Fermionic implementation of projected entangled pair states algorithm

Pižorn, Iztok; Verstraete, Frank

doi:10.1103/physrevb.81.245110

Cited by 53 publications

(46 citation statements)

References 24 publications

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“…We refer to these wavefunctions collectively as correlator product states here. These wavefunctions were constructed as a way to generalize the density matrix renormalization group (DMRG) 11,12 to higher dimensions, while avoiding the computational cost of recent classes of tensor network wavefunctions such as the projected entangled pair states (PEPS) [13][14][15][16][17][18][19][20][21][22][23] or multiscale entanglement renormalization ansatz (MERA) 24,25 . The correlator product state directly approximates a wavefunction as a product of correlator amplitudes.…”

Section: Introductionmentioning

confidence: 99%

Nonstochastic algorithms for Jastrow-Slater and correlator product state wave functions

et al. 2011

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Jastrow-Slater and correlator product state wavefunctions, two classes of quantum many-body wavefunctions, are commonly studied using Monte Carlo methods with the associated drawbacks of stochastic error. Here we show that efficient non-stochastic algorithms for these wavefunctions exist, both for observable evaluation and for optimization. The algorithms rely on the structure of these states as a product of local, commuting, invertible operators acting on a simple reference wavefunction. We describe the non-stochastic energy evaluation and optimization algorithms, and demonstrate them with applications to the Heisenberg and spinless and full Hubbard models. Our results demonstrate that the non-stochastic algorithms yield optimized wavefunctions and energies very close to those obtained with the variational Monte Carlo algorithm. Such algorithms provide new criteria for identifying new classes of wavefunctions for efficient computational simulation.

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Section: Introductionmentioning

confidence: 99%

Nonstochastic algorithms for Jastrow-Slater and correlator product state wave functions

et al. 2011

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“…Several approaches have been developed in order to implement the fermionic statistics at the level of TN algorithms [42][43][44][45][46], in the end being all equivalent (see, e.g., Ref. [47] for some examples of this).…”

Section: Fermionic Tensor Networkmentioning

confidence: 99%

Kitaev honeycomb tensor networks: Exact unitary circuits and applications

Schmoll

Orús

2017

Phys. Rev. B

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The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivial physical properties such as topological quantum order, abelian and non-abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different "solution ingredients" of the Kitaev honeycomb model can be accounted for in the TN language, namely: Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. We also show how the properties of two-point correlation functions follow easily. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d Projected Entangled Pair State (PEPS) with finite bond dimension. The results in this paper can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.

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“…The projected entangled-pair state (PEPS) [21][22][23][24][25][26][27][28][29][30] generalizes the MPS, whereas D > 1 versions of TTN 31,32 and MERA 33-39 also exist. Among those generalizations, PEPS and MERA stand out for offering efficient representations of many-body wave functions, thus leading to scalable simulations in D > 1 dimensions; and, importantly, for also being able to address systems that are beyond the reach of quantum Monte Carlo approaches due to the so-called sign problem, including frustrated spins 30,39 and interacting fermions [40][41][42][43][44][45][46][47][48][49][50] .…”

Section: Introductionmentioning

confidence: 99%

Tensor Network States and Geometry

2011

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Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D = 1 dimensions, as well as projected entangled pair states (PEPS) in D > 1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D = 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary lawthat is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D > 1 dimensions.

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Fermionic implementation of projected entangled pair states algorithm

Cited by 53 publications

References 24 publications

Nonstochastic algorithms for Jastrow-Slater and correlator product state wave functions

Nonstochastic algorithms for Jastrow-Slater and correlator product state wave functions

Kitaev honeycomb tensor networks: Exact unitary circuits and applications

Tensor Network States and Geometry

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