Matrix product state representations

By David Pérez-García, Frank Verstraete, Michael M. Wolf, J. I. Cirac. 2006

1,222 citations recorded by the source corpus.

This work gives a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.

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

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. Matthew Fisher
6. Fernando G. S. L. Brandão
7. J. Ruhman
8. J. Dubail

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Andrew J. Daley
3. Pasquale Calabrese
4. Matthew Fisher
5. V. Khemani
6. J. Cardy
7. J. Dubail
8. S. Vijay

Citations in the corpus

• A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Suitable conditions that imply a unique MPS representation can be found in [221, 263].”
• Unsupervised Generative Modeling Using Matrix Product States (2017, background): “In particular, matrix product states (MPS) consist of a subset of TN where the tensors are arranged in a one-dimensional geometry [25].”
• Efficient Variational Quantum Simulator Incorporating Active Error Minimization (2016, methodology): “Variational methods have numerous applications in the numerical study of many-body quantum systems: for example, density functional theory [41], the matrix product state method [42], and simulating molecular dynamics using the variational principle [43].”
• Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, neutral): “Also, in quantum physics the ALS algorithm is called one-site DMRG, while the Modified ALS is known as the two-site DMRG (for more detail, see Part 2 and [128, 201, 215, 232, 267, 271, 280] and references therein”
• Tensor Ring Decomposition (2016, background): “Although the similar concept has been mentioned and called MPS or tensor chain in few literatures [29], [42], [53], the algorithms and properties have not well explored yet.”
• Local random quantum circuits are approximate polynomial-designs: numerical results (2012, methodology): “To this aim we use a result of Nachtergaele [64], originally proposed to lower bound the spectral gap of frustration-free local Hamiltonians with a ground space spanned by matrix-product states [31, 69].”
• Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains (2013, background): “5 ([16]): If the mps matrices are site-independent (and thus fulfill Eqn.”
• T-Basis: a Compact Representation for Neural Networks (2020, background): “Another structure of TNs, with arguably better generalization capabilities (Wang et al., 2017), is Tensor-Ring (TR) (Perez-Garcia et al., 2007; Khoromskij, 2011; Espig et al., 2011; Zhao et al., 2016).”
• Random Quantum Circuits (2022, background): “The minimal cut is a way of bounding entanglement in any tensor network (150, 151, 152, 153, 154).”