Parameterization of Tensor Network Contraction

We present a conceptually clear and algorithmically useful framework for parameterizing the costs of tensor network contraction. Our framework is completely general, applying to tensor networks with arbitrary bond dimensions, open legs, and hyperedges. The fundamental objects of our framework are rooted and unrooted contraction trees, which represent classes of contraction orders. Properties of a contraction tree correspond directly and precisely to the time and space costs of tensor network contraction. The properties of rooted contraction trees give the costs of parallelized contraction algorithms. We show how contraction trees relate to existing tree-like objects in the graph theory literature, bringing to bear a wide range of graph algorithms and tools to tensor network contraction. Independent of tensor networks, we show that the edge congestion of a graph is almost equal to the branchwidth of its line graph.

[1]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[2]  Igor L. Markov,et al.  Simulating Quantum Computation by Contracting Tensor Networks , 2008, SIAM J. Comput..

[3]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height , 1991, WG.

[4]  David R. Wood,et al.  The treewidth of line graphs , 2012, J. Comb. Theory, Ser. B.

[5]  Eugene Dumitrescu,et al.  Tree tensor network approach to simulating Shor's algorithm , 2017, 1705.01140.

[6]  Alán Aspuru-Guzik,et al.  qTorch: The quantum tensor contraction handler , 2017, PloS one.

[7]  Daniel Bienstock,et al.  On embedding graphs in trees , 1990, J. Comb. Theory, Ser. B.

[8]  John R. Gilbert,et al.  Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree , 1995, J. Algorithms.

[9]  Dimitrios M. Thilikos,et al.  Derivation of algorithms for cutwidth and related graph layout parameters , 2002, J. Comput. Syst. Sci..

[10]  John A. Gunnels,et al.  Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits , 2017, 1710.05867.

[11]  H. Neven,et al.  Simulation of low-depth quantum circuits as complex undirected graphical models , 2017, 1712.05384.

[12]  Dimitrios M. Thilikos,et al.  Constructive Linear Time Algorithms for Branchwidth , 1997, ICALP.

[13]  M. I. Ostrovskii Minimal congestion trees , 2004, Discret. Math..

[14]  Zeph Landau,et al.  Quantum Computation and the Evaluation of Tensor Networks , 2008, SIAM J. Comput..

[15]  Hans L. Bodlaender A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[16]  J. Biamonte,et al.  Tensor Networks in a Nutshell , 2017, 1708.00006.

[17]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[18]  Rupak Biswas,et al.  A flexible high-performance simulator for the verification and benchmarking of quantum circuits implemented on real hardware , 2018 .

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

[20]  Scott Aaronson,et al.  Complexity-Theoretic Foundations of Quantum Supremacy Experiments , 2016, CCC.

[21]  Paul D. Seymour,et al.  Graph minors. X. Obstructions to tree-decomposition , 1991, J. Comb. Theory, Ser. B.

[22]  Travis S. Humble,et al.  Benchmarking treewidth as a practical component of tensor network simulations , 2018, PloS one.

[23]  John A. Gunnels,et al.  Pareto-Efficient Quantum Circuit Simulation Using Tensor Contraction Deferral , 2017 .

[24]  Paul D. Seymour,et al.  Tour Merging via Branch-Decomposition , 2003, INFORMS J. Comput..

[25]  Yaoyun Shi,et al.  Classical Simulation of Intermediate-Size Quantum Circuits , 2018, 1805.01450.