Improving the efficiency of variational tensor network algorithms

Glen Evenbly and Robert N. C. Pfeifer

Phys. Rev. B 89, 245118 – Published 12 June 2014

Abstract

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 $T$ , we prove that if the environment of a single tensor from the network can be evaluated with computational cost $κ$ , then the environment of any other tensor from $T$ can be evaluated with identical cost $κ$ . Moreover, we describe how the set of all single tensor environments from $T$ can be simultaneously evaluated with fixed cost $3 κ$ . The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a multiscale entanglement renormalization Ansatz for the ground state of a one-dimensional quantum system, where they are shown to substantially reduce the computation time.

Received 7 February 2014
Revised 13 May 2014

DOI:https://doi.org/10.1103/PhysRevB.89.245118

Authors & Affiliations

Glen Evenbly^1,* and Robert N. C. Pfeifer^2,†

¹Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
²Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada N2L 2Y5

^*evenbly@caltech.edu
^†rpfeifer@perimeterinstitute.ca

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Vol. 89, Iss. 24 — 15 June 2014

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Figure 1
(a) An example of a closed tensor network, here consisting of tensors ${A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7}}$ , evaluates to a scalar $B$ under contraction of its internal indices. (b) An open tensor network is obtained by removing $A_{1}$ from the closed network in diagram (a); this open network evaluates to a tensor $Γ_{A_{1}}$ that we call the environment of $A_{1}$ . (c) An open tensor network, here consisting of tensors ${A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7}}$ , describes a quantum state $| ψ 〉$ on the lattice $L$ . (d) Given the Hamiltonian $\hat{H}$ defined on $L$ , the closed tensor network $〈 ψ | \hat{H} | ψ 〉$ evaluates to the energy $E$ of the tensor network state $| ψ 〉$ with regard to $\hat{H}$ . (e) The environment $Γ_{A_{1}}$ of tensor $A_{1}$ from the closed tensor network $〈 ψ | \hat{H} | ψ 〉$ is evaluated. (f) The environment $Γ_{A_{1}}$ and tensor $A_{1}$ are contracted to give the energy of state $| ψ 〉$ . (g) A closed tensor network representing the overlap $〈 ϕ | ψ 〉$ of two tensor network states $| ϕ 〉$ and $| ψ 〉$ . (h) The environment $Γ_{A_{1}}$ of tensor $A_{1}$ from the closed tensor network in diagram (g) is evaluated.
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Figure 2
(a) Two tensors $A_{1}$ and $A_{2}$ are contracted, through summation over index $β$ , to give a new single tensor $B_{1}$ . The indices labeled $α$ , $β$ , $γ$ , and $δ$ have dimensions $| α |$ , $| β |$ , $| γ |$ , and $| δ |$ , respectively. Following Eq. (4), the computational cost of performing this contraction is $cost : (A_{1}, A_{2}) = | α | \times | β | \times | γ | \times | δ |$ . (b) A contraction tree representing the tensor contraction from diagram (a). (c) A closed tensor network $T$ is contracted to a scalar $B_{6}$ following a sequence of pairwise tensor contractions (indicated by dashed ovals). (d) Representation of the tensor contractions of diagram (c) as a contraction tree. Edges of the graph represent tensors, and vertices of the tree represent the contraction of two tensors to give a new tensor. The open edges represent the original tensors of the network $T$ , while internal edges represent the tensors obtained during intermediate stages of the contraction. The root vertex, depicted as a square, represents the contraction of two tensors into a scalar, denoted $B_{6}$ .
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Figure 3
(a) A closed tensor network $T$ of three tensors ${A_{i}, A_{2}, A_{3}}$ , with $χ_{i j}$ representing the dimension of the index connecting tensors $A_{i}$ and $A_{j}$ . (b) Contraction trees representing three different ways to contract the network of diagram (a) to a scalar. Each of these contraction orders has the same leading-order cost [see Eq. (6)]—the costs differ only in the contraction at the root vertex, which is never greater than that associated with the preceding trivalent vertex. (c) Two contraction trees that differ only within the dotted circles (relating to the contraction order of their final three tensors). The result from diagram (b) implies that the cost associated with each of these contraction orders differs only in the contraction at the root vertex (represented by the square).
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Figure 4
(a) A contraction tree $C_{1}$ that could be used to evaluate the environment $Γ_{A_{1}}$ of tensor $A_{1}$ (by performing all contractions excluding that associated with the root vertex). (b) A contraction tree $C_{2}$ that could be used to evaluate the environment $Γ_{A_{2}}$ of tensor $A_{2}$ . Since both contraction trees are in the same family, it follows that the cost of contracting for $Γ_{A_{1}}$ is identical to that of $Γ_{A_{2}}$ . Notice that the contractions associated with vertices $V_{1}$ , $V_{2}$ , and $V_{5}$ represent identical binary tensor contractions in both $C_{1}$ and $C_{2}$ ; it follows that, in the evaluation of both $Γ_{A_{1}}$ and $Γ_{A_{2}}$ together, these binary tensor contractions need only be performed once and the result may be reused.
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Figure 5
(a), (b) The two closed tensor networks that arise in the optimization of a 1D binary MERA [9]. The dotted ovals indicate the single tensor environments that need to be computed during each iteration of the optimization. (c) Plot of error in the ground-state energy compared with the iteration number during optimization of a binary MERA for the ground state of the critical Ising model on a 1D lattice of 72 sites. The solid line represents the new algorithm where all environments are computed simultaneously, whereas the dashed line represents the traditional algorithm where environments are computed sequentially. The algorithm based upon simultaneous computation of environments takes more iterations to converge to the same energy. (d) In this diagram, the errors in the ground-state energy from diagram (c) are plotted against time, and the advantages of the new algorithm become apparent. Although the original algorithm requires fewer iterations to converge, these iterations take significantly longer than those of the new algorithm, with the new algorithm requiring approximately $40 %$ less computing time than the original algorithm to reach the chosen error threshold of one part in $10^{- 5}$ .
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