Quantum Annealing-Based Sampling

• Quantum Annealing-Based Sampling is a method that exploits the quantum evolution of system Hamiltonians to generate representative samples from complex, high-dimensional energy landscapes modeled by Ising and QUBO formulations.
• It employs innovative techniques such as SPVAR, inhomogeneous annealing schedules, and hybrid classical–quantum protocols to enhance sample diversity and mitigate hardware-induced biases.
• Applications span combinatorial optimization, statistical physics simulations, and machine learning, offering rapid sampling and improved solution diversity in practical, noisy environments.

Quantum Annealing-Based Sampling encompasses a spectrum of algorithms, hardware realizations, and hybrid strategies that exploit quantum annealing (QA) devices or QA-inspired protocols to generate representative samples from complex, high-dimensional energy landscapes. While quantum annealing was initially developed for combinatorial optimization, its stochastic output—comprising multiple spin or bitstring configurations per run—renders it a tool for sampling from distributions related to classical Ising and QUBO models, Boltzmann weights, or ground-state spaces. The interplay of quantum fluctuations, control noise, and open-system effects yields a diverse array of sampling behaviors, ranging from rare-event exploration to (sometimes biased) coverage of degenerate manifolds. These capabilities underpin applications in combinatorial enumeration, thermal simulation, machine learning model training, and probabilistic inference.

1. Foundations: Quantum Annealing as a Sampling Process

Quantum annealing is implemented as a time-dependent evolution under a Hamiltonian of the form

$$H(t) = A(s(t))\,H_{\rm driver} + B(s(t))\,H_{\rm problem}, \quad s(t) \in [0,1]$$

with $A(0) \gg B(0)$, $A(1) \ll B(1)$. The standard driver is a transverse field $H_{\rm driver} = -\sum_{i=1}^N \sigma_i^x$, inducing off-diagonal quantum fluctuations, and the problem Hamiltonian $H_{\rm problem}$ encodes an Ising or QUBO cost function. Physical QA hardware, such as D-Wave's Chimera or Pegasus topologies, or gate-model emulators via Suzuki–Trotter discretization, realizes this evolution via a finite-speed schedule over microseconds to milliseconds scales (Karimi et al., 2016, Henke et al., 2024, Bhave et al., 2023).

Upon completion, a projective measurement in the computational (z-) basis yields a classical string; repeated anneals sample a distribution over configurations. In the open-system, finite temperature, and nonadiabatic limit relevant to practical devices, this output is not a strict ground-state projector but rather a noisy, low-temperature Gibbs-like sampler (Vuffray et al., 2020, Nelson et al., 2021, Pelofske, 5 Nov 2025).

2. Statistical Properties and Effective Sampling Distribution

The output distribution of a physical quantum annealer, $\nu(s)$, is well-captured as a mixture of Boltzmann distributions at a hardware-specific effective inverse temperature $\beta_{\rm eff}$, subject to spurious fields and couplings from calibration noise: $$\nu(s) \approx \frac{\exp[-\beta_{\rm eff} H_{\text{eff}}(s)]}{Z(\beta_{\rm eff})}$$ where $H_{\text{eff}}$ incorporates systematic and random errors in local field biases and coupler strengths. Spurious terms arise predominantly at second order in control noise, creating non-native interactions and biasing sampling away from the clean target Gibbs ensemble (Vuffray et al., 2020). For small, hardware-native Ising instances, the total-variation distance from the intended Gibbs distribution can be made $\lesssim 5\%$ by tuning the energy-scale and anneal time; $\beta_{\rm eff}$ can be manipulated via these controls to target a broad range of effective temperatures (Nelson et al., 2021, Pelofske, 5 Nov 2025).

This paradigm extends to complex frustrated systems, such as the ANNNI model, where record-low sampling errors (TVD $\sim 3\times 10^{-4}$) and low temperatures ($\beta\sim 32$) have been achieved by careful hardware parameter optimization (Pelofske, 5 Nov 2025), and to kernel learning tasks in regression via QA-sampled random Fourier features (Hasegawa et al., 13 Jan 2026).

3. Ground-State Sampling, Diversity, and Fairness Issues

A central theme in QA-based sampling is the exploration of degenerate ground-state manifolds. In the zero-temperature adiabatic limit, the quantum annealer produces superpositions in the ground-subspace; projective measurement samples these with probabilities set by the lowest-eigenvector of the driver Hamiltonian projected onto the ground manifold (Zhang et al., 2017, Kumar et al., 2020). In generic instances, especially with the standard transverse-field driver, amplitudes are highly non-uniform—some states are "hard-suppressed" (never observed), while others are over-represented (Könz et al., 2018). This failure of fair sampling is intrinsic to the sparsity of the driver and not merely a hardware imperfection.

While perturbative addition of higher-order drivers (quadratic, cubic) can occasionally reduce bias in toy problems, they do not generically restore uniformity for real-world spin glasses; only "dense" all-to-all drivers would suffice, which are not experimentally feasible (Könz et al., 2018). The output ground-state bias can be quantified via the total-variation distance, Kullback–Leibler divergence, or explicit bias metrics $b(\mathbf{p})$ (Zhang et al., 2017).

Hybrid classical-quantum schemes offer two mitigation strategies:

1. Hybrid Sampling: Aggregating samples from simulated annealing (SA), classical Monte Carlo, and QA runs reduces statistical bias and covers rare solutions that may be exponentially suppressed by either method alone (Zhang et al., 2017).
2. Reverse Annealing and Diagonal Perturbations: Modifying the terminal stages of annealing by introducing random diagonal perturbations (reverse annealing-inspired schedules) can restore approximate fairness, samplng each degenerate ground state with nearly equal probability as proven perturbatively and validated experimentally (Kumar et al., 2020).

4. Algorithmic Innovations and Enhanced Scheduler Strategies

Several algorithmic and scheduler-level advances improve sampling quality and ground-state coverage:

• Sample Persistence Variable Reduction (SPVAR/ISPVAR):

By analyzing the persistence $p_i$ of spin values over low-energy samples, highly persistent variables are identified and fixed, recursively reducing problem dimensionality. This protocol enables 84–96% of variables to be frozen on large random Ising instances, dramatically increasing success rates and solution multiplicity, particularly when combined with classical pre-processing such as roof duality and weak persistency (Karimi et al., 2016).

• Inhomogeneous/Algorithmic Annealing Schedules:

Inducing non-uniform, space–time separated annealing fronts allows the systematic redistribution of defect nucleation events, enabling higher diversity among sampled near-optimal states. Applying independent critical fronts to different system clusters (algorithmic quantum phase transitions) yields up to 40% increase in solution diversity (number of statistically independent low-energy "basins") and reduces the fraction of hard-to-sample instances by more than 25% in 2D spin glasses (Mohseni et al., 2021).

• Discretized QA and Hybrid Classical–Quantum Schedules (DiQA/QASA):

Discretizing the QA process for gate-model quantum devices (DiQA) and using short-depth quantum circuits to initialize classical simulated annealing enables retention of quantum sampling benefits at dramatically reduced overall classical runtime (Bhave et al., 2023). The QASA framework provides equivalent or better solution quality with savings of 30–50% in classical SA steps.

5. Practical Applications in Machine Learning and Statistical Physics

Quantum-annealing–based sampling has been integrated as a Markov-chain Monte Carlo (MCMC) subroutine to accelerate mixing and improve sample quality, especially for models and datasets with severe mode separation (Arai et al., 12 Feb 2025, Korenkevych et al., 2016). In training Boltzmann machines and deep neural networks, raw or briefly post-processed QA samples eliminate the need for long burn-in periods; QA MCMC seeds enable rapid crossing of high-energy barriers in frustrated or multimodal landscapes, outperforming classical CD/PCD and generating richer support in generative use cases (Adachi et al., 2015, Korenkevych et al., 2016, Koshka et al., 2019).

QA-based samplers also target thermal and importance sampling of complex physical systems. For example, full-lattice importance sampling of U(3) gauge theory in the strong-coupling dual representation utilizes QA to generate sub-lattice Boltzmann–like histograms, stitched together via Metropolis–Hastings corrections. This protocol, validated on 2D and 4D volumes, matches classical thermodynamic observables and statistical measures, demonstrating the scalability of QA-based sampling for problems in lattice quantum field theory (Kim et al., 2024).

Diabatic QA, where annealing is halted before full adiabatic completion, provides controllable Boltzmann sampling in the high-temperature regime, with the effective temperature governed by the annealing rate and largely independent of system size in the paramagnetic phase (Gyhm et al., 2024). This supports rapid sampling for variational inference and negative-phase updates in machine learning models.

6. Controversies, Bias Correction, and Open Problems

Intrinsic bias in quantum-annealing–based sampling remains a subject of technical concern and ongoing research (Zhang et al., 2017, Könz et al., 2018). Standard TF-QA fails at fair sampling except in trivial cases; even with hardware-native and symmetric Ising problems, strong suppression of subsets of ground states occurs, limiting applicability to domains demanding uniform enumeration. Reverse annealing schemes, programmable diagonal perturbations, or engineered hybrid quantum–classical protocols have made significant progress, but scaling to large, realistic instances and nontrivial bias fields is challenging. Post-processing, including classical cluster updates, reweighted sampling, or Glauber melts, can partially ameliorate bias but require additional classical computation (Könz et al., 2018, Kumar et al., 2020, Yamamoto et al., 2019).

Fair sampling by quantum annealers, essential for model counting, SAT filtering, or precise entropy estimation, remains an open technological and algorithmic problem. Further, the coupon-collector scaling of uniform sampling (requiring $O(m \log m)$ anneals for $m$-degenerate instances) sets a fundamental complexity lower bound for exhaustive ground-state enumeration (Kumar et al., 2020).

Noise characterization and mitigation in hardware remain important; while coherent unitary evolution and parametric tuning enable low-TVD sampling in some regimes, sample-to-sample field noise and quadratic spurious couplings fundamentally limit perfect thermalization on current devices (Vuffray et al., 2020, Nelson et al., 2021, Pelofske, 5 Nov 2025).

7. Outlook and Impact

Quantum-annealing–based sampling is established as a versatile, hardware-accelerated approach for drawing approximate samples from complex discrete distributions, with unique abilities to explore rare-event regions inaccessible to local classical methods. Continued advancements in hybrid quantum–classical algorithms, robust bias correction, embedded pre/post-processing (such as SPVAR, reverse annealing, MCMC seeding), and adoption of inhomogeneous time-dependent protocols, combined with ongoing hardware improvements (lower noise, better connectivity, more flexible drivers), are expected to solidify QA-based sampling as a valuable tool for the physical sciences, machine learning, combinatorial optimization, and data-driven inference (Karimi et al., 2016, Zhang et al., 2017, Mohseni et al., 2021, Bhave et al., 2023, Kumar et al., 2020, Pelofske, 5 Nov 2025, Hasegawa et al., 13 Jan 2026).