Time evolution of projected entangled pair states in the single-layer picture

Pižorn, Iztok; Wang, Ling; Verstraete, Frank

doi:10.1103/physreva.83.052321

Cited by 35 publications

(33 citation statements)

References 26 publications

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“…The Single-Layer (SL) algorithm for the computation of the norm ψ ψ 〈 was presented in [31]. This method takes into account the bra-ket structure of the sandwich, and maintains it and hence the positive character of the environment while the contraction of the network progresses from one edge.…”

Section: Single-layermentioning

confidence: 99%

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Unifying projected entangled pair state contractions

2014

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The approximate contraction of a tensor network of projected entangled pair states (PEPS) is a fundamental ingredient of any PEPS algorithm, required for the optimization of the tensors in ground state search or time evolution, as well as for the evaluation of expectation values. An exact contraction is in general impossible, and the choice of the approximating procedure determines the efficiency and accuracy of the algorithm. We analyze different previous proposals for this approximation, and show that they can be understood via the form of their environment, i.e. the operator that results from contracting part of the network. This provides physical insight into the limitation of various approaches, and allows us to introduce a new strategy, based on the idea of clusters, that unifies previous methods. The resulting contraction algorithm interpolates naturally between the cheapest and most imprecise and the most costly and most precise method. We benchmark the different algorithms with finite PEPS, and show how the cluster strategy can be used for both the tensor optimization and the calculation of expectation values. Additionally, we discuss its applicability to the parallelization of PEPS and to infinite systems.In the past few years, tensor network states (TNS) have been revealed as a very promising choice for the numerical simulation of strongly correlated quantum many-body systems. A sustained effort has led to significant conceptual and technical advancement of these methods, e.g. in .In the case of one-dimensional systems, matrix product states (MPS) are the variational class of TNS underlying the density matrix renormalization group (DMRG) [23]. Insight gained from quantum information theory has allowed the understanding of DMRGʼs enormous success at approximating ground states of spin chains, and the extension of the technique to dynamical problems [1-6] and lattices of more complex geometry [7][8][9].Projected entangled pair states (PEPS) [7] constitute a family of TNS that naturally generalizes MPS to spatial dimensions larger than 1 and arbitrary lattice geometry. Like MPS, PEPS incorporate the area law by construction, which makes them a very promising variational ansatz for strongly correlated systems which might not be tractable by other means, e.g. frustrated or fermionic states where quantum Monte Carlo methods suffer from the sign problem. Although originally defined for spin systems, PEPS have been subsequently formulated for fermions [24][25][26][27], and their potential in the numerical simulation of fermionic phases has been demonstrated [27][28][29][30]. But in contrast to the case for MPS, even local expectation values cannot be computed exactly in the case of PEPS. This is because the exact evaluation of the TN that represents the observables has an exponential cost in the system size. The same difficulty affects the contraction of the TN that surrounds a given tensor, the so-called environment, required for the local update operations in the course of optimization algorithms. It is neve...

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Section: Single-layermentioning

confidence: 99%

“…6 when ′ d takes on its largest possible value. In [31,33] it was proposed to set = = ′ ″ d D D , in which case the number of operations scales only like D O( ) 7 , and a clear computational gain compared to the original contraction can be expected.…”

Section: Single-layermentioning

confidence: 99%

Unifying projected entangled pair state contractions

2014

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“…For our decoder, this effectively requires the contraction of a three-dimensional, rather than a two-dimensional, tensor network. Algorithms for such calculations have been developed in the context of condensed-matter physics, [35][36][37][38][39][40] and could potentially be applied here.…”

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confidence: 99%

Linear-time general decoding algorithm for the surface code

Darmawan¹,

Poulin²

2018

Phys. Rev. E

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A quantum error correcting protocol can be substantially improved by taking into account features of the physical noise process. We present an efficient decoder for the surface code which can account for general noise features, including coherences and correlations. We demonstrate that the decoder significantly outperforms the conventional matching algorithm on a variety of noise models, including non-Pauli noise and spatially correlated noise. The algorithm is based on an approximate calculation of the logical channel using a tensor-network description of the noisy state.

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“…Plugging ω ȟ ( ) into the Boltzmann weights (2) guarantees that the random approximation scheme will provide rigorous upper bounds on the ground state energy of H. The use of ω ȟ ( ) effectively steers Markov-chains away from pointless PEPS. As a result, none of the instability issues addressed in [16] has to be faced.…”

Section: Discussionmentioning

confidence: 99%

“…Importantly, ȟ is by construction a rigorous upper bound on the ground state energy of H for any Hamiltonian. As a result, the instability issue studied in [16] simply never shows up. We again consider distributions of the form given by equation (2), with h replaced with ȟ; strong ergodicity and reversibility are still satisfied.…”

Section: Extension To Two Dimensionsmentioning

confidence: 99%

Simulated annealing for tensor network states

Iblisdir

2014

New J. Phys.

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Markov chains for probability distributions related to matrix product states and one-dimensional Hamiltonians are introduced. With appropriate 'inverse temperature' schedules, these chains can be combined into a simulated annealing scheme for ground states of such Hamiltonians. Numerical experiments suggest that a linear, i.e., fast, schedule is possible in non-trivial cases. A natural extension of these chains to two-dimensional settings is next presented and tested. The obtained results compare well with Euclidean evolution. The proposed Markov chains are easy to implement and are inherently sign problem free (even for fermionic degrees of freedom).

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Time evolution of projected entangled pair states in the single-layer picture

Cited by 35 publications

References 26 publications

Unifying projected entangled pair state contractions

Unifying projected entangled pair state contractions

Linear-time general decoding algorithm for the surface code

Simulated annealing for tensor network states

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