Quantum annealers as programmable thermal machines

Technical Summary: Quantum Annealers as Programmable Thermal Machines

Problem Statement
While quantum annealers (specifically D-Wave systems) are routinely benchmarked by solution quality, sampling capabilities, and time-to-solution, their thermodynamic performance remains largely uncharacterized. Existing literature has established that these devices operate as quantum thermal machines, particularly in reverse annealing cycles where they absorb and release heat. However, a comprehensive framework to quantify energy exchange, entropy production, and work without full access to the environment or a detailed microscopic bath model is lacking. The authors address the need to characterize the D-Wave processor not just as an optimizer, but as a programmable thermal machine whose operating mode (heater, accelerator, engine, or refrigerator) can be controlled and diagnosed through energy statistics.

Methodology
The authors treat the D-Wave quantum annealer as a closed thermodynamic cycle driven by a reverse annealing protocol. The experimental workflow involves:

1. Initialization: Preparing a classical Gibbs state of the programmed Ising Hamiltonian ($H_p$) at a controlled inverse temperature $\beta_1$.
2. Evolution: Executing a reverse annealing cycle where the annealing parameter $s(t)$ ramps from $s=1$ to a turning point $s_p$ and back to $s=1$. This ensures the initial and final Hamiltonians are identical ($H(0)=H(1)=H_p$).
3. Measurement: Performing a two-point energy measurement to determine the change in problem energy, $\Delta E_1 = E_p(\sigma(\tau)) - E_p(\sigma(0))$.
4. Analysis:
   • Effective Thermometry: Fitting the output spin configurations to the programmed Ising Hamiltonian using pseudo-likelihood to estimate an effective environment inverse temperature, $\beta_2$ (and corresponding temperature $T_2$).
   • Thermodynamic Bounds: Utilizing the Exchange Fluctuation Theorem and Thermodynamic Uncertainty Relations (TURs). Since only $\Delta E_1$ is directly observable, the authors derive rigorous lower bounds on entropy production ($\Sigma$), heat exchange ($Q$), and work ($W$) using only the first two moments of $\Delta E_1$ and the fitted $\beta_2$.
   • Classification: Mapping the operating regime based on the signs of the average processor energy change ($\langle \Delta E_1 \rangle$), environment energy change ($\langle \Delta E_2 \rangle$), and work ($\langle W \rangle$).

Experiments were conducted on D-Wave Advantage (Pegasus topology) and Advantage2 (Zephyr topology) systems using one-dimensional Ising chains and two-dimensional Pegasus instances (Random Uniform and Corrupted Biased Ferromagnet).

Key Contributions

1. Unified Thermodynamic Classification: The paper demonstrates that by varying the initial preparation temperature ($\beta_1$) and the reverse annealing turning point ($s_p$), the same hardware can be programmed to operate in distinct thermodynamic regimes:
   • Heater: Both the processor and environment gain energy; work is supplied.
   • Accelerator: The processor loses energy while the environment gains energy; work is supplied to assist relaxation.
   • Refrigerator: The drive forces energy exchange against the spontaneous thermal bias.
   • Engine/Accelerator: In regions where the available bounds do not uniquely separate work extraction from driven acceleration, the authors assign a combined "Engine/Accelerator" label to avoid overinterpreting the data. While engine-compatible operation is theoretically allowed, the experimental bounds do not uniquely distinguish it from accelerator operation.
2. Device-Agnostic Bounds: The authors provide a method to place lower bounds on entropy production, heat, work, and power using only the energy changes reported by the Quantum Processing Unit (QPU), without requiring a specific model of the environment.
3. Operational Thermometry: The study introduces a built-in thermometry diagnostic where the fitted effective temperature $T_2$ serves as a sensitive indicator of the thermal state of the computational degrees of freedom, distinct from the cryogenic refrigerator plate temperature.

Results

• Phase Diagrams: The authors mapped thermodynamic phase diagrams in the $(\beta_1, s_p)$ plane. For 1D chains, the boundary between heater and accelerator regimes shifts and smooths with annealing time ($\tau$), but the fundamental classification remains consistent.
• Critical Phenomenology: A distinct feature in the effective temperature $T_2$ and work bounds was observed around $s_p \approx 0.6$, corresponding to the critical region of the transverse field Ising model where the driver and problem energy scales compete. This region exhibits enhanced energy fluctuations and dissipation.
• Scaling: For 1D chains, the TUR-based bounds on total work and power scale as $L \log L$ (where $L$ is chain length), while per-spin quantities scale as $\log L$. This reflects the infrared dominance of critical fluctuations in one dimension.
• Problem Dependence: The thermodynamic response is instance-dependent. Two-dimensional instances with rugged energy landscapes (e.g., Corrupted Biased Ferromagnet) exhibit broader active windows and higher effective temperatures compared to smoother Random Uniform instances.
• Hardware Comparison: The classification procedure transfers between Advantage and Advantage2 processors, though the specific locations of mode boundaries and fitted temperatures vary with hardware generation and problem embedding.

Significance and Claims
The paper claims to provide a framework that connects quantum optimization, probabilistic computing, and statistical physics simulation with energy-aware assessment. Its significance lies in:

• Beyond Solution Quality: It adds a thermodynamic layer to benchmarking, distinguishing between driven refinement (accelerator), net heating (heater), and heat pumping (refrigerator/engine).
• Programmability: It establishes that the thermodynamic role of a quantum annealer is programmable; users can steer the device between different energy flow regimes by adjusting schedules and initial states.
• Diagnostic Utility: The effective temperature $T_2$ and energy bounds serve as diagnostic tools to detect changes in relaxation, freeze-out, or calibration without needing external sensors.
• Energy Efficiency: By quantifying the energetic consequences of different schedules, the work offers a route to co-designing algorithms and hardware schedules for both accuracy and energy efficiency.

The authors explicitly state that thermodynamic labels alone do not determine computational performance (e.g., finding the global optimum). Instead, the thermodynamic mode provides context for how a solution was reached (e.g., via relaxation or driven exploration), which must be evaluated alongside standard metrics like success probability and time-to-solution. The work does not claim to solve the optimization problem itself but rather to characterize the physical process of the annealer.