Valence-bond states for quantum computation

Verstraete, Frank; Cirac, J. I.

doi:10.1103/physreva.70.060302

Cited by 412 publications

(565 citation statements)

References 27 publications

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“…MPS are at the heart of numerical methods such as the Density Matrix Renormalization Group (DMRG) [72], and have also been used recently as a key theoretical tool in the successful efforts made by several groups to classify one-dimensional symmetry-protected topological phases of matter [73][74][75][76]. Their higher-dimensional analogs, the TPS (also called "PEPS" for Projected Entangled-Pair States [77]) are currently being used in attempts to classify higher-dimensional topological phases (see for instance [78]). The basic idea of such a program is that any gapped phase of matter is believed to be accurately represented by one (or more) trial state, like an MPS for D large enough, or some higher-dimensional generalization (see for instance [79]).…”

Section: Discussion: Entanglement Trial Wavefunctions Conformal mentioning

confidence: 99%

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

2012

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We consider the trial wavefunctions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e. short-range correlations only) inside the quantum Hall droplet, and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk 2d chiral CFT which underlies the trial wavefunction. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation.This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition A ∪ B in real-space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization group arguments. We prove that generalized screening implies that the entanglement Hamiltonian HE = − log ρA is isospectral to an operator that is local along the cut between A and B. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wavefunctions given by conformal blocks and Tensor Product States, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of px ± ipy paired superfluids are treated in a similar fashion in the appendix.

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Section: Discussion: Entanglement Trial Wavefunctions Conformal mentioning

confidence: 99%

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

2012

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“…This is very good news if we want to create families of variational ground states: it suffices to approximate well the local properties of all translational invariant states. The family of matrix product states ͑MPSs͒ 3,21,22 and generalizations to higher dimensions ͑PEPS͒ 18,23,24 were exactly created with this property in mind; the amazing accuracy of renormalization group algorithms is precisely related to the fact that the convex set under consideration can be very well approximated with the reduced density operators of MPS. Both Wilson's numerical renormalization group 1 and density matrix renormalization group ͑DMRG͒ 2,25 methods can indeed be reformulated as variational methods within the MPS.…”

Section: Ground States As Convex Problemsmentioning

confidence: 99%

Matrix product states represent ground states faithfully

2006

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We quantify how well matrix product states approximate exact ground states of one-dimensional quantum spin systems as a function of the number of spins and the entropy of blocks of spins. We also investigate the convex set of local reduced density operators of translational invariant systems. The results give a theoretical justification for the high accuracy of renormalization group algorithms and justifies their use even in the case of critical systems.

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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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Valence-bond states for quantum computation

Cited by 412 publications

References 27 publications

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

Edge-state inner products and real-space entanglement spectrum of trial quantum Hall states

Matrix product states represent ground states faithfully

Tensor Network States and Geometry

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