman problem instances embedded on the D-Wave Advantage2 system, showing improvements in energy, success probability, and chain integrity over random and greedy baselines. A key finding is that TuRBO provides robust performance up to modest problem sizes within embedding-feasible regimes, while scalability is ultimately limited by embedding overhead and ICE-type hardware noise, motivating future embedding-aware and hybrid mitigation strategies. Overall, the paper demonstrates a practical, scalable pathway to improve QA performance in NISQ settings and points to concrete avenues for extending applicability to industrial optimization tasks.

Abstract

Quantum annealing (QA) is a practical model of adiabatic quantum computation, already realized on hardware and considered promising for combinatorial optimization. However, its performance is critically dependent on the annealing schedule due to hardware decoherence and noise. Designing schedules that account for such limitations remains a significant challenge. We propose a trust region Bayesian optimization (TuRBO) framework that jointly tunes annealing time and Fourier-parameterized schedules. Given a fixed embedding on a quantum processing unit (QPU), the framework employs Gaussian process surrogates with expected improvement to balance exploration and exploitation, while trust region updates refine the search around promising candidates. The framework further incorporates mechanisms to manage QPU runtime and enforce feasibility under hardware constraints efficiently. Simulation studies demonstrate that TuRBO consistently identifies schedules that outperform random and greedy search in terms of energy, feasible solution probability, and chain break fraction. These results highlight TuRBO as a resource-efficient and scalable strategy for annealing schedule design, offering improved QA performance in noisy intermediate-scale quantum regimes and extensibility to industrial optimization tasks.

Figures (4)

• Figure 1: D-Wave Quantum Annealing Framework. The process begins with casting the problem into a QUBO form, followed by its translation into a logical graph. This graph is then embedded onto the hardware topology through minor embedding, after which the annealing dynamics defined by $A(s)$ and $B(s)$ are executed. Finally, the qubits are measured to obtain classical bitstring solutions.
• Figure 2: TuRBO for annealing schedule on quantum annealer. (Left Box) QA workflow: fixed minor embedding, annealing with schedule $s(t)$, and measurement of qubit states. (Right Box) TuRBO: a Gaussian process surrogate models the objective energy $E(\boldsymbol{s})$, the acquisition function (Expected Improvement) selects new candidates within a trust region, and the surrogate is iteratively updated to refine the approximation and improve annealing schedules. The annealing schedule is parameterized in a Fourier basis. At each iteration, an initial parameter $\boldsymbol{s}$ is updated to a new parameter $\boldsymbol{s}'$ by TuRBO, which is then mapped to a device-valid schedule through amplitude clipping and discretization before being executed on the quantum annealer.
• Figure 3: Comparison of normalized annealing schedules for $N{=}7$ and $N{=}8$ under fixed vs. joint optimization across TuRBO, RS, and GS. Line styles distinguish fixed (solid) vs. joint (dashed) control, and colors correspond to optimization methods.
• Figure 4: Comparison of optimization performance between noisy and noiseless SQA at $N{=}9$ and $N{=}10$. Subplots (a) and (b) show the evolution of energy across iterations, while (c) and (d) present the corresponding success probabilities.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1: Computational Complexity of TuRBO (single-point) for Fourier-based Schedule Optimization
• Remark 5
• Lemma 1: Per-TR GP training cost
• proof
• Lemma 2: Acquisition maximization cost
• proof