Thermodynamic significance of QUBO encoding on quantum annealers

TL;DR

This work shows that QUBO encodings for constrained optimization, exemplified by Job Shop scheduling, are not neutral preprocessors but actively shape the quantum annealing landscape and its thermodynamic cost. By introducing a two-parameter penalty family $p_{ m sum}$ and $p_{ m pair}$, the authors identify regime boundaries in encoding space that delineate feasible from infeasible ground states and correspond to distinct dissipation patterns. Combining reverse-annealing experiments on a D-Wave Advantage with open-system adiabatic-master-equation simulations, they derive TUR-based bounds on entropy production, work, and heat, revealing that stronger penalties can increase irreversibility and reduce thermodynamic efficiency, while too-weak penalties leave low-energy infeasible states that skew sampling. The results advocate a thermodynamics-aware, co-design approach to QUBO encoding, aiming to minimize dissipation while preserving a resolvable energy scale, with broad implications for NISQ-era quantum annealers and constrained optimization tasks.

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.

Figures (8)

• Figure 1: Workflow of the paper. The Job Shop Scheduling Problem is mapped to a QUBO Hamiltonian characterized by tunable penalty parameters $psum$ (for one-hot constraints) and $ppair$ (for precedence constraints). We perform a parallel analysis of this encoding: assessing computational accuracy via solution probability (top branch) and thermodynamic cost via work $\langle W \rangle$, heat $\langle Q \rangle$, and entropy production $\langle \Sigma \rangle$ (bottom branch). The results demonstrate that phase transitions in the problem's computational hardness coincide with distinct thermodynamic signatures, establishing that optimal encoding parameters are those that minimize work dissipation during the annealing schedule.
• Figure 2: The left plot shows the difference between the best feasible solution and the best infeasible solution. Here the dashed line marks the region where the solution to the QUBO is a valid solution to the original Job Shop problem. The right plot shows the difference between the worst feasible and best infeasible solutions, with the border between the "split" and "unsplit" regimes marked in black.
• Figure 3: Probability that three different QUBO solvers provided by D-Wave find the optimal solution to the QUBO built from the 6 variable Job Shop instance. As the penalty values change, the identity of optimal solution to the QUBO changes. This is illustrated in the bottom right panel, with region I having a single optimal solution which is also feasible, and regions II and III having two or three degenerate infeasible solutions respectively. The black lines correspond to degeneracies between these sets of solutions.
• Figure 4: Thermodynamics of the Job Shop problem on D-Wave Advantage before the critical region ($\bar{s}=0.15$). Job Shop (10 qubits) executed via reverse annealing on the D-Wave Advantage system. The processor is initialized in a classical Gibbs state at inverse temperature $\beta_1=10$ and reverse annealed for $t_a=10~\mu$s down to $\bar{s}=0.15$ (before the quantum critical region) and back to $s=1$. Each panel is plotted versus the QUBO penalty weights $(p_{\rm pair},p_{\rm sum})$ controlling precedence and one-hot constraints. (a) mean processor energy change $\langle\Delta E_1\rangle=\langle E_{1,f}-E_{1,i}\rangle$; (b) TUR-based lower bound on the average entropy production \ref{['eq:entropy-bound']}; (c) work bound \ref{['eq:work-bound']}; (d) heat bound \ref{['eq:heat-bound']}; (e) thermodynamic efficiency \ref{['efficiency']}; (f) effective bath inverse temperature $\hat{\beta}_2$ obtained from pseudo-likelihood estimation \ref{['eq:beta-estimator']}. The maps reveal a pronounced regime boundary driven primarily by $p_{\rm sum}$: strengthening constraint penalties reorganizes the energy landscape and produces a concomitant change in dissipation (entropy/work/heat) and efficiency, directly demonstrating that QUBO encoding parameters act as thermodynamic control knobs on hardware.
• Figure 5: Thermodynamics of the Job Shop problem on D-Wave Advantage near the critical region ($\bar{s}=0.27$). Job Shop (10 qubits) executed via reverse annealing on the D-Wave Advantage system. The processor is initialized in a classical Gibbs state at inverse temperature $\beta_1=10$ and reverse annealed for $t_a=10~\mu$s down to $\bar{s}=0.27$ (near the quantum critical region) and back to $s=1$. Each panel is plotted versus the QUBO penalty weights $(p_{\rm pair},p_{\rm sum})$ controlling precedence and one-hot constraints. (a) mean processor energy change $\langle\Delta E_1\rangle=\langle E_{1,f}-E_{1,i}\rangle$; (b) TUR-based lower bound on the average entropy production \ref{['eq:entropy-bound']}; (c) work bound \ref{['eq:work-bound']}; (d) heat bound \ref{['eq:heat-bound']}; (e) thermodynamic efficiency \ref{['efficiency']}; (f) effective bath inverse temperature $\hat{\beta}_2$ obtained from pseudo-likelihood estimation \ref{['eq:beta-estimator']}. Relative to $\bar{s}=0.15$, the entropy/work/heat landscapes display enhanced spatial variability and sharper local features, consistent with increased susceptibility to thermal noise and decoherence near small-gap dynamics. Nevertheless, the same $p_{\rm sum}$-dominated regime structure persists, indicating that encoding-induced spectral transitions remain visible thermodynamically even in the most noise-sensitive operating region.