Classical Thermometry of Quantum Annealers

Abstract

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.

Figures (11)

• Figure 1: Model and Embedding (a) A ring of antiferromagnetically arranged spins with a domain wall (red) comprising aligned neighboring pair of spins. (b) Real space rendering of a domain wall distributions at high($j_{\text{enc}}=1$) and low ($j_{\text{enc}}=0.1$) coupling in two configurations measured from the QPU for the same system size. (c) The empirical domain wall probability distribution (bar plots) obtained from annealer measurements along with the most similar theoretical domain wall probability distribution (line plots) is shown for two coupling strength: $N_{\text{qb}} = 301$, $\tau = 10 \mu s$, on Advantage2_System1.1, $j_{\text{enc}}=0.75,0.25$). Note the poor agreement at strong couplings, small times, for this machine, discussed in the text. (d) Empirically domain-wall density $\langle n_{\text{dw}} \rangle$ (dots) at different extracted effective temperatures $t_{\text{eff}}$ for Advantage_system4.1 as shown on top of the analytical expressions (solid lines) over a range of system sizes. (e) An embedding of a 1D ring onto D-Wave's Zephyr Architecture, without chaining, one qubit per spin.
• Figure 2: Effective Temperatures and TVD Deviations from Gibbs Distributions for the four machines. Each sub-figure corresponds to a machine and contains 3D Surface plots of the effective temperature ($t_{\text{eff}}$), and TVD ($\epsilon$) ploted versus the inverse encoded energy coupling $1/j_{\text{enc}}$ and order of magnitude of system size $\log_{10}(N_{\text{qb}})$ or annealing time $\tau$. The small gold colored regions in the plots for $\epsilon$ for Advantage2 devices correspond to zero temperature samples, for which $\epsilon$ is not plotted: as explained in the text, strong energy couplings and small system sizes lead to inapplicable sampling distributions and a breakdown of our thermometry methods, because the ground state is always reached.
• Figure 3: Effective Temperature $t_{\text{eff}}$ plotted vs. $1/j_{\text{enc}}$ (a) and $T_{\text{machine}}/J_{\text{phys}}$ (b) for the four machines. The data points are averaged over sizes $N_{\text{qb}} \ge 100$, and the error bar shows the very small fluctuation in $t_{\text{eff}}$ across different sizes $N_{\text{qb}}$, with the lower (upper) bound corresponding to the smaller (larger) $t_{\text{eff}}$. Note that data across different machines collapse around similar at the same physical coupling $J_{\text{phys}}$, with the exception of Advantage2_system1.1. The slope of panel (b) is the dimensionless ($J_{\text{phys}}$ is measured in units of temperature) constant $\alpha$ of Eq. (\ref{['Eq:Teff']},\ref{['eq:alpha_ratio']},\ref{['Eq:Teff2']}). Plots of $t_{\text{eff}}$ vs. $\tau$ on QPU Advantage2_system1.1 programmed at $j_{\text{enc}}=0.75$ for difference values of $N_{\text{qb}}$ (c) or for $N_{\text{qb}}=1001$ and different values of $J_{\text{phys}}$ (d) show monotonic decrease of the effective temperature converging to asymptotic values at large annealing time, .
• Figure 4: Extracted Parameters for the Effective Temperature The temperature offset $\bar{t}_{\text{eff}}$ (a) and the slope $\alpha$ (b) obtained from fitting Eq. (\ref{['Eq:Teff2']}) to data from annealing $T$ for all four machines, are plotted vs. the annealing time $\tau$. As in Fig.. \ref{['fig:cuts']}b,c, the values are monotonically decreasing in annealing time, saturating at long annealing.
• Figure S1: Effective Temperatures and TVD Deviations from Gibbs Sampling for the four machines at annealing time of $\tau= 1000 \mu s$. Each sub-figure corresponds to a different machine and contains 3D Surface plots of the effective temperature ($t_{\text{eff}}$), and TVD ($\epsilon$) plotted against the inverse encoded energy coupling $1/j_{\text{enc}}$ and order of magnitude of system size $\log_{10}(N_{\text{qb}})$. The small gold colored regions in the plots for $\epsilon$ for Advantage2 devices correspond to zero temperature samples, for which $\epsilon$ is not plotted: as explained in the text, strong energy couplings and small system sizes lead to inapplicable sampling distributions and a breakdown of our thermometry methods, because the ground state is always reached.