Inspired by the success of Boltzmann Machines based on classical Boltzmann
distribution, we propose a new machine learning approach based on quantum
Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the
non-commutative nature of quantum mechanics, the training process of the
Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the
problem by introducing bounds on the quantum probabilities. This allows us to
train the QBM efficiently by sampling. We show examples of QBM training with
and without the bound, using exact diagonalization, and compare the results
with classical Boltzmann training. We also discuss the possibility of using
quantum annealing processors like D-Wave for QBM training and application.

本文提出了一种新的基于量子玻尔兹曼分布的机器学习方法，即量子玻尔兹曼机。通过对量子概率引入边界，可以使该机器学习模型的训练高效进行，同时也探讨了使用 D-Wave 等量子退火处理器进行训练和应用的可能性。

量子玻尔兹曼机

Quantum Boltzmann Machine

We extend a quantum model in Hilbert space developed in Aerts (2007a) into a
quantum field theoric model in Fock space for the modeling of the combination
of concepts. Items and concepts are represented by vectors in Fock space and
membership weights of items are modeled by quantum probabilities. We apply this
theory to model the disjunction of concepts and show that the predictions of
our theory for the membership weights of items regarding the disjunction of
concepts match with great accuracy the complete set of results of an experiment
conducted by Hampton (1988b). It are the quantum effects of interference and
superposition of that are at the origin of the effects of overextension and
underextension observed by Hampton as deviations from a classical use of the
disjunction. It is essential for the perfect matches we obtain between the
predictions of the quantum field model and Hampton's experimental data that
items can be in superpositions of `different numbers states' which proves that
the genuine structure of quantum field theory is needed to match predictions
with experimental data.

这篇论文将 Aerts（2007a）在 Hilbert 空间中开发的量子模型扩展到 Fock 空间中的量子场理论模型，旨在建模概念的组合，并将概念表示为 Fock 空间中的向量。该理论的应用使得能够模拟概念的析取，结果表明该模型的预测与 Hampton（1988b）实验的结果完全吻合，其量子效应的干涉和叠加是 Hampton 观察到的过度和欠度现象的原因，为了解释实验数据，必须采用真正的量子场理论结构。