$κ$-Parametrized Boltzmann Distribution Sampling on Quantum Approximate Optimization Algorithm

Abstract

Generative models use training data to produce new samples that resemble the original, with applications in classification and recommendation systems. Restricted Boltzmann Machines (RBMs) are at the core of generative models but suffer from the computational intractability of training, which relies on sampling an unknown Boltzmann Distribution (BD). The Quantum Approximate Optimization Algorithm (QAOA) can assist RBM training, potentially providing benefits over strictly classical strategies. One particular problem is determining the quantum circuit depth for an arbitrary problem size that yields the expected sampling results. In this study, we evaluate QAOA-assisted sampling of BDs using Qiskit's Aer Simulator by analysing the inverse effective temperature of the distribution, $κ$, for tractable problem sizes. The results provide numerical evidence, based on Qiskit Aer simulations, that parameterized annealing-scheduled QAOA (PAQ-QAOA) can approximate RBM Boltzmann sampling for the tractable instances considered, while hardware performance and scaling remain open questions.

The paper asks whether annealing-scheduled QAOA can approximate the joint Boltzmann distribution of a restricted Boltzmann machine (RBM), and whether an effective inverse temperature $\kappa$ can identify circuit parameters yielding the target distribution. It uses a transverse-field mixer and an RBM-derived Ising cost Hamiltonian, with fixed linear-schedule angles $\beta_k=(1-k/p)\Delta t$ and $\gamma_k=(k/p)\Delta t$, rather than variational angle optimization. Qiskit Aer simulations examine one randomly initialized RBM instance at each of $n=6$ and $n=19$ qubits, with $p=10$ layers. For each instance, $\Delta t$ is swept over 1,000 uniformly spaced points in $[0,\pi]$, using $10^6$ shots per point. Sampled distributions are compared with exactly computed target probabilities through KL divergence. An effective temperature is fitted by maximum likelihood to $P_\kappa(x)\propto e^{-\kappa E(x)}$, equivalently matching the sampled mean energy. The simulations distinguish approximately uniform, intermediate non-adiabatic, and ground-state-concentrated sampling regimes. At six qubits, sampled KL-divergence values closely follow the thermal-model prediction, with the minimum near $\kappa=1$. At 19 qubits, they depart from that prediction and the minimum shifts away from unity. The study does not establish a predictive relationship between system size, depth, time step, and $\kappa$; choosing sampling parameters remains unresolved. Reported evidence is limited to tractable simulator instances, with one instance per size. Hardware performance, scaling, shot requirements, and the origin of the larger-instance deviations remain open.