Realizing Quantum Boltzmann Machines Through Eigenstate Thermalization

TL;DR

This work introduces an ETH-based framework to train quantum Boltzmann machines by sampling local observables through quantum quenches, avoiding full quantum thermal state preparation. By coupling a small thermometer system, it estimates the effective temperature and enables gradient estimation for QBMs on NISQ devices. Numerical results show local thermalization of gradient observables, training performance competitive with exact QBMs, and robustness to noise, with RBMs generally underperforming in the tested regimes. The approach has potential to enhance variational quantum algorithms and offers a path toward practical QBM-based generative modeling on near-term hardware.

Abstract

Quantum Boltzmann machines are natural quantum generalizations of Boltzmann machines that are expected to be more expressive than their classical counterparts, as evidenced both numerically for small systems and asymptotically under various complexity theoretic assumptions. However, training quantum Boltzmann machines using gradient-based methods requires sampling observables in quantum thermal distributions, a problem that is NP-hard. In this work, we find that the locality of the gradient observables gives rise to an efficient sampling method based on the Eigenstate Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient method for training quantum Boltzmann machines on near-term quantum devices. Furthermore, under realistic assumptions on the moments of the data distribution to be modeled, the distribution sampled using our algorithm is approximately the same as that of an ideal quantum Boltzmann machine. We demonstrate numerically that under the proposed training scheme, quantum Boltzmann machines capture multimodal Bernoulli distributions better than classical restricted Boltzmann machines with the same connectivity structure. We also provide numerical results on the robustness of our training scheme with respect to noise.

Figures (11)

• Figure 1: (a) An example Boltzmann machine. The units are coupled with interaction weights $w_{ij}$. Each unit also has a local bias field $b_i$. (b) An example RBM. Samples are drawn from the visible units, and correlations between visible units are created through couplings with the hidden layer. The visible units are coupled with the hidden units through interaction weights $w_{\upsilon\eta}$. Each unit also has a local bias field $b_i$. (c) An example QBM with a semi-restricted architecture. The units are coupled with interaction weights $w_{ij}$. Each unit also has a local bias field $b_i$. Furthermore, off-diagonal fields and interactions are included in the Hamiltonian (see Sec. \ref{['sec:bm']}).
• Figure 2: An example QBM/ thermometer combination. The thermometer is weakly coupled to the QBM such that temperature measurements of the thermometer approximately agree with those of the entire system (see Sec. \ref{['sec:thermal']}).
• Figure 3: A typical fit of the Berry--Robnik distribution to the energy level spacing distribution of our trained QBM/ thermometer combination. The trained model was a restricted transverse Ising model with $\frac{\overline{\varGamma}}{\sqrt{w_{\textrm{int}}^2}}=1$ (see Sec. \ref{['sec:ergodicity']}), six visible units, one hidden unit, and two thermometer units.
• Figure 4: The Berry--Robnik interpolation parameter $\rho$ plotted as a function of the normalized mean single-site transverse field $\frac{\overline{\varGamma}}{\sqrt{w_{\textrm{int}}^2}}$ (see Sec. \ref{['sec:ergodicity']}). The trained models were restricted transverse Ising models with six visible units, one hidden unit, and two thermometer units. Error bars denote one standard error over five instances.
• Figure 5: The median error in gradient observable using our QBM/ thermometer scheme for multiple Hamiltonian models as a function of the number of visible units of the system (see Sec. \ref{['sec:ergodicity']}). The lack of a convergence in the thermodynamic limit for the semi-restricted transverse Ising model is most likely due to a nonvanishing variance of the energy expectation value of the system in the thermodynamic limit (see Appendix \ref{['sec:qbm_quench_therm']}). The studied models each had one hidden unit and two thermometer units; for greater detail on the studied systems, see Appendix \ref{['sec:systems']}. Error bars denote one standard error over five instances and over all gradient observables.