Matrix product states represent ground states faithfully

By Frank Verstraete, J. I. Cirac. 2005

628 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/7e649332969fe445bfd13dd348401f5d2da87ea6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. Nahum
2. Andrew J. Daley
3. Pasquale Calabrese
4. J. Cardy
5. J. Ruhman
6. Fernando G. S. L. Brandão
7. T. Takayanagi
8. B. Skinner

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Andrew J. Daley
3. Pasquale Calabrese
4. J. Cardy
5. J. Ruhman
6. B. Skinner
7. Fernando G. S. L. Brandão
8. T. Takayanagi

Citations in the corpus

• Geometry of Quantum States: Index (2007, background): “35 Much more can be said on the topic of why and how MPS represent ground states faithfully [924].”
• Exponential Decay of Correlations Implies Area Law (2012, background): “It is interesting to analyse how Theorem 1 fits together with the entanglement properties of random states and quantum expander states, since these have been thought of giving an obstruction to a statement of a similar flavour [8, 21, 24, 26, 31–35].”
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States (2017, background): “In many cases of interest, an initial approximation of the ground state can be obtained, either by optimizing simpler wave functions or by first applying DMRG to optimize a MPS.”
• Quantum Entanglement in Neural Network States (2017, background): “It has been proved that a bounded R´enyi entropy of all the orders in 1D necessarily guarantees an efficient MPS representation [86] (note that counterexample does exist if only the von Neumann entropy is bounded [87]).”
• Nonequilibrium steady states of matrix-product form: a solver's guide (2007, background): “Recently, matrix product states have received a lot of attention in the context of quantum information as they are able to describe highly entangled states [75].”
• Quantum spin liquid states (2016, background): “In practice, the low-energy states of gapped local Hamiltonians in one dimension can be efficiently represented by MPSs with a finite value of M (Hastings, 2007; Verstraete and Cirac, 2006).”
• Entanglement growth in quench dynamics with variable range interactions (2013, background): “This leads to a linear increase in bipartite entanglement in time, so that the dynamics cannot be efficiently computed in existing classical simulations beyond short times [16,17].”
• Entanglement entropy and conformal field theory (2009, methodology): “This is of relevance for the convergence of the algorithms based on matrix product states [27, 28].”