Annealing-based Quantum Boltzmann Machines

• Annealing-based QBMs are quantum statistical models defined by non-commuting Hamiltonians that leverage an annealing process instead of classical Gibbs sampling.
• They utilize clamped Hamiltonians and quantum Gibbs states for efficient training in applications like reinforcement learning and generative modeling.
• Various architectures, including restricted, semi-restricted, and fully-visible QBMs, address hardware calibration and error-mitigation challenges in current quantum annealers.

Searching arXiv for recent and foundational work on annealing-based Quantum Boltzmann Machines.

Annealing-based Quantum Boltzmann Machines (QBMs) are Boltzmann-machine models whose sampling, training, or inference loop is organized around an annealing process rather than purely classical Gibbs updates. In the strict formulation, a QBM is a quantum statistical model defined by a non-commuting Hamiltonian, typically of transverse-field Ising form, with learning expressed through quantum Gibbs states, quantum free energies, and expectation values of observables. In a broader annealing-based usage, the term also covers classical or semi-classical Boltzmann machines whose negative phase is estimated from quantum annealers, simulated quantum annealing, or annealing-inspired samplers, including regimes in which raw annealer outputs are interpreted as the native distribution of a QBM rather than as a distorted approximation to a classical Boltzmann law (Amin et al., 2016, Crawford et al., 2016, Korenkevych et al., 2016).

1. Hamiltonian and probabilistic formulation

A classical Boltzmann machine is specified by an Ising energy over binary variables. The quantum generalization replaces hidden binary variables by qubits and the energy function by a quantum Hamiltonian. In the transverse-field Ising formulation, a QBM is defined by

$$H = - \sum_{a} \Gamma_a \sigma^x_a - \sum_{a} b_a \sigma^z_a - \sum_{a,b} w_{ab} \sigma^z_a \sigma^z_b,$$

with Gibbs state

$$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$

For visible configuration $\mathbf v$, the model probability is obtained by projecting the visible subsystem and tracing over the remainder. When $\Gamma_a \to 0$, the Hamiltonian becomes diagonal in the computational basis and the QBM reduces to a classical Boltzmann machine (Amin et al., 2016).

Annealing-based work often uses a clamped Hamiltonian, especially in supervised or reinforcement-learning settings. In the deep QBM formulation for state-action inputs, visible units are fixed and only hidden qubits remain dynamical: $$\mathcal{H}_{\mathbf v}= -\sum_{v \in V,\, h \in H} w^{vh}v \,\sigma^z_h -\sum_{\{v, v'\} \subseteq V } w^{vv'} v v' -\sum_{\{h, h'\} \subseteq H} w^{hh'} \sigma^z_h\sigma^z_{h'} - \Gamma \sum_{h \in H} \sigma_h^x .$$ The associated free energy is

$F(\mathbf v) = -\frac{1}{\beta}\ln Z_{\mathbf v}$

and serves as the basic scalar quantity for learning and inference. In these clamped formulations, the visible sector remains classical, while the hidden sector and the free-energy functional carry the quantum character (Crawford et al., 2016).

Several model families coexist within the annealing-based literature. Restricted Boltzmann machines (RBMs) retain bipartite visible-hidden structure and classical conditionals; deep Boltzmann machines (DBMs) introduce hidden-hidden couplings across layers; semi-restricted QBMs allow visible-visible couplings but no hidden-hidden couplings; and recent fully-visible QBMs remove hidden units altogether and place the expressive burden on a non-commuting fully-visible Hamiltonian. A central subtlety is that, once the Hamiltonian is non-diagonal, clamping no longer reproduces the true conditional distribution in general; this is one of the sharp conceptual differences between classical Boltzmann machines and genuine QBMs (Amin et al., 2016, Tüysüz et al., 2024).

2. Annealing as the sampling engine

The standard annealing primitive is a transverse-field Ising evolution

$$\mathcal H(t)= - \sum_{i, j} J_{ij} \sigma^z_i \sigma^z_j - \sum_i h_i \sigma^z_i - \Gamma(t) \sum_i \sigma^x_i,$$

with a large initial transverse field that is gradually reduced. In simulated quantum annealing (SQA), this dynamics is approximated by a Suzuki-Trotter mapping to a classical Ising model in one extra dimension, so that samples of a lifted classical system approximate the quantum Gibbs state of the original Hamiltonian. This is the mechanism used to sample deep QBMs for reinforcement learning, with near-zero final transverse field reproducing classical DBM sampling and non-negligible final transverse field producing genuinely quantum QBM sampling (Crawford et al., 2016).

Annealing-based QBM research is not confined to physical quantum annealers. Hamiltonian-simulation approaches based on the Eigenstate Thermalization Hypothesis replace explicit thermal-state preparation by quench dynamics and local measurements: under ETH, long-time expectations of local observables approximate their thermal values, which is sufficient because QBM gradients depend on low-weight local operators. This yields a route that is annealing-like in function—thermal expectation estimation—but is realized through unitary dynamics rather than through an open-system annealer (Anschuetz et al., 2019).

A recurring empirical theme is that annealer outputs need not coincide with a classical Boltzmann distribution of the programmed Ising energy. On frustrated-cluster-loop benchmarks, raw quantum-annealer samples can be extremely close to classical Boltzmann distributions in some symmetric cases and significantly non-Boltzmann in heterogeneous cases; those deviations are attributed to transverse field, freeze-out, environmental coupling, and finite temperature. This motivates interpreting the device either as an imperfect classical sampler or as a sampler for a QBM-like quantum distribution (Korenkevych et al., 2016).

Because annealer outputs are hardware-distorted, effective-temperature and calibration issues are central. One line of work models samples with a single effective inverse temperature $\beta_{\rm eff}$; another introduces separate internal parameters for interaction terms, visible biases, and hidden biases, or even one parameter per bias, and updates these online from the same samples used for RBM training. In parallel, Nested Quantum Annealing Correction (NQAC) has been shown to reduce effective temperature and improve training, even though the decoded output can become less Gibbs-like at larger nesting levels and longer anneal times (Goto et al., 2023, Li et al., 2019).

Annealing-based sampling also extends to quantum-inspired devices. SimCIM, a numerical simulation of a coherent Ising machine, was used as a Boltzmann generator for partition-function estimation and BM training, while supervised transfer-learning work employed simulated annealing as a stand-in for quantum annealing in a QBM classifier. These studies preserve the algorithmic role of annealing-based sampling even when the sampler itself is classical (Ulanov et al., 2019, Schuman et al., 2023).

3. Learning rules, objectives, and information geometry

The exact negative log-likelihood gradient of a QBM is complicated by non-commutativity: the derivative of $e^{-H}$ cannot be reduced to a classical expectation unless $H$ commutes with its derivatives. A standard workaround is the Golden-Thompson bound, which replaces the intractable quantum likelihood by an upper bound involving clamped Hamiltonians. For diagonal parameters such as $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$0 and $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$1, this yields update rules of the familiar positive-phase minus negative-phase form, but with quantum expectations: $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$2 This makes annealing-based training feasible because both clamped and unclamped expectations can, in principle, be sampled rather than computed exactly (Amin et al., 2016).

In free-energy-based reinforcement learning, the negative free energy of a clamped BM or QBM is used as a Q-function approximator: $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$3 For QBMs the required gradients are expectations of observables in the quantum Gibbs state. In the transverse-field formulation used for maze domains, SQA provides estimates of $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$4, $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$5, and the effective free energy, which then enter TD-style updates for visible-hidden and hidden-hidden weights (Crawford et al., 2016).

Recent theory recasts fully-visible QBM learning as an operational expectation-matching problem and studies it with the quantum relative entropy as loss. In that setting, the gradient has the particularly simple form

$$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$6

and stochastic gradient descent is proved to require at most a polynomial number of Gibbs states. The same work proves that pre-training on a subset of Hamiltonian parameters can only lower the sample-complexity bounds, with explicit pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. These results establish a trainability contrast between fully-visible QBMs and earlier restricted constructions (Coopmans et al., 2023).

The geometry of parameterized thermal states has also been made explicit. For a Hamiltonian $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$7 and Gibbs state $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$8, the Fisher-Bures and Kubo-Mori information matrices can be written in closed form and used to define natural-gradient updates for QBM training. This places annealing-based QBMs on an information-geometric footing: the annealer supplies approximate thermal states, while the outer-loop optimizer can incorporate the intrinsic geometry of the thermal-state manifold rather than relying on Euclidean gradient descent (Patel et al., 2024).

4. Architectures and expressive power

Annealing-based QBMs appear in several architectural forms. Deep QBMs for reinforcement learning are graphically close to DBMs: visible state units connect to a first hidden layer, visible action units connect to a last hidden layer, and hidden layers couple only to adjacent layers. This structure was emphasized as compatible with D-Wave-like hardware because layered bipartite blocks resemble Chimera-style connectivity. In the corresponding clamped Hamiltonians, visible units are removed from the sampling graph and replaced by effective fields on neighboring hidden qubits (Crawford et al., 2016).

Semi-restricted QBMs occupy an intermediate position between RBMs and fully unrestricted BMs. In Amin’s formulation, visible-visible couplings are permitted but hidden-hidden couplings are absent; in anomaly-detection work, the adopted QBM architecture likewise contains one visible layer, one hidden layer, all visible-hidden couplings, no hidden-hidden couplings, and full lateral connectivity among visible units. This architecture is motivated by the expectation that richer visible-side couplings can improve generative modeling without the full cost of unrestricted hidden-sector interactions (Amin et al., 2016, Stein et al., 2023).

The most direct recent challenge to classical expressivity comes from fully-visible QBMs with non-commuting 2-local Hamiltonians. These models were shown numerically to learn distributions “typically associated with higher-dimensional systems,” and the study explicitly found that non-commuting terms and Hamiltonian connectivity improve learning capabilities. In particular, low-dimensional fully-visible QBMs can represent higher-dimensional correlation structure more effectively than classical fully-visible BMs with the same basic connectivity resources. This suggests that, under fixed hardware graph constraints, non-commutativity can act as an additional representational resource rather than merely as a sampling complication (Tüysüz et al., 2024).

A related trainability distinction is now standard. Restricted QBMs with non-commuting Hamiltonians have recurrent trainability issues, while fully-visible QBMs have emerged as a more tractable option because recent results establish sample-efficient learning with relative-entropy-based objectives. This suggests that annealing-based implementations are likely to be most practical when built around fully-visible or otherwise structurally simplified Hamiltonians, rather than around heavily latent restricted constructions (Tüysüz et al., 2024, Coopmans et al., 2023).

5. Application areas

The earliest sustained application area was generative learning with annealer-assisted negative phases. Fully visible Chimera-structured Boltzmann machines were trained on D-Wave hardware, and the central finding was that raw QA samples can be useful both as seeds for short classical Gibbs chains and, when taken directly with $$\rho = Z^{-1} e^{-H}, \qquad Z=\mathrm{Tr}(e^{-H}).$$9, as the correct gradient estimator for maximizing a variational lower bound on the log-likelihood of a quantum model. On frustrated, high-barrier synthetic problems, QA-seeded MCMC improved gradient quality relative to CD and PCD, while raw QA training was interpreted explicitly as QBM training (Korenkevych et al., 2016).

Reinforcement learning is the canonical annealing-based QBM application. In maze navigation, negative free energy was used to approximate $\mathbf v$0, with state and action represented by clamped visible units and hidden qubits sampled by SQA. The resulting QBM-RL method numerically outperformed RBM-RL and slightly outperformed DBM-RL on clear mazes, windy mazes, and larger $\mathbf v$1 mazes, while classical SA and SQA gave essentially identical performance for DBM sampling when the final transverse field was negligible (Crawford et al., 2016).

The multi-agent extension preserved the same QBM free-energy principle but added an experience replay buffer and separate policy and target QBMs, in direct analogy with DQN stabilizers. These additions improved stability in single-agent $\mathbf v$2 and $\mathbf v$3 domains and enabled two-agent $\mathbf v$4 grid domains that earlier QBM-based work had not solved rationally, while larger multi-agent domains remained limited by QBM and QPU size (Müller et al., 2021).

Annealing-based sampling has also been used purely as an inference aid. In hybrid RBM inference, D-Wave samples were used to initialize Gibbs chains. The empirical pattern was that QA initialization improved sampling quality relative to random initialization for a fixed number of Gibbs updates, but that the benefit vanished as the amount of classical post-processing increased. This is a recurring theme in annealer-assisted BM work: the annealer primarily improves the early part of the mixing trajectory rather than changing the asymptotic fixed point (Kālis et al., 2023).

Outside RL, annealing-based QBMs have been embedded in hybrid supervised pipelines. In a transfer-learning setting for COVID-CT-MD, a frozen ResNet-18 backbone and compression layer produced 64-dimensional features that were binarized and passed to a supervised QBM trained with simulated annealing as a stand-in for quantum annealing. Relative to a classical feed-forward baseline of comparable scale, the QBM configuration consistently outperformed in test accuracy and AUC-ROC-Score and required fewer epochs, although absolute test accuracy remained under $\mathbf v$5 and test AUC-ROC around $\mathbf v$6 (Schuman et al., 2023).

Unsupervised anomaly detection provides another example. On an Endpoint Detection and Response inspired synthetic dataset with 21 visible units, a semi-restricted QBM trained with simulated annealing reached an F1-score of $\mathbf v$7, compared with $\mathbf v$8 for an RBM baseline. When the same trained architecture was run on D-Wave hardware, performance dropped to $\mathbf v$9 on 2000Q and $\Gamma_a \to 0$0 on Advantage 4.1, leading to the conclusion that either more accurate classical simulators or substantially more QPU time would be needed to replicate the simulation results on quantum hardware (Stein et al., 2023).

A broader annealing-based role is visible in quantum-inspired Boltzmann generation. SimCIM was used to sample high-dimensional Boltzmann distributions, train fully visible BMs, and estimate partition functions via annealed importance sampling. On BAS, SimCIM with temperature correction tracked exact-gradient learning better than MCMC, and on binarized MNIST plus label bits it yielded a final classification accuracy of approximately $\Gamma_a \to 0$1, demonstrating that annealing-based negative-phase estimation can remain useful even outside strict QBM formulations (Ulanov et al., 2019).

6. Hardware realism, controversies, and future directions

A central controversy is what distribution an annealer is actually sampling. One coherent viewpoint treats QA outputs followed by $\Gamma_a \to 0$2 Gibbs sweeps as approximate classical BM training; another treats raw QA outputs at $\Gamma_a \to 0$3 as samples from a QBM-like distribution and therefore as legitimate gradients for a variational lower bound on quantum log-likelihood. The distinction is not semantic: it determines whether annealer bias is regarded as an error to be corrected or as part of the model class itself (Korenkevych et al., 2016).

Another recurring misconception is that successful annealer-assisted learning requires faithful sampling from the programmed logical Gibbs distribution. NQAC experiments on BAS and coarse-grained MNIST complicate that picture: longer anneal times and higher nesting levels improved training performance and reduced effective temperature, yet the decoded output distribution often became less Gibbs-like with respect to the logical Hamiltonian. Similarly, hybrid inference studies found that QA samples mainly help before long classical refinement, and QBM reinforcement-learning work reported numerical gains without claiming rigorous quantum speedup (Li et al., 2019, Kālis et al., 2023, Crawford et al., 2016).

The present hardware bottlenecks are concrete. Sparse native connectivity forces embedding, chains, and penalty tuning; effective temperatures are problem-dependent and often require calibration; chain breaks, programming noise, and finite control precision distort the realized Hamiltonian; and QPU size constrains visible dimension, hidden-layer size, and hence environment complexity in RL and generative modeling. These limitations are explicit in RBM calibration work, multi-agent QBM reinforcement learning, and anomaly-detection experiments on D-Wave hardware (Goto et al., 2023, Müller et al., 2021, Stein et al., 2023).

Current theory points toward a more focused near-term agenda. Fully-visible QBMs trained with relative entropy admit polynomial sample-complexity bounds and avoid barren plateaus in that setting, while natural-gradient methods now provide explicit Fisher-Bures and Kubo-Mori geometries for thermal-state optimization. This suggests that the most viable annealing-based QBM programs are likely to combine structurally simple but non-commuting Hamiltonians, rigorous observable-based training objectives, and hardware-aware calibration or error-mitigation layers (Coopmans et al., 2023, Patel et al., 2024).

At the same time, annealing-based modeling is beginning to move beyond strict Boltzmann statistics. Variational Quantum Annealing for quantum chemistry explicitly describes itself as resembling QBMs while generalizing beyond Boltzmann distributions: the annealer is used as a sampler, and its empirical statistics define a variational ansatz even when the final state is not an exact Gibbs state. A plausible implication is that future annealing-based QBM research will increasingly treat hardware-generated distributions as native model families in their own right, rather than solely as approximations to ideal thermal states (Yip et al., 19 Mar 2025).