Quantum Boltzmann Machines (QBMs) are a natural adaption of BMs to the quantum computing framework. Instead of an energy function with nodes being represented by binary spin values, QBMs define the underlying network using a Hermitian operator, a parameterized Hamiltonian $$ \begin{equation*} H_{\theta}=\sum_{i=0}^{p-1}\theta_ih_i, \end{equation*} $$ with \( \theta\in\mathbb{R}^p \) and \( h_i=\bigotimes_{j=0}^{n-1}\sigma_{j, i} \) for \( \sigma_{j, i} \) acting on the \( j^{\text{th}} \) qubit.
The network nodes are characterized by the Pauli matrices \( \sigma_{j, i} \).
It should be noted that the qubits which determine the model output are referred to as visible and those which act as latent variables as hidden qubits. The aim of the model is to learn Hamiltonian parameters such that the resulting Gibbs state reflects a given target system. In contrast to Boltzmann Machines, this framework allows the use of quantum structures which are potentially inaccessible classically. Equivalently to the classical model, QBMs are suitable for discriminative as well as generative learning.