Quantum Machine Learning Matrix Product States

Matrix product states minimize bipartite correlations to compress the classical data representing quantum states. Matrix product state algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics. Matrix product states have a further range of applications in machine learning. Finding matrix product states is in general a computationally challenging task, a computational task which we show quantum computers can accelerate. We present a quantum algorithm which returns a classical description of a $k$-rank matrix product state approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires $O(n\cdot k^2)$ quantum gates, yielding sufficient conditions for our quantum variational algorithm to terminate in polynomial-time.

[1]  Guifre Vidal,et al.  Entanglement Renormalization: An Introduction , 2009, 0912.1651.

[2]  F. Verstraete,et al.  Renormalization and tensor product states in spin chains and lattices , 2009, 0910.1130.

[3]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[4]  Alán Aspuru-Guzik,et al.  The theory of variational hybrid quantum-classical algorithms , 2015, 1509.04279.

[5]  B. M. Fulk.  MATH , 1992 .

[6]  Jason Morton,et al.  Tensor Network Contractions for #SAT , 2014, Journal of Statistical Physics.

[7]  G. Evenbly,et al.  Tensor Network States and Geometry , 2011, 1106.1082.

[8]  S Montangero,et al.  Quantum multiscale entanglement renormalization ansatz channels. , 2008, Physical review letters.

[9]  G. Vidal.  Class of quantum many-body states that can be efficiently simulated. , 2006, Physical review letters.

[10]  Masashi Sugiyama,et al.  Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives , 2017, Found. Trends Mach. Learn..

[11]  J. Eisert.  Entanglement and tensor network states , 2013, 1308.3318.

[12]  Ville Bergholm,et al.  Categorical quantum circuits , 2010, 1010.4840.

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

[14]  J. D. Biamonte,et al.  Algebraically contractible topological tensor network states , 2011, 1108.0888.

[15]  Marco Lanzagorta,et al.  Tensor network methods for invariant theory , 2012, 1209.0631.

[16]  Jacob Biamonte,et al.  Charged string tensor networks , 2017, Proceedings of the National Academy of Sciences.

[17]  Roman Orus,et al.  A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States , 2013, 1306.2164.

[18]  U. Schollwoeck.  The density-matrix renormalization group in the age of matrix product states , 2010, 1008.3477.

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

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

[21]  Jacob D. Biamonte,et al.  Categorical Tensor Network States , 2010, ArXiv.

[22]  Román Orús,et al.  Advances on tensor network theory: symmetries, fermions, entanglement, and holography , 2014, 1407.6552.

[23]  S. R. Clark,et al.  Unifying neural-network quantum states and correlator product states via tensor networks , 2017, 1710.03545.

[24]  Khadija Iqbal,et al.  An introduction , 1996, Neurobiology of Aging.

[25]  Anmer Daskin.  Quantum Principal Component Analysis , 2015 .

[26]  Mikko Möttönen,et al.  Quantum circuits for general multiqubit gates. , 2004, Physical review letters.

[27]  Andrzej Cichocki,et al.  Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions , 2016, Found. Trends Mach. Learn..

[28]  G. Vidal.  Entanglement renormalization. , 2005, Physical review letters.