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International Journal of Quantum Information

A complete restricted Boltzmann machine on an adiabatic quantum computer

Lorenzo Rocutto, Enrico Prati

Published Jun 1, 2021• 13 citations
Publisher

Abstract

Boltzmann Machines constitute a paramount class of neural networks for unsupervised learning and recommendation systems. Their bipartite version, called Restricted Boltzmann Machine (RBM), is the most developed because of its satisfactory trade-off between computability on classical computers and computational power. Though the diffusion of RBMs is quite limited as their training remains hard. Recently, a renewed interest has emerged as Adiabatic Quantum Computers (AQCs), which suggest a potential increase of the training speed with respect to conventional hardware. Due to the limited number of connections among the qubits forming the graph of existing hardware, associating one qubit per node of the neural network implies an incomplete graph. Thanks to embedding techniques, we developed a complete graph connecting nodes constituted by virtual qubits. The complete graph outperforms previous implementations based on incomplete graphs. Despite the fact that the learning rate per epoch is still slower with respect to a classical machine, the advantage is expected by the increase of number of nodes which impacts on the classical computational time but not on the quantum hardware based computation.

DOI
10.1142/s0219749921410033

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Boltzmann Machines constitute a paramount class of neural networks for unsupervised learning and recommendation systems. Their bipartite version, called Restricted Boltzmann Machine (RBM), is the most developed because of its satisfactory trade-off between computability on classical computers and computational power. Though the diffusion of RBMs is quite limited as their training remains hard. Recently, a renewed interest has emerged as Adiabatic Quantum Computers (AQCs), which suggest a potential increase of the training speed with respect to conventional hardware. Due to the limited number of connections among the qubits forming the graph of existing hardware, associating one qubit per node of the neural network implies an incomplete graph. Thanks to embedding techniques, we developed a complete graph connecting nodes constituted by virtual qubits. The complete graph outperforms previous implementations based on incomplete graphs. Despite the fact that the learning rate per epoch is still slower with respect to a classical machine, the advantage is expected by the increase of number of nodes which impacts on the classical computational time but not on the quantum hardware based computation.
Abstract Restricted Boltzmann machines (RBMs) constitute a class of neural networks for unsupervised learning with applications ranging from pattern classification to quantum state reconstruction. Despite the potential representative power, the diffusion of RBMs is quite limited since their training process proves to be hard. The advent of commercial adiabatic quantum computers (AQCs) raised the expectation that the implementations of RBMs on such quantum devices can increase the training speed with respect to conventional hardware. Here, the feasibility of a complete RBM on AQCs is demonstrated, thanks to an embedding that associates the nodes of the neural networks to virtual qubits. A semantic quantum search is implemented thanks to a reverse annealing schedule. Such an approach exploits more information from the training data, mimicking the behavior of the classical Gibbs sampling algorithm. The semantic training is shown to quickly raise the sampling probability of a subset of the set of the configurations. Even without a proper optimization of the annealing schedule, the RBM semantically trained achieves good scores on reconstruction tasks. The development of such techniques paves the way toward the establishment of a quantum advantage of adiabatic quantum computers, especially given the foreseen improvement of such hardware.
Abstract Quantum machine learning promises to revolutionize traditional machine learning by efficiently addressing hard tasks for classical computation. While claims of quantum speed-up have been announced for gate-based quantum computers and photon-based boson samplers, demonstration of an advantage by adiabatic quantum annealers (AQAs) is open. Here we quantify the computational cost and the performance of restricted Boltzmann machines (RBMs), a widely investigated machine learning model, by classical and quantum annealing. Despite the lower computational complexity of the quantum RBM being lost due to physical implementation overheads, a quantum speed-up may arise as a reduction by orders of magnitude of the computational time. By employing real-world cybersecurity datasets, we observe that the negative phase on sufficiently challenging tasks is computed up to 64 times faster by AQAs during the exploitation phase. Therefore, although a quantum speed-up highly depends on the problem’s characteristics, it emerges in existing hardware on real-world data.
Restricted Boltzmann machines are common machine learning models that can utilize quantum annealing devices in their training processes as quantum samplers. While this approach has shown promise as an alternative to classical sampling methods, the limitations of quantum annealing hardware, such as the number of qubits and the lack of connectivity between the qubits, still pose a barrier to wide-scale adoption. We propose the use of multiple software techniques such as dropout method, passive labeling, and parallelization techniques for addressing these hardware limitations. The study found that using these techniques along with quantum sampling showed comparable results to its classical counterparts in certain contexts, while in others the increased complexity of the sampling process hindered the performance of the trained models. This means that further research into the behavior of quantum sampling needs to be done to apply quantum annealing to training tasks of more complicated RBM models.
Abstract Despite the anticipated speed‐up of quantum computing, the achievement of a measurable advantage remains subject to ongoing debate. Adiabatic Quantum Computers (AQCs) are quantum devices designed to solve quadratic uncostrained binary optimization (QUBO) problems, but their intrinsic thermal noise can be leveraged to train computationally demanding machine learning algorithms such as the Boltzmann Machine (BM). Despite an asymptotic advantage is expected only for large networks, a limited quantum speed up can be already achieved on a small BM is shown, by exploiting parallel adiabatic computation. This approach exhibits a 8.6‐fold improvement in wall time on the Bars and Stripes dataset when compared to a parallelized classical Gibbs sampling method, which has never been outperformed before by quantum approaches.
We demonstrate how to leverage thermal noise within an Adiabatic Quantum Computer (AQC) to train a complete Boltzmann Machine (BM). In contrast to previous implementations, our fully-connected BM includes hidden units and we sample both negative and positive statistics from the AQC. By training the network on a standard benchmark Bars and Stripes dataset, the maximally parallelized classical algorithm takes 11.4 times longer to reach the same likelihood as the quantum algorithm, indicating that AQCs may expedite BM training. The ability of AQCs to sample Boltzmann-distributed configurations quickly has the potential to impact BMs training as well as influence the field of stochastic machine learning.
Restricted Boltzmann Machines (RBMs) are widely used probabilistic undirected graphical models with visible and latent nodes, playing an important role in statistics and machine learning. The task of structure learning for RBMs involves inferring the underlying graph by using samples from the visible nodes. Specifically, learning the two-hop neighbors of each visible node allows for the inference of the graph structure. Prior research has addressed the structure learning problem for specific classes of RBMs, namely ferromagnetic and locally consistent RBMs. In this paper, we extend the scope to the quantum computing domain and propose corresponding quantum algorithms for this problem. Our study demonstrates that the proposed quantum algorithms yield a polynomial speedup compared to the classical algorithms for learning the structure of these two classes of RBMs.