matrix product states

matrix product states
Name	Matrix product states
Abbreviation	MPS
Field	Quantum many-body physics; Condensed matter physics
Introduced	1992
Creators	Steven R. White; Richard Feynman (conceptual precursors)
Related	Density matrix renormalization group; Tensor network states; Quantum entanglement

matrix product states

Matrix product states are a class of variational quantum many-body wavefunctions used to represent low-entanglement states in one-dimensional systems. They underpin numerical methods such as the Density matrix renormalization group and connect to analytical frameworks in Statistical mechanics and Quantum information theory. MPS formalism links to concepts developed in work by Steven R. White, applications in studies by Umesh Chandrachud and formalizations related to Fannes, Nachtergaele, and Wielandt results.

Introduction

Matrix product states arose from efforts to efficiently encode quantum states of lattice systems with local interactions, notably in algorithms pioneered by Steven R. White and theoretical developments influenced by ideas from Richard Feynman and results used in Bethe ansatz studies. MPS serve as a bridge between computational schemes like the Density matrix renormalization group and rigorous constructs used in proofs by Fannes, Nachtergaele, and Woronowicz for one-dimensional gapped systems. They relate to entanglement area laws studied by John Preskill, Patrick Hayden, and Masahito Ueda, and have been applied in numerical studies by groups including Umesh Vazirani collaborators and experimental proposals by Immanuel Bloch.

Mathematical definition

An MPS represents a many-body state on an N-site chain by a product of site-dependent tensors A^(i) with physical index σ_i and bond indices of dimension D, as formalized in works by Fannes, Nachtergaele, and Werner. The coefficients of the global wavefunction in a local basis are given by a trace or open chain contraction: ψ_{σ1...σN} = Tr[A^(1)_{σ1} A^(2)_{σ2} ... A^(N)_{σN}], a representation exploited in proofs by Elliott Lieb and Helen Haight on representability and approximation. Bond dimension D controls expressivity; scaling of entanglement entropy with log(D) connects to bounds proven by Hastings and discussed by Eran Sela and Michael Levin. Injective and non-injective MPS classifications reference results by Cirac and Perez-Garcia.

Construction and canonical forms

MPS can be brought into canonical forms—left-canonical, right-canonical, and mixed-canonical—using singular value decompositions, a technique refined in algorithmic contexts by Ian McCulloch and Ulrich Schollwöck. Gauge transformations among tensor representations follow theorems by Perez-Garcia, Cirac, and Verstraete on canonicalization and uniqueness modulo unitary equivalence. The Schmidt decomposition at bipartitions yields orthonormal tensors and diagonal singular-value matrices, foundations built upon work by Nicolas Laflorencie and proofs by Hastings about correlation decay. Matrix product operator forms and purification constructions draw on operator algebra results by Araki and numerical practices developed in Steven R. White’s implementations.

Physical applications

MPS are extensively used to study ground states and low-energy excitations of one-dimensional quantum spin chains such as models by Heisenberg, Ising, and Haldane, and to characterize symmetry-protected topological phases investigated by F. D. M. Haldane, Xiao-Gang Wen, and Ashvin Vishwanath. They model dynamics in quench and transport problems treated in studies by Markus Greiner and Immanuel Bloch experiments, and serve in describing correlated fermionic systems related to work by Giovanni Mussardo and Frank Wilczek. MPS also underpin entanglement scaling analyses in conformal systems studied by John Cardy and are used in proposals for quantum simulation platforms by Rainer Blatt and Mikhail Lukin.

Numerical algorithms and simulations

Algorithms leveraging MPS include the original Density matrix renormalization group by Steven R. White, time-evolving block decimation developed by Guifre Vidal, and variational optimization methods refined by Ulrich Schollwöck and Ian McCulloch. Time evolution schemes such as TEBD and time-dependent variational principle approaches reference contributions by Guifre Vidal, Jutho Haegeman, and Frank Verstraete. Finite-temperature algorithms use matrix product operator techniques and purification methods applied by A. E. Feiguin and P. Schmitteckert. Krylov subspace and Lanczos-based MPS solvers relate to numerical analysis traditions from John von Neumann and implementations in community codes influenced by groups at Oak Ridge National Laboratory and Max Planck Institute.

Extensions and related tensor network states

Extensions generalize MPS to higher dimensions and different geometries, yielding tensor network states such as projected entangled pair states associated with Frank Verstraete and J. Ignacio Cirac, multi-scale entanglement renormalization ansatz developed by Guifre Vidal, and tree tensor networks utilized in works by Román Orús and F. Verstraete. Continuous matrix product states were introduced for quantum field applications in papers by Jutho Haegeman and Frank Verstraete, connecting to continuum limit studies by Giuseppe Mussardo and Alexander Zamolodchikov. Matrix product operators and operator spreading analyses link to studies by Pieter W. Brouwer and Markus Grüner.

Computational complexity and limitations

Representability and hardness results for MPS approximation tie to computational complexity results by Hastings and hardness proofs referencing Leonid Levin-style reductions; finding global minima in variational MPS spaces can be NP-hard in certain instances discussed by Eisert and Zwolak. Limitations arise in representing highly entangled states such as volume-law states relevant to Sachdev or in critical systems beyond logarithmic scaling, issues analyzed by Calabrese and Cardy. Scaling of bond dimension required for faithful approximation connects to lower bounds proved by Hastings and open problems posed by Avi Wigderson in complexity-theoretic contexts.

Category:Quantum many-body physics