Tomography and generative training with quantum Boltzmann machines

Abstract

The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide new methods of training quantum Boltzmann machines, which are a class of recurrent quantum neural network. Our work generalizes existing methods and provides new approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of quantum state tomography that not only estimates a state but provides a prescription for generating copies of the reconstructed state. Classical Boltzmann machines are incapable of this. Finally we compare small non-stoquastic quantum Boltzmann machines to traditional Boltzmann machines for generative tasks and observe evidence that quantum models outperform their classical counterparts.Introduction-The Boltzmann machine is a widely used type of recurrent neural net that, unlike the feed forward neural nets used in many applications, is capable of generating new examples of the training data [1]. This makes it an excellent model to use in cases where data is missing. We focus on Boltzmann machines because, of all neural net models, the Boltzmann machine is perhaps the most natural one for physicists. It models the input data as if it came from an Ising model in thermal equilibrium. The goal of training is then to find the Ising model that is most likely to reproduce the input data which is known as a training set.The close analogy between this model and physics has made it a natural fit for quantum computing and quantum annealing. A number of proposals have been put forward for accelerating Boltzmann machines in current generation quantum annealers [2-4] and quantum computers [5], the latter showing polynomial speedups relative to classical training [6]. While these methods showed that quantum technologies can train Boltzmann machines more accurately and at lower cost than classical methods, the question of whether transitioning from an Ising model to a quantum model for the data would provide substantial improvements.This question is addressed in [7], wherein a new method for training Boltzmann machines is provided that uses transverse Ising models in thermal equilibrium to model the data. While such models are trainable and can outperform classical Boltzmann machines, the training procedure proposed therein suffers two drawbacks. First, it is unable to learn quantum terms from classical data. Second, the transverse Ising models considered are widely believed to be simulatable using quantum Monte-Carlo methods. This means that such models are arguably not quantum and as such the benchmarks they give do not necessarily apply to manifestly quantum models. Here we rectify these issues by giving new training methods that do not suffer these drawbacks and illustrate their performance for models that are manifestly quantum.The first, and arguably most important, task when approach...

Quantum Boltzmann Machine

“…For classical data sets training a model with hidden variables, the objective function of Ref. [43] involves a computationally difficult trace, in contrast to our bound [45]. Thus, the two approaches offer complementary strategies for the unsupervised learning of classical and quantum data distributions using quantum Boltzmann machines.…”

“…Whereas the present paper focuses only on learning classical probability distributions, Ref. [43] provides a strategy to also learn a quantum distribution, including the ability to train the transverse field. In the case of classical data sets, when the model is fully visible, the objective function is exactly the same as the bound introduced in this paper [44].…”

“…Recently, Kieferova and Wiebe [43] proposed the relative entropy between the density matrix of the data and that of the quantum system as an objective function for training a QBM. Whereas the present paper focuses only on learning classical probability distributions, Ref.…”

Adversarial quantum circuit learning for pure state approximation

“…For example, the quantum principal component analysis [2] requires universal fault-tolerant hardware in order to implement the necessary SWAP operations. As another example, the quantum Boltzmann machine [3,4] requires the preparation of highly non-trivial thermal states. Moreover, those approaches provide limited control over the level of approximation.…”

“…This process of approximately reconstructing a quantum state is already known to physicists under the name of quantum state tomography. Indeed, there already exist proposals of generative models for tomography such as the quantum principal component analysis [2] and the quantum Boltzmann machine [3,4]. Other machine learning approaches for tomography have been formulated using the different framework of probably approximately correct learning [5,6].…”