A recent experiment [Le and Pelofske, arXiv:2512.19685] estimated the density of states and partition function of a 25-spin ±J Ising instance by reading raw energy histograms of D-Wave annealer samples, and highlighted a striking unexplained observation: the fastest anneals (8ns, within the coherent timescale of the device) gave the best estimates, reaching a log-relative error of 7.6 × 10−6. We show that no anomaly remains once two known ingredients are combined. First, a diabatic anneal is approximately a Boltzmann sampler whose inverse temperature βe is set by the annealing rate [Gyhm et al., arXiv:2409.18126]; we verify this in exact closed-system simulations at the experimental size n = 25, across two schedules, to sub-percent accuracy in the fast regime, and we measure the sampled ensembles to be uniform within energy shells to high precision, validating the energy-only description. Second, the raw-histogram estimator is unbiased only for a flat sampler: its bias is exactly φ(β + βe) − φ(βe) − φ(β), with φ(x) = ln⟨e−xE⟩unif, which grows linearly in βe while low-energy enrichment buys no compensating variance reduction for a Gaussian-like bulk. The fastest anneals are therefore accurate because they are nearly flat samplers sitting on the shot-noise floor; slow or strongly coupled anneals degrade exactly as observed; the interior optima in the reported cumulative sweeps are the bias–variance crossover of histogram pooling; and the headline 7.6 × 10−6 is consistent with the order statistics of a grid search over shot-noise-limited cells. The same analysis yields a free upgrade: because the schedule predicts each cell’s temperature, multi-histogram reweighting (WHAM) applies with rate-derived weights, converting the anneal-time ladder into a temperature ladder, removing the pooling bias by up to three orders of magnitude in our simulations, and extending accurate estimation toward temperatures where flat sampling fails. We state what the closed-system account does not explain as concrete, falsifiable predictions for reanalysis of the hardware data. AI use and validation: This work was produced in close collaboration with an AI research assistant (Anthropic's Claude, Fable 5), directed by the author, who set the research direction, the standards of evidence, and all decisions about what is claimed and published. The model performed analysis, simulation, and drafting under a registered protocol: hypotheses, tolerances, and kill criteria were committed to the repository before the experiments were run; validation scripts and their outputs are included in this deposit; and all results are reproducible from the code and data archived here.
A recent experiment [Le and Pelofske, arXiv:2512.19685] estimated the density of states and partition function of a 25-spin ±J Ising instance by reading raw energy histograms of D-Wave annealer samples, and highlighted a striking unexplained observation: the fastest anneals (8ns, within the coherent timescale of the device) gave the best estimates, reaching a log-relative error of 7.6 × 10−6. We show that no anomaly remains once two known ingredients are combined. First, a diabatic anneal is approximately a Boltzmann sampler whose inverse temperature βe is set by the annealing rate [Gyhm et al., arXiv:2409.18126]; we verify this in exact closed-system simulations at the experimental size n = 25, across two schedules, to sub-percent accuracy in the fast regime, and we measure the sampled ensembles to be uniform within energy shells to high precision, validating the energy-only description. Second, the raw-histogram estimator is unbiased only for a flat sampler: its bias is exactly φ(β + βe) − φ(βe) − φ(β), with φ(x) = ln⟨e−xE⟩unif, which grows linearly in βe while low-energy enrichment buys no compensating variance reduction for a Gaussian-like bulk. The fastest anneals are therefore accurate because they are nearly flat samplers sitting on the shot-noise floor; slow or strongly coupled anneals degrade exactly as observed; the interior optima in the reported cumulative sweeps are the bias–variance crossover of histogram pooling; and the headline 7.6 × 10−6 is consistent with the order statistics of a grid search over shot-noise-limited cells. The same analysis yields a free upgrade: because the schedule predicts each cell’s temperature, multi-histogram reweighting (WHAM) applies with rate-derived weights, converting the anneal-time ladder into a temperature ladder, removing the pooling bias by up to three orders of magnitude in our simulations, and extending accurate estimation toward temperatures where flat sampling fails. We state what the closed-system account does not explain as concrete, falsifiable predictions for reanalysis of the hardware data. AI use and validation: This work was produced in close collaboration with an AI research assistant (Anthropic's Claude, Fable 5), directed by the author, who set the research direction, the standards of evidence, and all decisions about what is claimed and published. The model performed analysis, simulation, and drafting under a registered protocol: hypotheses, tolerances, and kill criteria were committed to the repository before the experiments were run; validation scripts and their outputs are included in this deposit; and all results are reproducible from the code and data archived here.
Three recent experimental studies of D-Wave annealers each report an unexplained effective-temperature phenomenon: fitted inverse temperatures whose dependence on annealing time is left empirical, with sampling quality degrading at a critical frustration point [Pelofske, PRR 8, 033280 (2026)]; a coupling-independent temperature offset t-bar in [0.2,0.4] that violates the assumed T_eff ∝ 1/J scaling [Grattan et al., arXiv:2512.03162]; and hardware inverse temperatures a "consistent" factor of 5–7 above closed-system theory, attributed to uncontrollable hardware effects [Kim et al., PRE 113, 035302 (2026)]. We show that a single account, built from the diabatic effective-temperature relation of Gyhm et al. evaluated on the vendor's published fast-anneal schedules and one thermal bath, organizes all three. First, on real schedules the coherent inverse temperature is nearly linear in annealing time — not ∝ sqrt(t_a) as for linearly vanishing drivers — because the transverse field decays exponentially and dies by s ≈ 0.6; exact statevector simulation confirms the first-order integral to 0.4–4.2% in its validity window. Second, a units identity, beta_cyclic(t_a) = beta_correct(t_a/2pi), resolves the factor of 5–7: the published theoretical reference curves in this lane are quantitatively consistent with schedule data evaluated in cyclic rather than angular frequency units. The foundational paper's own schedule evaluations carry the rescale: its earlier single evaluation, beta ≈ 0.54 at 5 ns, equals our correct-convention value at t_a/2pi (0.539), and all 24 of its current tabulated beta and c_2 values that we can recompute from vendor schedule tables are reproduced at t_a/2pi to within 0.11%; a free-rescale fit to Kim et al.'s published simulation curve gives r = 6.40, with 2pi = 6.283 inside the fit's shallow region; what remains after the correction is consistent with open-system cooling. Third, the coupling-independent offset is a residual-excitation floor read through a domain-wall thermometer, t-bar ≈ 2/ln(1/rho): the implied kink densities (9e-5 to 4e-3) overlap published kink measurements, the excitation-floor form beats the constant-offset (noise) model on AIC for all four reported machines, and the logarithm compresses two and a half decades of rho into the narrow observed band, explaining its apparent universality. In the coherent nanosecond regime the closed-system law is an upper envelope on hardware inverse temperature; the shortfall is a candidate in-situ decoherence diagnostic, machine-resolved, and at the one matched cell where both machines are coherent the measured cross-machine ratio (2.63) agrees with the parameter-free prediction at the few-percent level. All hypotheses, tolerances, and kill criteria were registered before data extraction; failed and undecidable gates are reported alongside the successes. No quantum-advantage claim is made. AI use and validation: This work was produced in close collaboration with AI research assistants, directed by the author, who set the research direction, the standards of evidence, and all decisions about what is claimed and published. Anthropic's Claude (Fable 5) performed the analysis, simulation, extraction, and drafting; xAI's Grok (via Cursor) provided independent pre-publication review; and Anthropic's Claude Opus 5.5 performed a post-publication adversarial audit of the manuscript against its pinned evidence, with findings and dispositions logged in the project ledger and incorporated in subsequent versions. The registered-verdict protocol is designed so that no result rests on trusting a model's output: hypotheses, tolerances, and kill criteria were committed to the repository before any data was extracted; failed and undecidable gates are reported in the paper; every numeric claim traces to pinned, machine-readable extraction files and analysis code included in this deposit; figure data was extracted from the source PDFs' vector drawing operations rather than read by a model; and verbatim quotes were verified against raw page dumps.
We evaluate using programmable superconducting flux qubit D-Wave quantum annealers to approximate the partition function of Ising models. We propose the use of two distinct quantum annealer sampling methods: chains of Monte Carlo-like reverse quantum anneals, and standard linear-ramp quantum annealing. The control parameters used to attenuate the quality of the simulations are the effective analog energy scale of the J coupling, the total annealing time, and for the case of reverse annealing the anneal-pause. The core estimation technique is to sample across the energy spectrum of the classical Hamiltonian of interest, and therefore obtain a density of states estimate for each energy level, which in turn can be used to compute an estimate of the partition function with some sampling error. This estimation technique is powerful because once the distribution is sampled it allows thermodynamic quantity computation at arbitrary temperatures. On a 25 spin ±J hardware graph native Ising model we find parameter regimes of the D-Wave processors that provide comparable result quality to two standard classical Monte Carlo methods, Multiple Histogram Reweighting and Wang-Landau. Remarkably, we find that fast quench-like anneals can quickly generate ensemble distributions that are very good estimates of the true partition function of the classical Ising model; on a Pegasus graph-structured QPU we report a logarithmic relative error of 7.6×10−6, from 171,000 samples generated using 0.2 seconds of QPU time with an anneal time of 8 nanoseconds per sample which is interestingly within the closed system dynamics timescale of the superconducting qubits.
Motivated by recent experiments in which specific thermal properties of complex many-body systems were successfully reproduced on a commercially available quantum annealer, we examine the extent to which quantum annealing hardware can reliably sample from the thermal state in a specific basis associated with a target quantum Hamiltonian. We address this question by studying the diagonal thermal properties of the canonical one-dimensional transverse-field Ising model on a D-Wave 2000Q quantum annealing processor. We find that the quantum processor fails to produce the correct expectation values predicted by Quantum Monte Carlo. Comparing to master equation simulations, we find that this discrepancy is best explained by how the measurements at finite transverse fields are enacted on the device. Specifically, measurements at finite transverse field require the system to be quenched from the target Hamiltonian to a Hamiltonian with negligible transverse field, and this quench is too slow. The limitations imposed by such hardware make it an unlikely candidate for thermal sampling, and it remains an open question what thermal expectation values can be robustly estimated in general for arbitrary quantum many-body systems.
Energy-based generative models, such as restricted Boltzmann machines (RBMs), require unbiased Boltzmann samples for effective training. Classical Markov chain Monte Carlo methods, however, converge slowly and yield correlated samples, making large-scale training difficult. We address this bottleneck by applying the analytic relation between annealing schedules and effective inverse temperature in diabatic quantum annealing. By implementing this prescription on a quantum annealer, we obtain temperature-controlled Boltzmann samples that enable RBM training with faster convergence and lower validation error than classical sampling. We further identify a systematic temperature misalignment intrinsic to analog quantum computers and propose an analytical rescaling method that mitigates this hardware noise, thereby enhancing the practicality of quantum annealers as Boltzmann samplers. In our method, the model's connectivity is set directly by the qubit connectivity, transforming the computational complexity inherent in classical sampling into a requirement on quantum hardware. This shift allows the approach to extend naturally from RBMs to fully connected Boltzmann machines, opening opportunities inaccessible to classical training methods.
Quantum annealers are emerging as programmable, dynamical experimental platforms for probing strongly correlated spin systems. Yet key thermal assumptions, chiefly a Gibbs-distributed output ensemble, remain unverified in the large-scale regime. Here, we experimentally and quantitatively assess Gibbs sampling fidelity across system sizes spanning over three orders of magnitude. We explore a wide parameter space of coupling strengths, system sizes, annealing times, and D-wave hardware architectures. We find that the naively assumed scaling law for the effective temperature requires a non-negligible, coupling-independent offset that is robust across machines and parameter regimes, quantifying residual non-thermal effects that still conform to an effective Gibbs description. These non-idealities are further reflected in a systematic discrepancy between the physical temperature inferred from the sampled ensemble and the nominal cryogenic temperature of the device. Our results systematically assess the viability of quantum annealers as experimental platforms for probing classical thermodynamics, correct previous assumptions, and provide a physically grounded thermometry framework to benchmark these machines for future thermodynamic experiments.
Reverse annealing initializes a programmable quantum annealer in a chosen classical state, reintroduces quantum fluctuations down to a pause point, and returns to readout; repeated cycles form a configurable dissipative dynamics whose schedule knobs are routinely treated as a temperature. On mixed-frustration 12-qubit Ising instances on two D-Wave generations (Advantage2 and Advantage_system6.4), we measure schedule-driven redistribution of the subsystem readout and calibrate each schedule's effective scales, using exhaustive enumeration of the 212 configurations, a locked analysis plan with a prior-knowledge disclosure, and regenerated classical reference families. First, schedule shape redistributes basin occupation: 11 of 20 random instances respond to a single-pause-to-two-pause change (15 of 20 for long-hold; block permutation, FDR-controlled), with shifts up to 38 percentage points and changes of dominant-configuration identity. Second, the single-qubit probe temperature does not transfer to the many-body problem: moment-matched inverse temperatures span 0.63-9.1 against a probe value of 7.22, and pause depth and duration each shift the scale on most instances (19/20 and 15/20). Third, no observation requires more than the tested classical families: with update budgets extended below one sweep per cycle, all 33 (device, instance, schedule) cells are reproduced within the tested family (minimum intersection-union p=0.76), and schedule-aware spin-vector Monte Carlo brings the late-window statistics within semiclassical reach. A pre-registered linear predictor fails on held-out instances, and an earlier pilot's two-pause enhancement claim is contradicted (Fisher p=0.0013). Reverse-anneal schedules are instance-specific basin-occupation controls whose effective scales must be calibrated per instance and schedule; nothing here licenses a quantum-mechanistic reading.