Figure 1: Conceptual illustration comparing classical samplers – Metropolis–Hastings and parallel tempering – with DCQS. While classical samplers rely on thermal fluctuations to traverse the energy landscape, which can lead to poor sampling in the low-temperature regime, DCQS employs quantum fluctuations guided by counterdiabatic protocols, enabling more efficient access to low-energy configurations.
Digitized Counterdiabatic Quantum Sampling (DCQS)
Boltzmann sampling lies at the heart of numerous industrial problems — from molecular simulation and drug discovery to financial risk analysis and energy grid optimization.
Classical algorithms such as Metropolis–Hastings and parallel tempering have been refined over decades and are among the most powerful tools for sampling complex energy landscapes [1]. However, even these mature techniques encounter severe slowdowns in a particularly important regime — low temperatures, where energy barriers grow steep and relevant configurations become exponentially hard to access.
Kipu Quantum introduces a new paradigm: Digitized Counterdiabatic Quantum Sampling (DCQS) [2] — a hybrid classical-quantum algorithm that leverages quantum dynamics to overcome this bottleneck and achieve quantum-advantage level runtime and sampling efficiency for low-temperature Boltzmann sampling on current quantum hardware.
What is Digitized Counterdiabatic Quantum Sampling?
At low temperatures, Boltzmann distributions are dominated by low-energy configurations. Classical samplers rely on thermal fluctuations to explore this landscape but become inefficient at low temperatures, making important configurations increasingly hard to discover.
DCQS addresses this limitation by using quantum fluctuations instead, which can overcome high energy barriers by quantum tunneling to access low-energy regions more efficiently.
Counterdiabatic protocols, combined with an iteratively refined bias field, are employed to steer the system toward these relevant configurations, suppressing unwanted excitations. DCQS then reconstructs accurate low-temperature Boltzmann distributions using classical reweighting techniques — achieving efficient, low-depth sampling on today’s quantum computers.
Validated at Scale on Quantum Hardware
The algorithm was validated experimentally on IBM’s Marrakesh and Fez quantum processors, scaling up to 156 qubits. Across benchmarks including disordered Ising and spin-glass Hamiltonians, we found cases where DCQS outperforms state-of-the-art classical sampler parallel tempering in the low-temperature regime. Parallel tempering required three orders of magnitude more samples to match the accuracy of DCQS, establishing an approximate 2× runtime quantum-advantage [2].
Figure 2: Comparison of DCQS and parallel tempering performance. (a) Energy distributions of DCQS and (b) energy distribution of parallel tempering for a challenging 156 qubit problem with three-body interactions. (c) Runtime necessary to accurately reproduce the low-temperature Boltzmann distribution. DCQS achieves a runtime advantage reaching the target accuracy in 5 seconds. Meanwhile, a high-performance parallel tempering implementation generating 8.7 million samples per second requires at least 8 seconds across 5 different runs. This demonstrates a tangible quantum acceleration for sampling on current quantum hardware.
Industrial Relevance and Applications
Efficient Boltzmann sampling is at the heart of numerous industrially relevant use cases.
- Pharmaceuticals: Boltzmann sampling governs how molecular systems occupy conformations at equilibrium. Accurate sampling of protein–ligand configurations determines binding affinities, drug efficacy, and pathway kinetics [3,4]. DCQS accelerates access to these rare low-energy states, potentially reducing the computational cost of molecular dynamics and docking pipelines.
- Finance: Many financial models depend on exploring distributions of portfolio states or stochastic processes under uncertainty. Boltzmann sampling enables efficient estimation of rare but impactful events in risk and derivative pricing models [5]. By improving sampling efficiency, DCQS can accelerate stress testing — the evaluation of portfolio resilience under extreme market conditions — and enhance portfolio optimization in Monte Carlo simulations.
- Energy and Logistics: Optimization problems such as grid scheduling or logistics routing can be formulated as minimizing energy-like cost functions. Sampling from Boltzmann distributions over possible configurations enables probabilistic search for near-optimal solutions [6]. Quantum sampling enhances exploration of rugged cost landscapes, offering speedups in optimization pipelines for energy dispatch and industrial planning.
- Materials Science: The low-temperature phases of materials are determined by rare low-energy configurations in complex atomic or spin systems. Boltzmann sampling provides access to equilibrium distributions that describe stability, defect formation, or phase transitions [7]. Efficient Boltzmann sampling can aid the design of new materials and magnetic systems.
- Machine Learning: Many generative and probabilistic models, such as Boltzmann machines or diffusion-based samplers, rely on efficient exploration of energy landscapes to learn equilibrium-like distributions [8]. DCQS can serve as a quantum-accelerated sampler, improving convergence in energy-based models and offering a foundation for quantum-enhanced learning architectures.
Outlook
DCQS demonstrates that hybrid classical-quantum strategies can already deliver practical value. Its robustness to noise, low circuit depth, and iterative bias-field scheme make it ideal for near-term deployment. Looking forward, integration with machine-learning-based Boltzmann generators or hybrid Markov chain Mote Carlo pipelines could enable broader quantum-enhanced modeling in chemistry, finance, and materials engineering, paving the way towards industrial quantum advantage.
References
[1] D. J. Earland and M. W. Deem, “Parallel tempering: Theory, applications, and new perspectives”, Phys. Chem. Chem. Phys. 7, 3910 (2005).
[2] N. N. Hegade, N. L. Kortikar, B. A. Bhargava, J. F. R. Hernandez, A. G. Cadavid, P. Chandarana, S. V. Romero, S. Kumar, A. Simen, A.-M Visuri, E. Solano, and P. A. Erdman, “Digitized Counterdiabatic Quantum Sampling”, arXiv:2510.26735 (2025).
[3] D. Frenkel and B. Smit, “Understanding Molecular Simulation”, Academic Press (2002).
[4] F. Noé, S. Olsson, J. Köhler, and H. Wu “Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning”, Science 365, eaaw1147 (2019).
[5] P. Glasserman,“Monte Carlo Methods in Financial Engineering”, Springer New York (2013).
[6] D. Sahu, N., R. Chaturvedi, S. Prakash, T. Yang, R. S. Rathore, and I. Alsolbi, “Optimizing energy and latency in edge computing through a Boltzmann driven Bayesian framework for adaptive resource scheduling”, Sci. Rep. 15, 30452 (2025).
[7] J.M. Yeomans, “Statistical Mechanics of Phase Transitions”, Oxford University Press (1993).
[8] J. Ngiam, Z. Chen, P. W. Koh, and A. Y. Ng, “Learning deep energy models”, Proc. Int. Conf. Mach. Learn. (2011).