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arXiv:2601.04402 · New Journal of Physics

Thermodynamic significance of QUBO encoding on quantum annealers

Emery Doucet, Zakaria Mzaouali, Reece Robertson, Bartłomiej Gardas, Sebastian Deffner, Krzysztof Domino

quant-phSubmitted Jan 7, 2026• Announced Jan 9, 2026• 1 citation
arXivHTMLPDF

Abstract

Quadratic unconstrained binary optimization (QUBO) is the standard interface to quantum annealers, yet a single constrained task admits many QUBO encodings whose penalty choices reshape the energy landscape experienced by hardware. We study a Job Shop Scheduling instance using a two-parameter family of encodings controlled by penalty weights $p_{\rm sum}$ (one-hot/sum constraints) and $p_{\rm pair}$ (precedence constraints). Sweeping $(p_{\rm sum},p_{\rm pair})$, we observe sharp transitions in feasibility and solver success across classical annealing-inspired heuristics and on a D-Wave Advantage processor. Going beyond solution probability, we treat the annealer as an open thermodynamic system and perform cyclic reverse-annealing experiments initialized from thermal samples, measuring the stochastic processor energy change. From the first two moments of this energy change we infer lower bounds on entropy production, work, and exchanged heat via thermodynamic uncertainty relations, and corroborate the observed trends with adiabatic master equation simulations. We find that the same encoding transitions that govern computational hardness also reorganize dissipation: weak penalties generate low-energy infeasible manifolds, while overly strong penalties suppress the effective problem energy scale and increase irreversibility, reducing the thermodynamic efficiency. Our results establish QUBO penalties as thermodynamic control knobs and motivate thermodynamics-aware encoding strategies for noisy intermediate-scale quantum annealers.

Journal reference
2026, New J. Phys. 28 054512
DOI
10.1088/1367-2630/ae6e98
Comments
17 pages, 8 figures

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Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.
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Resource constrained project scheduling and related allocation tasks can be expressed as Quadratic Unconstrained Binary Optimization (QUBO) models and submitted to quantum annealers. This paper develops the QUBO formulation in detail: choice of binary variables, linear and quadratic objective terms, conversion of precedence and resource constraints into penalty functions, and calibration of penalty weights. The resulting Hamiltonian is embedded onto quantum hardware, annealed, and post-processed. Hybrid classical–quantum workflows that wrap the QUBO solver inside Monte Carlo risk analysis or a digital twin are examined. Supporting process flowcharts and energy-landscape figures illustrate every stage. The treatment remains grounded in construction and logistics applications while remaining transferable to other combinatorial domains such as ULD configuration. Keywords: QUBO formulation; quantum annealing; Ising model; penalty methods; resource-constrained project scheduling; hybrid quantum-classical; minor embedding; Monte Carlo; digital twin.
This paper investigates several Quadratic Unconstrained Binary optimisation (QUBO) formulations for the parallel machine scheduling problem P m | | C max with the goal of assessing their suitability for quantum annealing and related heuristic optimisation methods. Three QUBO models are constructed, ranging from formulations derived from classical integer and one-hot encodings to a more compact load-balancing based representation. Extensive computational experiments using Simulated Annealing (SA) and Quantum Annealing (QA) reveal substantial differences in scalability and solution quality among the formulations. The alternative load-balanced model consistently yields near-optimal makespans, typically within 1%–3% of the theoretical lower bound, highlighting its stability and effectiveness compared to the more penalty sensitive formulations, whose performance deteriorates for larger instances. Sensitivity analyses further reveal that the appropriate selection of penalty weights and problem-specific parameter tuning significantly influences both solution accuracy and computational effort in QUBO formulations. While current quantum hardware limitations restrict experimentation to small problem sizes, the findings illustrate how compact QUBO designs can better exploit quantum optimisation frameworks. The insights gained in this study provide practical modelling guidelines for future quantum scheduling research and lay the groundwork for extending QUBO-based approaches to more complex scheduling environments. • Three QUBO formulations are compared for identical parallel machine scheduling. • A compact load-balancing QUBO avoids auxiliary variables. • Load balancing yields makespans on average within 1–3%. • Both simulated and quantum annealing produce near-optimal schedules. • Penalty selection and encoding strongly affect scalability and solution quality.
We simulate Quantum Annealing on a variational manifold defined by a parametric family of wavefunctions represented by a Restricted Boltzmann Machine architecture. By iteratively lowering the transverse field and optimizing the neural network parameters, we prove the effectiveness of our methods for a challenging class of real-world Quadratic Unconstrained Binary Optimization (QUBO) instances, namely the Job Shop Scheduling Problem. Our methods outperform a straightforward optimization of the same problem without simulated quantum fluctuations. Despite the QUBO model being equivalent to a frustrated Ising spin glass, we obtain exact solutions up to 50-100 binary variables and empirically outperform results obtained with D-Wave quantum annealers.
Heterogeneous HPC workflow scheduling under multiple hard constraints poses a challenging combinatorial optimization problem. Classical exact solvers provide optimal solutions but often face scalability limitations, motivating interest in quantum-inspired Quadratic Unconstrained Binary Optimization (QUBO) formulations as alternative optimization paradigms. This work presents a systematic and reproducible empirical evaluation of QUBO-based scheduling methods against established classical baselines, including MILP, CP-SAT, GA, and HEFT. We evaluate three QUBO variants, single-run simulated annealing, multi-attempt annealing, and a layered QAOA-inspired schedule, together with hybrid enhancement strategies on both ground-truth validation workflows (3–4 tasks) and synthetic scaling instances (5–20 tasks). All solvers are assessed through a unified evaluation pipeline that explicitly tracks feasibility, makespan, and resource utilization under progressive constraint activation and controlled penalty sweeps. Results show that all approaches recover the expected optimal makespan on validation instances, confirming formulation correctness. However, feasibility degradation emerges for specific QUBO variants as constraint interactions intensify, particularly when communication costs are introduced. Penalty sensitivity analysis reveals a sharp feasibility threshold for QUBO-SA, where insufficient penalties consistently fail and moderate-to-strong penalties restore feasibility. Scaling experiments delineate the regimes in which classical solvers and heuristics remain robust across all tested sizes, while QUBO-SA loses feasibility beyond 15 tasks and the QAOA-inspired variant beyond 10 tasks. Overall, the study provides a clear empirical characterization of the reliability boundaries of quantum-inspired QUBO formulations for heterogeneous HPC scheduling and identifies the regimes where classical approaches remain preferable under current formulations and solver capabilities.