Comparison of Use of a 2000 Qubit D-Wave Quantum Annealer and MCMC for Sampling, Image Reconstruction, and Classification

The next generation D-Wave quantum computer (QC) having more than 2000 qubits was demonstrated to be adequate for embedding graphs of restricted Boltzmann machines (RBMs) having the connectivity between RBM units (though still limited) already sufficient for successful training by classical algorithms. As a verification of the validity and quality of the employed embedding approach, the RBM was trained by classical contrastive divergence (CD), while the D-Wave was initially used only in the recognition step for reconstruction of incomplete test images (8 × 8 bars-and-stripes), and for their classification. The label classification errors by the D-Wave compared favorably to those obtained by classical Gibbs sampling, indicating that the QC was successfully finding the most energetically favorable combinations of the unknown visible RBM units/labels for the given fixed incomplete input image. The valid RBM embedding was further applied to investigate opportunities for using QCs in the RBM training, and specifically for replacing the classical Gibbs technique in generating a representative sample from the RBM model distribution. Statistical comparison of samples obtained by the Gibbs technique versus 10000 sample states generated from the D-Wave revealed significant differences in the observed outcomes. The D-Wave samples were found insufficiently representative of the model distribution, specifically by missing many local valley found by the Gibbs sampling. On the other hand, the D-Wave possessed an ability to find local valleys of the configuration space that were consistently missed by the classical sampling algorithms, which could potentially be very interesting for sampling applications.

[1]  M. Sipser,et al.  Quantum Computation by Adiabatic Evolution , 2000, quant-ph/0001106.

[2]  Fabián A. Chudak,et al.  The Ising model : teaching an old problem new tricks , 2010 .

[3]  Kristel Michielsen,et al.  Spanning Tree Calculations on D-Wave 2 Machines , 2016 .

[4]  Yuhong Yang,et al.  Information Theory, Inference, and Learning Algorithms , 2005 .

[5]  Geoffrey E. Hinton,et al.  Learning and relearning in Boltzmann machines , 1986 .

[6]  Rupak Biswas,et al.  Determination and correction of persistent biases in quantum annealers , 2015, Scientific Reports.

[7]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[8]  Andrew Lucas,et al.  Ising formulations of many NP problems , 2013, Front. Physics.

[9]  Paul Smolensky,et al.  Information processing in dynamical systems: foundations of harmony theory , 1986 .

[10]  Geoffrey E. Hinton,et al.  An Efficient Learning Procedure for Deep Boltzmann Machines , 2012, Neural Computation.

[11]  Daniel A. Lidar,et al.  Evidence for quantum annealing with more than one hundred qubits , 2013, Nature Physics.

[12]  Yaroslav Koshka,et al.  Determination of the Lowest-Energy States for the Model Distribution of Trained Restricted Boltzmann Machines Using a 1000 Qubit D-Wave 2X Quantum Computer , 2017, Neural Computation.

[13]  Christoph Koch,et al.  Multiple Query Optimization on the D-Wave 2X Adiabatic Quantum Computer , 2015, Proc. VLDB Endow..

[14]  Geoffrey E. Hinton,et al.  Deep Boltzmann Machines , 2009, AISTATS.

[15]  Rupak Biswas,et al.  Estimation of effective temperatures in a quantum annealer and its impact in sampling applications: A case study towards deep learning applications , 2015 .

[16]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[17]  Yaroslav Koshka,et al.  Empirical investigation of the low temperature energy function of the Restricted Boltzmann Machine using a 1000 qubit D-Wave 2X , 2016, 2016 International Joint Conference on Neural Networks (IJCNN).

[18]  Steven H. Adachi,et al.  Application of Quantum Annealing to Training of Deep Neural Networks , 2015, ArXiv.

[19]  Christian Igel,et al.  Training restricted Boltzmann machines: An introduction , 2014, Pattern Recognit..

[20]  Yoshua Bengio,et al.  On the Challenges of Physical Implementations of RBMs , 2013, AAAI.