Bayesian deep learning on a quantum computer
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Bayesian deep learning on a quantum computer Zhikuan Zhao1,2,3 · Alejandro Pozas-Kerstjens4
· Patrick Rebentrost3 · Peter Wittek5,6,7,8
Received: 23 November 2018 / Accepted: 13 March 2019 / Published online: 15 May 2019 © Springer Nature Switzerland AG 2019
Abstract Bayesian methods in machine learning, such as Gaussian processes, have great advantages compared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected deep feedforward neural networks with Gaussian processes, allowing training without backpropagation. This connection enables us to leverage a quantum algorithm designed for Gaussian processes and develop a new algorithm for Bayesian deep learning on quantum computers. The properties of the kernel matrix in the Gaussian process ensure the efficient execution of the core component of the protocol, quantum matrix inversion, providing at least a polynomial speedup over classical algorithms. Furthermore, we demonstrate the execution of the algorithm on contemporary quantum computers and analyze its robustness with respect to realistic noise models. Keywords Bayesian methods · Quantum computing · Quantum algorithms · Quantum-enhanced AI · Experimental quantum computing
1 Introduction The Bayesian approach to machine learning provides a clear advantage over traditional techniques, namely, it provides information about the uncertainty in their predictions. But not only that, they have further advantages, including automated ways of learning structure and avoiding overfitting, Alejandro Pozas-Kerstjens
[email protected] 1
Department of Computer Science, ETH Zurich, Universit¨atstrasse 6, 8092 Z¨urich, Switzerland
2
Singapore University of Technology and Design, 8 Somapah Road, Singapore, 487372 Singapore
3
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117543 Singapore
4
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
5
Rotman School of Management, University of Toronto, M5S 3E6 Toronto, Canada
6
Creative Destruction Laboratory, M5S 3E6 Toronto, Canada
7
Vector Institute for Artificial Intelligence, M5G 1M1 Toronto, Canada
8
Perimeter Institute for Theoretical Physics, N2L 2Y5 Waterloo, Canada
a principled foundation (Ghahramani 2015), and robustness to adversarial attacks (Bradshaw et al. 2017; Grosse et al. 2017). The Bayesian framework has been making advances in various deep architectures (Blundell et al. 2015; Gal and Ghahramani 2016). Some recent advances made a connection between a quintessentially Bayesian model, Gaussian processes (GPs) (Rasmussen and Williams 2006), and deep feedforward neural networks (Lee et al. 2018; Matthews et al. 2018). Parallel to these developments, quantum technologies have been making advances in machine learning. A new breed of quantum neural
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