# Adiabatic Quantum Computation Applied to Deep Learning Networks

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## Abstract

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## 1. Introduction

#### 1.1. Boltzmann Machines

#### 1.2. Convolutional Neural Networks

#### 1.3. Spiking Neural Networks

#### 1.4. Challenges

## 2. Related Work

## 3. Approach and Data

#### 3.1. Adiabatic Quantum Computation

#### 3.2. The Superconducting Quantum Adiabatic Processor

#### 3.3. Implementing a Boltzmann Machine on D-Wave

## 4. Results

## 5. Alternative Approaches

#### 5.1. HPC

#### 5.2. Neuromorphic

## 6. Discussion

#### Future Work

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Related Works for High Performance Computing

#### Appendix A.2. Related Works for Neuromorphic Computing

## Appendix B.

#### Appendix B.1. Description of High Performance Computing

#### Appendix B.2. Description of Neuromorphic Computing

_{2}) [53], tantalum-oxide (TaO

_{2}) [54], and titanium-oxide (TiO

_{2}) [55]. All of these memristive material stacks consist of an oxide layer sandwiched between two metallic layers. Depending on the polarity and magnitude of an applied voltage bias, the oxide layer transitions between being less or more conductive, providing the switching characteristics desirable for representing synaptic weights.

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**Figure 1.**A Boltzmann machine is divided into a visible layer, representing the data input, and a hidden layer, which represents latent factors controlling the data distribution. This diagram shows the restricted Boltzmann machine, or RBM, in which intralayer connections are prohibited. Each connection between units is a separate weight parameter which is discovered through training.

**Figure 2.**A convolutional neural network is composed of a series of alternating convolutional and pooling layers. Each convolutional layer extracts features from its preceding layer to form feature maps. These feature maps are then down-sampled by a pooling layer to exploit data locality. A perceptron, a simple type of classification network, is placed as the last layer of the CNN.

**Figure 3.**The connectivity in a CNN is sparse relative to the previously shown BM model. Additionally, the set of weights is shared between units, unlike in BMs. In this illustration we symbolize this with the red, green, and blue connections to show that each unit in the convolutional layer applies the same operation to different segments of the input.

**Figure 4.**Our LBM model added connectivity between units in the hidden layer, shown in red. RBMs prohibit such intralayer connections because they add too much computational complexity for classical machines. We represented the hidden layer (outlined in blue) on the D-Wave device. The connections between hidden units were 4-by-4 bipartite due to the device’s physical topology constraints.

**Figure 5.**The hidden layer from Figure 4 is represented in one of D-Wave’s chimera cells here, with the cell’s bipartite connectivity made more obvious. The input/visible units of the LBM are left on a classical machine. Their contributions to the activity of the hidden units is reduced to an activity bias (represented with ± symbols) on those units. Figure 6 shows the overall chimera topology of the D-Wave device.

**Figure 6.**Chimera graphs are composed of 8-qubit cells featuring bipartite connectivity. Each cell’s partition is connected to another partition in the adjacent cells.

**Figure 7.**An initial experiment to demonstrate LBM utility. Reconstruction error (sum of squared error) of BMs trained on simulated data using no intralayer connections and using random intralayer connections with a small (0.0001) hidden-to-hidden weight learning rate. Here we show five RBMs (

**red**) and five LBMs (

**blue**), and the results suggest even just the presence of relatively static intralayer connections gives LBMs a performance advantage over RBMs. We obtained these results from the quantum annealing simulator provided by D-Wave.

**Figure 8.**Reconstruction error and classification rate over 25 training epochs using 6000 MNIST images for training and 6000 for testing. Reconstruction error decreases as classification rate rises, confirming that the RBM learns the MNIST data distribution.

**Figure 9.**RBM and LBM performance on the MNIST digit classification task. The LBM tends to label the digits slightly better and produces lower reconstruction error than the RBM.

**Figure 10.**Comparison of RBM against LBM trained on neutrino data using a software simulator. Weights are randomly initialized from a normal distribution. The change in learning rate at epoch 5 is due to a change in the momentum parameter in the algorithm that is designed to speed the rate of training. The graph shows the mean performance of five different RBMs and five different LBMs and suggests the mean reconstruction error of RBM and LBM are significantly different.

**Figure 11.**Another comparison of RBM against LBM run on neutrino data using D-Wave hardware. Both the RBM and LBM are initialized from the same pre-trained model. The pre-trained model is an RBM run for three epochs on a classical machine. The graph shows the mean performance of five different RBMs and five different LBMs, suggesting the performance difference between RBM and LBM persists on hardware.

**Figure 12.**A comparison of the platforms, deep learning approaches, contributions, and significance of the result from the MNIST experiment.

**Figure 13.**A proposed architecture that shows howthe three approaches, quantum, HPC, and neuromorphic can be used to improve a deep learning approach. Image data can be analyzed using an HPC rapidly derived CNN with the top layers using an LBM on a quantum computer. The top layers have fewer inputs, and require greater representational capabilities which both play to the strength and limitations of a quantum approach. The temporal aspect of the data can be analyzed using an SNN. Finally, the image and temporal models will be merged to provide a richer and we believe a more accurate model, with an aim to be deployed in very low power neuromorphic hardware.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, J.; Spedalieri, F.M.; Yao, K.-T.; Potok, T.E.; Schuman, C.; Young, S.; Patton, R.; Rose, G.S.; Chamka, G. Adiabatic Quantum Computation Applied to Deep Learning Networks. *Entropy* **2018**, *20*, 380.
https://doi.org/10.3390/e20050380

**AMA Style**

Liu J, Spedalieri FM, Yao K-T, Potok TE, Schuman C, Young S, Patton R, Rose GS, Chamka G. Adiabatic Quantum Computation Applied to Deep Learning Networks. *Entropy*. 2018; 20(5):380.
https://doi.org/10.3390/e20050380

**Chicago/Turabian Style**

Liu, Jeremy, Federico M. Spedalieri, Ke-Thia Yao, Thomas E. Potok, Catherine Schuman, Steven Young, Robert Patton, Garrett S. Rose, and Gangotree Chamka. 2018. "Adiabatic Quantum Computation Applied to Deep Learning Networks" *Entropy* 20, no. 5: 380.
https://doi.org/10.3390/e20050380