64,554 research outputs found
Faster identification of optimal contraction sequences for tensor networks
The efficient evaluation of tensor expressions involving sums over multiple
indices is of significant importance to many fields of research, including
quantum many-body physics, loop quantum gravity, and quantum chemistry. The
computational cost of evaluating an expression may depend strongly upon the
order in which the index sums are evaluated, and determination of the
operation-minimising contraction sequence for a single tensor network (single
term, in quantum chemistry) is known to be NP-hard. The current preferred
solution is an exhaustive search, using either an iterative depth-first
approach with pruning or dynamic programming and memoisation, but these
approaches are impractical for many of the larger tensor network Ansaetze
encountered in quantum many-body physics. We present a modified search
algorithm with enhanced pruning which exhibits a performance increase of
several orders of magnitude while still guaranteeing identification of an
optimal operation-minimising contraction sequence for a single tensor network.
A reference implementation for MATLAB, compatible with the ncon() and
multienv() network contractors of arXiv:1402.0939 and arXiv:1310.8023
respectively, is supplied.Comment: 25 pages, 12 figs, 2 tables, includes reference implementation of
algorithm, v2.01. Update corrects the display of contraction sequences
involving single-tensor traces (i.e. where an index in the input appears
twice on the same tensor
The Cost of contract renegotiation: evidence from the local public sector
We construct and estimate a structural principal/agent model of contract renegotiation in the French urban transport sector in a context where operators are privately informed on their innate costs (adverse selection) and can exert cost-reducing managerial effort (moral hazard). This model captures two important features of the industry. First, only two types of contracts are used in practice by local public authorities to regulate the service: cost-plus and fixedprice contracts with positive subsidies. Second, these subsidies increase over time. Such increasing subsidies are consistent with the theoretical hypothesis that principals cannot commit not to renegotiate and contracts are renegotiationproof. We compare this situation to the hypothetical case with full commitment. The distribution of innate costs of operators is shifted upwards under this hypothetical scenario. The welfare gains of commitment are significant and
accrue mostly to operators. Estimates of the weights that local governments give to the operator´s profit in their objective functions and of the social value of the cost-reducing managerial effort are obtained as by-products
Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms
In the last decade, there has been a substantial amount of research in
finding routing algorithms designed specifically to run on real-world graphs.
In 2010, Abraham et al. showed upper bounds on the query time in terms of a
graph's highway dimension and diameter for the current fastest routing
algorithms, including contraction hierarchies, transit node routing, and hub
labeling. In this paper, we show corresponding lower bounds for the same three
algorithms. We also show how to improve a result by Milosavljevic which lower
bounds the number of shortcuts added in the preprocessing stage for contraction
hierarchies. We relax the assumption of an optimal contraction order (which is
NP-hard to compute), allowing the result to be applicable to real-world
instances. Finally, we give a proof that optimal preprocessing for hub labeling
is NP-hard. Hardness of optimal preprocessing is known for most routing
algorithms, and was suspected to be true for hub labeling
Improving the efficiency of variational tensor network algorithms
We present several results relating to the contraction of generic tensor
networks and discuss their application to the simulation of quantum many-body
systems using variational approaches based upon tensor network states. Given a
closed tensor network , we prove that if the environment of a
single tensor from the network can be evaluated with computational cost
, then the environment of any other tensor from can be
evaluated with identical cost . Moreover, we describe how the set of
all single tensor environments from can be simultaneously
evaluated with fixed cost . The usefulness of these results, which are
applicable to a variety of tensor network methods, is demonstrated for the
optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for
the ground state of a 1D quantum system, where they are shown to substantially
reduce the computation time.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation.
Software updated to v1.02: Resolved two scenarios in which multienv would
generate errors for valid input
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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