69,603 research outputs found
Tensor network method for reversible classical computation
We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs and outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.We thank Justin Reyes, Oskar Pfeffer, and Lei Zhang for many useful discussions. The computations were carried out at Boston University's Shared Computing Cluster. We acknowledge the Condensed Matter Theory Visitors Program at Boston University for support. Z.-C. Y. and C. C. are supported by DOE Grant No. DE-FG02-06ER46316. E.R.M. is supported by NSF Grant No. CCF-1525943. (Condensed Matter Theory Visitors Program at Boston University; DE-FG02-06ER46316 - DOE; CCF-1525943 - NSF)Accepted manuscrip
Higher moments of Banach space valued random variables
We define the :th moment of a Banach space valued random variable as the
expectation of its :th tensor power; thus the moment (if it exists) is an
element of a tensor power of the original Banach space.
We study both the projective and injective tensor products, and their
relation. Moreover, in order to be general and flexible, we study three
different types of expectations: Bochner integrals, Pettis integrals and
Dunford integrals.
One of the problems studied is whether two random variables with the same
injective moments (of a given order) necessarily have the same projective
moments; this is of interest in applications. We show that this holds if the
Banach space has the approximation property, but not in general.
Several sections are devoted to results in special Banach spaces, including
Hilbert spaces, and . The latter space is non-separable, which
complicates the arguments, and we prove various preliminary results on e.g.
measurability in that we need.
One of the main motivations of this paper is the application to Zolotarev
metrics and their use in the contraction method. This is sketched in an
appendix.Comment: 110 pages. v2: Typos corrected and new references adde
qTorch: The Quantum Tensor Contraction Handler
Classical simulation of quantum computation is necessary for studying the
numerical behavior of quantum algorithms, as there does not yet exist a large
viable quantum computer on which to perform numerical tests. Tensor network
(TN) contraction is an algorithmic method that can efficiently simulate some
quantum circuits, often greatly reducing the computational cost over methods
that simulate the full Hilbert space. In this study we implement a tensor
network contraction program for simulating quantum circuits using multi-core
compute nodes. We show simulation results for the Max-Cut problem on 3- through
7-regular graphs using the quantum approximate optimization algorithm (QAOA),
successfully simulating up to 100 qubits. We test two different methods for
generating the ordering of tensor index contractions: one is based on the tree
decomposition of the line graph, while the other generates ordering using a
straight-forward stochastic scheme. Through studying instances of QAOA
circuits, we show the expected result that as the treewidth of the quantum
circuit's line graph decreases, TN contraction becomes significantly more
efficient than simulating the whole Hilbert space. The results in this work
suggest that tensor contraction methods are superior only when simulating
Max-Cut/QAOA with graphs of regularities approximately five and below. Insight
into this point of equal computational cost helps one determine which
simulation method will be more efficient for a given quantum circuit. The
stochastic contraction method outperforms the line graph based method only when
the time to calculate a reasonable tree decomposition is prohibitively
expensive. Finally, we release our software package, qTorch (Quantum TensOR
Contraction Handler), intended for general quantum circuit simulation.Comment: 21 pages, 8 figure
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