Questions tagged [tensor-network]

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Number of operations in tensor networks [closed]

I cannot understand how the number of operations in these Tensor Networks differ and are $D^4$ and $D^5$. I'd appreciate any help. source: https://arxiv.org/abs/1306.2164
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Schmidt decomposition for Matrix Product State in canonical form

During a lecture on comp. Quantum Phyiscs the concept of Matrix Product States (MPS) (with open boundry conditions) was introduced in the form $$\vert\psi\rangle = \sum_{i_1,\dots,i_n}\psi_{i_1\dots ...
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Does the degeneracy in dominant eigenvalues of MPS transfer matrix necessarily mean long-range correlation?

A short-range correlated MPS state generally has one non-degenerate dominant eigenvalue in its transfer matrix, and the degeneracy in transfer matrix generally leads to long-range correlation (I'm ...
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Why is tensor networks applicable only to ground states?

When one uses tensor network as a wave function ansatz for a variational method, we usually use this scheme to find a ground state. Why can’t we apply the tensor network formalism to find any excited ...
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Why does a flat entanglement spectrum contradict to the holographic CFT vacuum?

As stated in the introduction of 1806.05007 (https://arxiv.org/pdf/1806.05007.pdf), a certain tensor network called the holographic code has a flat entanglement spectrum (ES), i.e. the reduced density ...
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What is meant exactly by “renormalization” in condensed matter physics, specifically in density matrix renormalization group (DMRG)?

I first encountered the concept of renormalization in the context of statistical physics. Here, the renormalization "group" is a set of transformations of the system such that the Hamiltonian $H(J,\...
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Partial trace of matrix product state

I have come accross a formula that puzzles me a bit in the proof of lemma 23 (page 32) of this paper. The authors start from a (translationally-invariant) matrix product state: $$\lvert\psi\rangle := ...
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43 views

How to left-normalize a Matrix Product State

If you are reading this, you should probably have some background in Tensor Networks or Matrix Products States or this will be insufficient information. So when putting a MPS into left (or right) ...
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Can a random product state be expressed as a MPS (Matrix product state)?

$ |\psi\rangle = \prod_{i=1}^{N}|s_{i}\rangle $ where, $|s_{i}\rangle = \cos\left (\frac{\theta_{i}}{2}\right )|\uparrow_{i}\rangle + \exp{(i\phi_{i})}\sin\left (\frac{\theta_{i}}{2}\right )|\...
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63 views

Reduction from PEPS to MERA possible?

Multi-scale entanglement renormalization ansatz (MERA) includes two kinds of isometric tensors, disentanglers and isometries. Thus tensors in MERA are by definition composed of isometries. Meanwhile ...
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Clarification on DMRG computational complexity

I was reading a paper on the density matrix renormalization group (https://arxiv.org/abs/1008.3477). In DMRG, we gradually grow a chain by inserting a unit cell at the center of the chain (for ...
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56 views

Finding eigenvalue and eigenvector of non-Hermitian matrix product operator

Suppose we have a matrix product operator (MPO) $X$ with a periodic boundary, which is not necessarily Hermitian. That is, $$X^{s_1\cdots s_n}_{s^{\prime}_1\cdots s^{\prime}_n}:=\mathrm{Tr}(G_1[s_1,...
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Is HaPPY code a certain type of MERA?

Pastawski, Yoshida, Harlow, and Preskill introduced the HaPPY code in their (now famous) paper, arXiv:1503.06237, as a way to model the AdS/CFT correspondence as a quantum error-correcting code. ...
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All infinite volume matrix product states are in different super-selection sectors?

Consider two matrix product states $\Psi_1,\Psi_2$, i.e. let them be described (schematically) by $$ \Psi_\alpha = \sum_{...i_n ... } \left(\ell_{\alpha}, \left[\prod_{i \in \mathbb{Z}} E_\alpha(i) \...
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Does a gapped, local and non-frustration-free 1D Hamiltonian have an exact MPS groundstate?

(Fact 1) It is known that any arbitrary matrix product state (MPS) is a unique groundstate of a gapped, local, and frustration-free parent Hamiltonian (Perez-Garcia et al. 2007). It is by Fact 1, that ...
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58 views

why do matrix product states work at critical point?

Matrix product states satisfy the entanglement area law, which should be a property of gapped states. But usually, MPS work well in 1D quantum phase transition problems. As far as I know, ...
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Can we define a parent Hamiltonian for a given Multiscale Entanglement Renormalization Ansatz (MERA) state?

Given a state represented by a MERA (Multiscale Entanglement Renormalization Ansatz), can we define a local Hamiltonian such that the MERA is the exact ground state of such a Hamiltonian? If this is ...
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92 views

Can any 1D critical state be represented by a MERA tensor network?

My understanding of the Multiscale Entanglement Renormalisation Ansatz (MERA) is that it is designed to represent highly entangled, but low complexity states. Is MERA capable of representing high ...
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Tensor networks and construction of PEPO

I have a basic understanding of how to construct the tensors of an MPO (matrix product operator), based on, in part, on PhysRevA.81.062337 (arXiv version), see their equations 5 and 6. I am looking ...
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Nature of the $W$-state in the thermodynamic limit

Consider a matrix product state on $\mathbb{C}^{d N}$: $$ \Psi = \sum_{\sigma_1,...\sigma_N} A_1(\sigma_1) ... A_N(\sigma_N) |\sigma_1 ... \sigma_N \rangle \quad \quad (\text{OBC MPS}) $$ with some ...
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Doing addition and subtraction with tensor diagrams?

Tensor diagrams are a beautiful and useful tool for making calculations with tensors, up until you need to contract with the sum or difference of two tensors, at which point it seems to become awful. ...
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Partial Transpose in Gapped Time-reversal Symmetric Spin Chains

Suppose you have a one-dimensional quantum spin system with on-site Hilbert spaces $\mathcal{S}$. Suppose there is an anti-unitary, anti-linear operator $C$ on $\mathcal{S}$ inducing an anti-linear, ...
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Physical Interpretation of the Spectrum of MPS Transfer Matrices

Take an injective, translation invariant MPS with transfer matrix $E = \sum_\sigma \overline{A^\sigma} \otimes A^\sigma$ (i am using the terminology of https://arxiv.org/abs/quant-ph/0410227 , eq. (6))...
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176 views

Spin-1 Heisenberg model, the AKLT model, and their ground states

I am reading literature on quantum spin chains and matrix product states, and I notice similar arguments regarding the spin-1 antiferromagnetic Heisenberg model, $H_{H} = \sum_i S_i \cdot S_{i+1}$, ...
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How can I simulate a ground state degenerate system numerically?

I'm using numerical method like DMRG to simulate ground state of correlated systems. But the degeneracy of the ground state has long bothered me: When degeneracy exists the ground state isn't unique. ...
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60 views

PEPS expression of Toric Code ground state

It is well known that the ground state of 2D Toric Code system can be represented by $D=2$ PEPS (Projected Entangled Pair States, i.e. the tensors in the 2D network are connected by 2-dimensional ...
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How does the complexity in Matrix Product states ansatz drop from $D^N$ to $ND$?

I have just started to read about DMRG and MPS. It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of ...
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362 views

Why do we use matrix product states?

Given a many body $\vert\psi\rangle$, we can express it in terms of a matrix product state. That is, $\vert\psi\rangle = \sum_{i,j..k}\psi_{i,j..k}\vert i,j..k\rangle$ can be rewritten as $\vert\...
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267 views

Is there any relation between density matrix renormalization group (DMRG) and renormalization group (RG)?

Probably I am going to receive many down-votes for this post but I really need to ask this question here. I am new to statistical mechanics. I wanted to learn Density Matrix Renormalization Group (...
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Resource recommendation: Tensor Networks

I want to learn tensor network methods for condensed matter systems. I went through some basic papers (i.e. 1,2) and come to know that there are many things (i.e. different math, tensors, ...
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230 views

Left and Right Eigenvectors of Transfer Matrix in Matrix Product States (MPS)

Let $$\lvert{\psi}\rangle=\sum_{i_1i_2...i_n}Tr(A^{[1]}_{i_1}A^{[2]}_{i_2}...A^{[n]}_{i_n})\lvert{i_1 i_2...i_n}\rangle$$ be a MPS, where $i_k=1,2...d$ and $A^{[k]}_{i_k}$ are $D\times D$ matrices ...
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The physical picture of uncle Hamiltonian?

Uncle Hamiltonian was built to show the complex relationship between MPS (Matrix Product State) states and Hamiltonians, which claims that for a block injective MPS state, we can build a local ...
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1answer
54 views

Constructing PEPS representation of an arbitrary quantum state

Given a quantum state we can construct its MPS (Matrix Product State) representation by doing a series of singular value decompositions. Given the freedom to choose arbitrary bond dimensions the MPS ...
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Tensor Network from Lattice

I read an article about tensor networks and they seem very interesting and a promising approach to studying the relations between entanglement, gravity and quantum information. How does this network ...
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If a state can be efficiently represented by a Projected Entangled Pair State (PEPS), can we prepare it physically?

If we can use PEPS (Projected Entangled Pair State) to represent a many body quantum state, can we generate it by a quantum computer? As far as I can understand, PEPS is dual to a quantum computer ...
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What is some good reference for tensor network?

I am a physicist who is now doing some research in quantum information, which might involve tensor network.
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How to code Tensor Networks?

I'm interested in learning tensor networks, I've been reading some introductory articles about this. The problem is that these articles mostly discuss the theoretical definitions for tensor networks ...
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What's the tensor network representation for local ground state?

It's well known that many topological phases can be represented using matrix product state and PEPS. Example, toric code, $H=-\sum_{v}A_v-\sum_p B_p $. My question is: What's the tensor network ...
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Relationship between the 'relevant states' and the states that can be generated by a quantum computer?

In the huge quantum state space, for example n-qubit space has a state number as $2^{2^n}$, usually we are interested in two different kinds of 'physical' states. Set 1: The relevant state which is ...
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Dependence of tensor network on underlying physical lattice structure

I have noticed that the majority of research on tensor networks has focused on lattice systems. For example, in this review, the author mentions using PEPS on different lattices such as triangular, ...
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Question about general direction of development of application of Tensor Networks to AdS/CFT

I am quite interested in the area of application of Tensor Networks(TNs) to AdS/CFT correspondence. I would like to clarify certain points in order to get better general picture. What is the main ...
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Tensor network Renormalization group

I am working on Tensor network Renormalization group. While I am trying to implement TRG on Honeycomb lattice I got stuck on pairing two three rank tensors and represent it as D^2*D^2 matrix. Can you ...
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How to compute the MPS representation for a sequence of asymptotically gapped quantum spin chains

Suppose I have a sequence of gapped, spin-$1/2$, translationally invariant quantum spin chains $\{H_1, H_2,H_3\cdots\}$ with interactions of range $\leq 2$ (i.e. no further than nearest-neighbors). ...
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Simple analytic examples of Multi-scale Entanglement Renormalization Ansatz (MERA)

I want to understand Multi-scale Entanglement Renormalization Ansatz (MERA) with very elementary examples. So far I could find references which are mostly based on numerics. It would be a great help ...
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Examples of Matrix Product States

Matrix product states (MPS) are a way of representing a (many-body) wavefunction. The method has been described, for example, in The density-matrix renormalization group in the age of matrix ...
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Is Schmidt Decomposition well defined for periodic matrix product representation(MPS)?

We are able to perform Schmidt decomposition for open-boundary MPSs with dimension of boundary bond $m_0 = 1$, $|\psi\rangle=\sum w_{a_l}|a_l\rangle_L|a_l\rangle_R$. Because we can make $|a_l\rangle_{...
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Exponential decay of correlation in PEPS

PEPS (Projected Entangled Pair State) is a tensor network that plays the same role in two dimensional lattice as MPS (Matrix Product State) plays in one dimensional spin chain. A good introduction can ...
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Difference between DMRG (density matrix renomalization group) and MPS (matrix product states)?

I am learning DMRG recently. I noticed there are many papers both in the DMRG approach and MPS (such as variational matrix product state (VMPS) by F.Verstraete and J.I.Cirac) approach. In my eyes, ...
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What is the 'area law' in the context of matrix product states?

I am trying to get into the topic of matrix product states by reading this: A practical introduction to tensor networks: Matrix product states and projected entangled pair states. R. Orús. Ann. ...
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Connection between bond-dimension of a matrix product state and entanglement

The bond dimension is the dimension of the truncated matrix product state (MPS). Let us assume that I am simulating some many-body system with high entanglement via the density matrix renormalization ...