Digitized Counterdiabatic Quantum Sampling

TL;DR

DCQS addresses the challenge of sampling Boltzmann distributions at low temperatures by combining digitized counterdiabatic driving with an adaptive bias-field strategy, followed by classical reweighting to obtain the Boltzmann observables. The method yields approximate finite-temperature Boltzmann distributions through a set of low-energy states, demonstrated on disordered 1D Ising systems and a 156-qubit higher-order spin-glass on IBM hardware, with a scalable figure of merit based on the KL divergence and total variation distance to quantify convergence. On both simulators and quantum hardware, DCQS requires far fewer samples than Metropolis-Hastings and outperforms parallel tempering in the low-temperature regime, delivering about a 2x runtime advantage and validating a practical route to Boltzmann sampling on contemporary quantum processors. The work suggests potential synergies with classical samplers and emphasizes robustness to NISQ noise, offering a path toward scalable thermal sampling in physics and machine learning contexts.

Abstract

We propose digitized counterdiabatic quantum sampling (DCQS), a hybrid quantum-classical algorithm for efficient sampling from energy-based models, such as low-temperature Boltzmann distributions. The method utilizes counterdiabatic protocols, which suppress non-adiabatic transitions, with an iterative bias-field procedure that progressively steers the sampling toward low-energy regions. We observe that the samples obtained at each iteration correspond to approximate Boltzmann distributions at effective temperatures. By aggregating these samples and applying classical reweighting, the method reconstructs the Boltzmann distribution at a desired temperature. We define a scalable performance metric, based on the Kullback-Leibler divergence and the total variation distance, to quantify convergence toward the exact Boltzmann distribution. DCQS is validated on one-dimensional Ising models with random couplings up to 124 qubits, where exact results are available through transfer-matrix methods. We then apply it to a higher-order spin-glass Hamiltonian with 156 qubits executed on IBM quantum processors. We show that classical sampling algorithms, including Metropolis-Hastings and the state-of-the-art low-temperature technique parallel tempering, require up to three orders of magnitude more samples to match the quality of DCQS, corresponding to an approximately 2x runtime advantage. Boltzmann sampling underlies applications ranging from statistical physics to machine learning, yet classical algorithms exhibit exponentially slow convergence at low temperatures. Our results thus demonstrate a robust route toward scalable and efficient Boltzmann sampling on current quantum processors.

Figures (15)

• Figure 1: Sketch of (a,b) classical methods for Boltzmann sampling and (c) digitized counterdiabatic quantum sampling (DCQS). (a) The classical Metropolis-Hasting algorithm suffers from a diverging mixing time at low temperatures, since it can get trapped in local minima. The black line pictorially represents the energy landscape of the classical Hamiltonian. Transitions to the local minima on the right are exponentially suppressed in the low-temperature limit. (b) Parallel tempering (PT) is a classical method that overcomes this limitation by running multiple "replicas" in parallel (one per row), each one at a different inverse temperature $\beta_i$. The left column represents the allowed transitions within the energy landscape (horizontal arrows), and between adjacent replicas (vertical arrows), while the right column shows the corresponding energy distributions. This leads to a better low-temperature sampling at the expense of an increased simulation cost. (c) DCQS relies on running quantum circuits with an iterative bias field strategy on a digital quantum computer (left), producing lower and lower energy states (center), which are then reweighed in order to reproduce low-temperature observables (right).
• Figure 2: Energy distribution obtained from DCQS for the spin glass model in Eq. \ref{['eq:spin_glass']} for $N=18$. The red distribution in panel (a) corresponds to the $1$st BF iteration with $b_i^{(1)}=0$, and the gray distribution in panel (b) to the $5$th iteration with iteratively updated $b_i^{(5)}$. The black thick line corresponds to the exact Boltzmann distribution with an effective temperature, reported on the plot, determined by matching the average energy of the DCQS distribution. The parameters are $n_\text{iter} =5$, $n_\text{shots} = 30\,000$, $w=1$, and $n_\text{cvar}=30\,000$.
• Figure 3: Energy distribution obtained from DCQS for the 1D Ising model in Eq. \ref{['eq:ising_h']} for $N=18$ using a state-vector simulation. The red distribution corresponds to the first iteration with $b_i^{(1)}=0$ and the gray distribution to the last BF iteration with iteratively updated $b_i^{(5)}$. The parameters are $n_\text{iter} =5$, $n_\text{shots} = 1000$, $w=0.5$, and $n_\text{cvar}=20$.
• Figure 4: Observables computed for the 1D Ising model with $N=18$: (a) magnetization, (b) the connected correlation function of Eq. \ref{['eq:correlation_function']}, (c) average energy, and (d) figure of merit $\ln \tilde{\mathcal{Z}}$ as functions of temperature. The black full lines correspond to exact results based on transfer matrix calculations. The dashed lines correspond to DCQS with the same parameters as in Fig. \ref{['fig:ising_18_energy']}, but with a number of shots per iteration reported in the legend. In panel (d) the exact partition function (black line) is used as the zero reference, while the dashed lines are the corresponding values of $\ln \tilde{\mathcal{Z}}_\text{DCQS}$. The difference between the full and dashed lines corresponds to the KL diverge between the reweighed DCQS and the exact Boltzmann distribution [see Eq. \ref{['eq:kl_tilde_p']}]. The gray vertical line denotes the low-temperature regime.
• Figure 5: Energy distribution obtained running DCQS for the 1D Ising model in Eq. \ref{['eq:ising_h']} for $N=124$ using the MIMIQ MPS simulator [panel (a)], and running on IBM Marrakesh [panel (b)]. The red distribution corresponds to the $1$st iteration, the gray distribution to the last BF iteration, and the black one to the states found applying post-processing to the IBM hardware data. DCQS parameters are $n_\text{iter}=5$, $n_\text{shots}=20\,000$, $w=1$, $n_\text{cvar}=20$, $n_\text{PP}=100$, and $n_\text{sweeps}=10$.