Statistical limits of supervised quantum learning

Within the framework of statistical learning theory it is possible to bound the minimum number of samples required by a learner to reach a target accuracy. We show that if the bound on the accuracy is taken into account, quantum machine learning algorithms for supervised learning---for which statistical guarantees are available---cannot achieve polylogarithmic runtimes in the input dimension. We conclude that, when no further assumptions on the problem are made, quantum machine learning algorithms for supervised learning can have at most polynomial speedups over efficient classical algorithms, even in cases where quantum access to the data is naturally available.

[1]  David Haussler,et al.  Learnability and the Vapnik-Chervonenkis dimension , 1989, JACM.

[2]  David I. Kaiser,et al.  Exploiting Faraday rotation to jam quantum key distribution via polarized photons , 2019, Quantum Inf. Comput..

[3]  David P. Woodruff,et al.  Faster Kernel Ridge Regression Using Sketching and Preconditioning , 2016, SIAM J. Matrix Anal. Appl..

[4]  Lorenzo Rosasco,et al.  On regularization algorithms in learning theory , 2007, J. Complex..

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

[6]  Masoud Mohseni,et al.  Quantum support vector machine for big feature and big data classification , 2013, Physical review letters.

[7]  S. Lloyd,et al.  Quantum-Enhanced Measurements: Beating the Standard Quantum Limit , 2004, Science.

[8]  Martin J. Wainwright,et al.  Randomized sketches for kernels: Fast and optimal non-parametric regression , 2015, ArXiv.

[9]  Guoming Wang.  Quantum Algorithm for Linear Regression , 2017 .

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

[11]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[12]  Michael J. McCourt,et al.  Stable Evaluation of Gaussian Radial Basis Function Interpolants , 2012, SIAM J. Sci. Comput..

[13]  S. Aaronson.  Read the fine print , 2015, Nature Physics.

[14]  S. Lloyd,et al.  Quantum metrology. , 2005, Physical review letters.

[15]  Iordanis Kerenidis,et al.  Learning with Errors is easy with quantum samples , 2017, Physical Review A.

[16]  M. Schuld,et al.  Prediction by linear regression on a quantum computer , 2016, 1601.07823.

[17]  Simone Severini,et al.  Quantum machine learning: a classical perspective , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[19]  Felipe Cucker,et al.  On the mathematical foundations of learning , 2001 .

[20]  Radha Pyari Sandhir,et al.  State-independent quantum key distribution with two-way classical communication , 2019, Quantum Inf. Comput..

[21]  A. Prakash,et al.  Quantum gradient descent for linear systems and least squares , 2017, Physical Review A.

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

[23]  Maris Ozols,et al.  Hamiltonian simulation with optimal sample complexity , 2016, npj Quantum Information.