A path towards quantum advantage in training deep generative models with quantum annealers

The development of quantum-classical hybrid (QCH) algorithms is critical to achieve state-of-the-art computational models. A QCH variational autoencoder (QVAE) was introduced in Ref. [1] by some of the authors of this paper. QVAE consists of a classical auto-encoding structure realized by traditional deep neural networks to perform inference to, and generation from, a discrete latent space. The latent generative process is formalized as thermal sampling from either a quantum or classical Boltzmann machine (QBM or BM). This setup allows quantum-assisted training of deep generative models by physically simulating the generative process with quantum annealers. In this paper, we have successfully employed D-Wave quantum annealers as Boltzmann samplers to perform quantum-assisted, end-to-end training of QVAE. The hybrid structure of QVAE allows us to deploy current-generation quantum annealers in QCH generative models to achieve competitive performance on datasets such as MNIST. The results presented in this paper suggest that commercially available quantum annealers can be deployed, in conjunction with well-crafted classical deep neutral networks, to achieve competitive results in unsupervised and semisupervised tasks on large-scale datasets. We also provide evidence that our setup is able to exploit large latent-space (Q)BMs, which develop slowly mixing modes. This expressive latent space results in slow and inefficient classical sampling, and paves the way to achieve quantum advantage with quantum annealing in realistic sampling applications.

[1]  Jacob biamonte,et al.  Quantum machine learning , 2016, Nature.

[2]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[3]  Chong Wang,et al.  Stochastic variational inference , 2012, J. Mach. Learn. Res..

[4]  Daniel A. Lidar,et al.  Solving a Higgs optimization problem with quantum annealing for machine learning , 2017, Nature.

[5]  Rob Thew,et al.  Quantum Science and Technology—one year on , 2018 .

[6]  Daniel A. Lidar,et al.  Adiabaticity in open quantum systems , 2007, 1508.05558.

[7]  M. W. Johnson,et al.  Quantum annealing with manufactured spins , 2011, Nature.

[8]  Mark W. Johnson,et al.  Observation of topological phenomena in a programmable lattice of 1,800 qubits , 2018, Nature.

[9]  R. Car,et al.  Theory of Quantum Annealing of an Ising Spin Glass , 2002, Science.

[10]  Xu Li-ping,et al.  Quantum Information Processing in Quantum Wires , 2004 .

[12]  M. Benedetti,et al.  Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning , 2015, 1510.07611.

[13]  Andrew M. Childs,et al.  Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision , 2015, SIAM J. Comput..

[14]  Eleanor G. Rieffel,et al.  Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers , 2017, 1703.03902.

[15]  Peter Wittek,et al.  Quantum Machine Learning: What Quantum Computing Means to Data Mining , 2014 .

[16]  Alán Aspuru-Guzik,et al.  A variational eigenvalue solver on a photonic quantum processor , 2013, Nature Communications.

[17]  Vicky Choi,et al.  Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design , 2010, Quantum Inf. Process..

[18]  S. Lloyd,et al.  Quantum principal component analysis , 2013, Nature Physics.

[19]  Rupak Biswas,et al.  Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers , 2017, Quantum Science and Technology.

[20]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[21]  H Neven,et al.  A blueprint for demonstrating quantum supremacy with superconducting qubits , 2017, Science.

[22]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[23]  Geoffrey E. Hinton,et al.  A Learning Algorithm for Boltzmann Machines , 1985, Cogn. Sci..

[24]  M. W. Johnson,et al.  Phase transitions in a programmable quantum spin glass simulator , 2018, Science.

[25]  M. Amin Searching for quantum speedup in quasistatic quantum annealers , 2015, 1503.04216.

[26]  I. Hen,et al.  Temperature Scaling Law for Quantum Annealing Optimizers. , 2017, Physical review letters.

[27]  H. Nishimori,et al.  Quantum annealing in the transverse Ising model , 1998, cond-mat/9804280.

[28]  Daniel A. Lidar,et al.  Quantum adiabatic machine learning , 2011, Quantum Inf. Process..

[29]  Akihisa Tomita,et al.  Quantum information processing with fiber optics: Quantum Fourier transform of 1024 qubits , 2005 .

[30]  Hailin Chen,et al.  A Semi-Supervised Method for Drug-Target Interaction Prediction with Consistency in Networks , 2013, PloS one.

[31]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[32]  F ROSENBLATT,et al.  The perceptron: a probabilistic model for information storage and organization in the brain. , 1958, Psychological review.

[33]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[34]  E. Rieffel,et al.  Power of Pausing: Advancing Understanding of Thermalization in Experimental Quantum Annealers , 2018, Physical Review Applied.

[35]  Roger Melko,et al.  Quantum Boltzmann Machine , 2016, 1601.02036.

[36]  Rupak Biswas,et al.  Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models , 2016, 1609.02542.

[37]  Seth Lloyd,et al.  Quantum algorithm for data fitting. , 2012, Physical review letters.

[38]  F. Petruccione,et al.  An introduction to quantum machine learning , 2014, Contemporary Physics.

[39]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[40]  Rocco A. Servedio,et al.  Random classification noise defeats all convex potential boosters , 2008, ICML '08.

[41]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[42]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[43]  M. Paternostro,et al.  Quantum-information processing with noisy cluster states (12 pages) , 2005 .

[44]  Yee Whye Teh,et al.  A Fast Learning Algorithm for Deep Belief Nets , 2006, Neural Computation.

[45]  Jack Raymond,et al.  Global Warming: Temperature Estimation in Annealers , 2016, Front. ICT.

[46]  G. Aeppli,et al.  Tunable quantum tunnelling of magnetic domain walls , 2001, Nature.

[47]  Todd A. Brun,et al.  Quantum Error Correction , 2019, Oxford Research Encyclopedia of Physics.

[48]  N. Zheludev,et al.  PHYSICAL REVIEW APPLIED 11 , 064016 ( 2019 ) Far-Field Superoscillatory Metamaterial Superlens , 2019 .

[49]  Daniel A. Lidar,et al.  Scalable effective temperature reduction for quantum annealers via nested quantum annealing correction , 2017, 1710.07871.

[50]  Bing Zhang,et al.  Semi-supervised learning improves gene expression-based prediction of cancer recurrence , 2011, Bioinform..

[51]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[52]  Geoffrey E. Hinton,et al.  The "wake-sleep" algorithm for unsupervised neural networks. , 1995, Science.

[53]  Ronald J. Williams,et al.  Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning , 2004, Machine Learning.

[54]  Vicky Choi,et al.  Minor-embedding in adiabatic quantum computation: I. The parameter setting problem , 2008, Quantum Inf. Process..