Questions tagged [tensor-network]
The tensor-network tag has no usage guidance.
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Obtaining a Matrix Product State (MPS) using Schmidt Decomposition for a Tripartite State
I understand that one method to derive an MPS representation of a quantum state involves applying the Schmidt decomposition $ N−1$ times. While I'm familiar with the diagrammatic notation, I wanted to ...
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MPS canonical form
If I express a MPS in its (left, right or anything else) canonical form, does this representation encode all Schmidt decompositions between a subsystem and its complement,rather than only the Schmidt ...
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What is a parent Hamiltonian? [closed]
The term is used throughout the literature but I was not able to find a definition or even a paper properly introducing the term. What does a Hamiltonian have to satisfy to be a parent Hamiltonian?
An ...
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Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula and tensor networks
While working with AdS/CFT, I am trying to look at the nature of the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula in AdS/CFT, which is the statement that $S(\rho _{A}|\sigma _{A})=S_{\text{bulk}}(\...
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Construct an operator using outer product of two MPS using TeNPy
I am fairly new to Matrix Product State (MPS) formalism, but I've used Density Matrix Renormalisation Group (DMRG) techniques before. I'm learning to use TeNPy, and a particular problem I am trying to ...
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Is there Difference Between 1D and 2D in Spin model?
The Motivation is That:In the Tensor Network method, they say 'time evolution MPS(Matrix Product State) work quite well in 1 Dimension'.
but as I think any 2D could be expressed by 1D
for example in ...
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Question about the 'reduced basis transformation'
I' ve been reading the review Ulrich Schollwöck: The density-matrix renormalization group in the age of
matrix product states (arXiv link)
and encountered with a question about the so called 'reduced ...
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$G$-injective MPS and symmetry-broken phases
First, a little bit of motivation. I was reading the paper "Matrix Product States and Projected Entangled Pair States" to try to learn more about MPS representations of symmetry broken ...
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iTEBD\iDMRG for gapless systems
I have learned that the Matrix Product State (MPS) formulation can only handle systems where the entropy scales up to logarithmically. For gapped systems, as the entanglement entropy is constant, a ...
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Intuitive understanding of bond dimensions
I am trying to understand what is bond dimension in tensor network more intuitively, by meaning of bond dimension I meant the tensor dimension that connects between tensors (in the example below the ...
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Eigenvectors of Matrix Product Operator (MPO)
I am starting to work with tensor network and I wanted to know how to get the eigenvectors of a Matrix Product Operator (MPO).
As far as I know when one is trying to diagonalize his Hamiltonian he ...
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What kind of object is a Matrix Product State?
I'm learning about Matrix Product States (MPS) and I'm confused about what type of object it is. An answer to this previous question
Examples of Matrix Product States
mentions that MPS are built up of ...
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Finding the MPS representation of translationally invariant ground states of TFIM
We would like to carry out the so called renormalization group (RG) transformation on quantum states (see the link for the relevant reference). In the implementation we need an matrix product state (...
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How to solve the eigenvalue problem of Matrix product operator (MPO) using Tensor network method?
I am new to the Tensor network approach. I need to solve the eigenvalue problem for a Matrix product Operator (MPO). What are some techniques, resources, softwares or packages available to do it? I am ...
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How to obtain the goal function of DMRG by Lagrange multiplier method?
The goal of DMRG is to minimize the expectation value of energy, which can be written as
$$ \frac{d}{d |\psi \rangle} \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle} = 0. \...
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iTEBD real time evolution for 3-body time evolution operator
I am trying to implement the iTEBD algorithm for real-time evolution of the PXP model. Here, $P$ is the projector onto the ground state, and $X$ is the Pauli spin matrices.
I know for the 2-body case, ...
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Normalization in tensor networks [closed]
I am trying to implement the iTEBD algorithm for the $PXP$ model, i.e, the hamiltonian is
$$H = \sum_iP_{i-1}X_iP_{i+1}.$$
Here $P$ is the projector onto the ground state and $X$ is the usual pauli x ...
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What are the advantages of tensor network algorithms over monte-carlo simulations in terms of time-evolution?
I understand that tensor networks and monte carlo simulations are based on completely different principles. However, to my knowledge both are used to simulate the time evolution of a system. Is there ...
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Making measurements on infinite tensor network chain
I am trying to implement the iTEBD algorithm for a particular model. I am primarily interested in measuring two quantities, the entanglement entropy across the midpoint bipartition and the correlation ...
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How do you calculate the entanglement entropy of a tensor network?
I found that the entanglement entropy can be calculated using the Schmidt coefficients of the state, using
$S = -\sum_i|\alpha_i|^2\log(|\alpha_i|^2)$
In the case of tensor networks, does this simply ...
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How to find canonical form of three site MPS?
I am trying to implement the iTEBD algorithm for a certain model, where the hamiltonian acts on three successive sites. This means that my time-evolution operator is a rank 6 tensor, acting on a rank ...
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Physical meaning of Schmidt states
Given a bipartite system AB (where A and B are the subsystems), a generic pure quantum state $|\psi\rangle$ on AB can be Schmidt-decomposed into:
\begin{equation}
|\psi\rangle = \sum_\alpha^{r}s_\...
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How to efficiently get the largest probabilities / amplitudes of a quantum state stored as an MPS?
Let's say, that we have the following pure, superposition state
$$ |\psi \rangle = \frac{1}{\sqrt{2}}|000001 \rangle + \frac{1}{2}|101101 \rangle + \frac{1}{2}|100100 \rangle $$
stored in the MPS form....
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Is there a zero correlation length spin-$1$ chain in the Haldane phase?
The ground state of the spin-$1$ AKLT model gives an example of a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological (SPT) phase, the Haldane phase. This state is a nice example of the ...
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Why Matrix Product State (MPS) representation provide fast computation?
I am interested in tensor networks and I am trying to understand why MPS (for example) provide an efficient representation of a quantum state. In order to transform the quantum state in MPS ...
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How does the proof for the area law for 1D systems work?
I am currently reading this paper in order to understand the proof of the area law for one dimensional, low energy systems such as 1D spin chains. The main area law theorem is given on page 13 and is ...
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Reshaping $U$ and $V^\dagger$ matrices resulting from an SVD into rank-3 tensors
Let's say, that we have a $6 \times 6$ matrix $M$. By conducting an SVD of $M$ we obtain $USV^\dagger$ matrices, where $U$ is of size $6 \times 6$ and is left-normalized, $S$ is diagonal with 6 ...
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Resource recommendation for time-dependent DMRG
I am looking for any references (journal article/ review /tutorial) to understand the TEBD (Time-evolution by block decimation) algorithm and code it up for a simple 1D model system. Broadly, I am ...
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Particle number conservation in matrix product state
I've been trying to understand how particle number conservation is enforced in matrix product state algorithms. As far as I understand, if the Hamiltonian commutes with the number operator, you can ...
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Overlap of Matrix Product States (Python)
I'd like to implement the TEBD (finite, real time evolution) by hand (in python) and want to compute the overlap of a reference MPS with the time evolved MPS. I want to regard a simple Ising Chain.
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Tensor contraction in matrix product state
From Matrix product state, the matrix product can be written as
$$
|\Psi\rangle = \sum_{\{s\}} \operatorname{Tr}\left[A_1^{(s_1)} A_2^{(s_2)} \cdots A_N^{(s_N)}\right] |s_1 s_2 \ldots s_N\rangle,
$$
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Mutual information of a tensor network
Suppose I've got a tensor network (TN) representing some bipartite quantum state, $|\Psi\rangle$. Using the Schmidt decomposition, I can write $|\Psi\rangle = \sum_{k=1}^r \sqrt{\lambda_k}|\chi_k\...
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Algorithm that checks if a subspace of states contains a product state
Suppose I have two identical qudits, the full Hilbert space is $\mathcal{H}=(\mathbb{C}^{d})^{\otimes 2}$. Say I'm given a supspace of states $\Lambda\subset \mathcal{H}$. What is the fastest ...
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Can bond dimension vary from bond to bond?
Consider a bipartite system composed of subsystems $A$ and $B$, with corresponding Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, spanned by $\{\chi_1,...,\chi_n\}$ and $\{\phi_1,...,\phi_m\}$, ...
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Tensor product operator of 2D ( for Tensor Network)
I have difficulty representing Tensor Product Operators (TPO) of 2D in a concrete form.
For example in 1D case, according to the tutorial in ITensor , the TPO of the Hamiltonian for the simple Ising ...
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Tensor Networks with Julia and implementing given Hamiltonian
I have this Hamiltonian: (ref: https://arxiv.org/abs/1302.5843)
I want to solve this Hamiltonian by using tensor networks. I wanted to make the implementation with ITensors, Julia. However, I am ...
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Ground State calculation for defined 2D Ising Model with tensor networks
I have a Hamiltonian and 2D spin-lattice system. I am trying to find a ground state configuration. Spin interactions are long-ranged so I am trying to use PEPS to approximate. My question is this:
...
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Tensor networks and sign problem
I've been wondering about why tensor networks are capable of avoiding MC sign problem (e.g. see arXiv:1611.04791 [hep-lat] and the references therein). I have seen many papers stating that the TNs are ...
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How to add two Matrix Product States of different bond dimensions?
If I have the MPS representation of two quantum states, how do I add them? Note that the bond -dimensions need not be the same for the two MPSs.
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Conserved charge in Density Matrix Renormalization Group (DMRG)
Currently I am facing a problem which relates to the conserved quantities in DMRG. I use old-fashioned DMRG (Steven White approach) to compute the ground state of certain models. However, the ground ...
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Relationship between Hartley entropy and local dimension
I am recently reading a paper about entanglement entropy. It mentions that if we consider a 1D spin chain and write a pure state in the matrix product state:
\begin{align}
|\psi\rangle = A^{\sigma_1}A^...
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Uniqueness of AKLT Ground State vs. SU(2) symmetry and Lieb-Schultz-Mattis theorem
I have a question in my mind regarding the uniqueness of AKLT ground state. Currently I am watching a video clip of MPS and I am curious why the AKLT ground state model is unique gapped ground state. ...
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Rigorous derivation of entanglement entropy bound of MPS
I am working on a summer project and am trying to prove the Von Neumann entropy of an MPS is bounded by a constant to gain some introduction. While this easily follows if we start with the MPS being ...
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How to translate from a state/density matrix formalism to matrix product state representation?
From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $A_i$ matrices? From all the examples I read, people just somehow "...
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Projected Entangled Pair States (PEPS) and classical statistical physics
In the paper arXiv:quant-ph/0601075 the authors introduce an interesting correspondence between Projected Entangled Pair States (PEPS) and classical statistical physics. Basically, for any locally-...
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Matrix product state representation for the "infinitely repulsive hardcore boson" state
Consider a one-dimensional spin-1/2 chain with $N$ spins, and let $|\psi\rangle$ be the equal weight superposition of all states with no adjacent spin-ups, e.g.
for $N=3$ with open-boundary, $|\psi_{N=...
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The dominant eigenvalue of the transfer matrix of a matrix product state
Consider a translation-invariant matrix product state
\begin{equation}
|\psi_L\rangle= \mathrm{Tr}[A(s_1)A(s_2)\ldots A(s_L)]|s_1 s_2\ldots s_L\rangle.
\end{equation}
I'm interested in the ...
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Truncation Problem of Density Matrix Renormalisation Group (DMRG)
I am wondering that is there any restrictions for the truncation in DMRG algorithm. Currently I am using DMRG to calculate ground state energy per site of a many-body system described by on-site ...
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Efficient MPS Description of a given quantum state
If we know the amplitudes of a (pure) quantum state wrt some basis, is there an algorithmic procedure to ensure an efficient MPS description (one with the lowest bond dimension) of the state ?
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Thermodynamic Limit of Entanglement-Entropy like quantities
Suppose i have, in one spatial dimension, a unique ground state $\Omega$ of a local, gapped, translational invariant Hamiltonian. Denote by $\sigma_s$ the density matrix of $\Omega$ on lattice sites $\...
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