Robustness Verification of Quantum Classifiers

Several important models of machine learning algorithms have been successfully generalized to the quantum world, with potential speedup to training classical classifiers and applications to data analytics in quantum physics that can be implemented on the near future quantum computers. However, quantum noise is a major obstacle to the practical implementation of quantum machine learning. In this work, we define a formal framework for the robustness verification and analysis of quantum machine learning algorithms against noises. A robust bound is derived and an algorithm is developed to check whether or not a quantum machine learning algorithm is robust with respect to quantum training data. In particular, this algorithm can find adversarial examples during checking. Our approach is implemented on Google’s TensorFlow Quantum and can verify the robustness of quantum machine learning algorithms with respect to a small disturbance of noises, derived from the surrounding environment. The effectiveness of our robust bound and algorithm is confirmed by the experimental results, including quantum bits classification as the “Hello World” example, quantum phase recognition and cluster excitation detection from real world intractable physical problems, and the classification of MNIST from the classical world.

[1]  Russ Tedrake,et al.  Evaluating Robustness of Neural Networks with Mixed Integer Programming , 2017, ICLR.

[2]  Mykel J. Kochenderfer,et al.  Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks , 2017, CAV.

[3]  Sanjit A. Seshia,et al.  Formal Analysis and Redesign of a Neural Network-Based Aircraft Taxiing System with VerifAI , 2020, CAV.

[4]  Marta Z. Kwiatkowska.  Safety Verification for Deep Neural Networks with Provable Guarantees (Invited Paper) , 2019, CONCUR.

[5]  N. Vitanov,et al.  Smooth composite pulses for high-fidelity quantum information processing , 2011 .

[6]  Soonwon Choi,et al.  Quantum convolutional neural networks , 2018, Nature Physics.

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

[8]  Weiming Xiang,et al.  Verification of Deep Convolutional Neural Networks Using ImageStars , 2020, CAV.

[9]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information: Frontmatter , 2010 .

[10]  Carl W. Helstrom,et al.  Detection Theory and Quantum Mechanics , 1967, Inf. Control..

[11]  Masakazu Kojima,et al.  Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations , 2003, Comput. Optim. Appl..

[12]  Lujo Bauer,et al.  On the Suitability of Lp-Norms for Creating and Preventing Adversarial Examples , 2018, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW).

[13]  David Wagner,et al.  Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods , 2017, AISec@CCS.

[14]  Antonio Criminisi,et al.  Measuring Neural Net Robustness with Constraints , 2016, NIPS.

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

[16]  Justin Emile Gottschlich,et al.  An Abstraction-Based Framework for Neural Network Verification , 2019, CAV.

[17]  D. M. Lucas,et al.  Scalable simultaneous multiqubit readout with 99.99% single-shot fidelity , 2009, 0906.3304.

[18]  John Preskill,et al.  Information-theoretic bounds on quantum advantage in machine learning , 2021, Physical review letters.

[19]  Stephen P. Boyd,et al.  CVXPY: A Python-Embedded Modeling Language for Convex Optimization , 2016, J. Mach. Learn. Res..

[20]  A. Zagoskin,et al.  Emergence and control of complex behaviors in driven systems of interacting qubits with dissipation , 2020, npj Quantum Information.

[21]  Javad Lavaei,et al.  Sparse Semidefinite Programs with Near-Linear Time Complexity , 2017, 2018 IEEE Conference on Decision and Control (CDC).

[22]  Sirui Lu,et al.  Quantum Adversarial Machine Learning , 2020, ArXiv.

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

[24]  R. Bertlmann,et al.  Bloch vectors for qudits , 2008, 0806.1174.

[25]  Jonathon Shlens,et al.  Explaining and Harnessing Adversarial Examples , 2014, ICLR.

[26]  Hans-J. Briegel,et al.  Machine learning \& artificial intelligence in the quantum domain , 2017, ArXiv.

[27]  K. Audenaert,et al.  Discriminating States: the quantum Chernoff bound. , 2006, Physical review letters.

[28]  Joongheon Kim,et al.  A Tutorial on Quantum Convolutional Neural Networks (QCNN) , 2020, 2020 International Conference on Information and Communication Technology Convergence (ICTC).

[29]  Alessio Lomuscio,et al.  An approach to reachability analysis for feed-forward ReLU neural networks , 2017, ArXiv.

[30]  Robin Blume-Kohout.  How distinguishable are two quantum processes? a.k.a. “What is the error rate of a quantum gate?” , 2017 .

[31]  Ievgeniia Oshurko.  Quantum Machine Learning , 2020, Quantum Computing.

[32]  J. Doug Tygar,et al.  Adversarial machine learning , 2019, AISec '11.

[33]  Hugo Ribeiro,et al.  Engineering fast high-fidelity quantum operations with constrained interactions , 2020, 2003.12096.

[34]  Daniel Kroening,et al.  Global Robustness Evaluation of Deep Neural Networks with Provable Guarantees for the Hamming Distance , 2019, IJCAI.

[35]  David Von Dollen,et al.  TensorFlow Quantum: A Software Framework for Quantum Machine Learning , 2020, ArXiv.

[36]  P. Wittek,et al.  Vulnerability of quantum classification to adversarial perturbations , 2019, Physical Review A.

[37]  Sanjit A. Seshia,et al.  VerifAI: A Toolkit for the Formal Design and Analysis of Artificial Intelligence-Based Systems , 2019, CAV.

[38]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[39]  D M Lucas,et al.  High-fidelity readout of trapped-ion qubits. , 2008, Physical review letters.

[40]  Dacheng Tao,et al.  Quantum noise protects quantum classifiers against adversaries , 2020, Physical Review Research.

[41]  Nana Liu,et al.  Optimal provable robustness of quantum classification via quantum hypothesis testing , 2021, npj Quantum Information.

[42]  Hartmut Neven,et al.  Classification with Quantum Neural Networks on Near Term Processors , 2018, 1802.06002.

[43]  John Watrous,et al.  Semidefinite Programs for Completely Bounded Norms , 2009, Theory Comput..

[44]  Weiming Xiang,et al.  NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems , 2020, CAV.

[45]  Aleksander Madry,et al.  Towards Deep Learning Models Resistant to Adversarial Attacks , 2017, ICLR.