| matrix product state | |
|---|---|
| Name | Matrix product state |
| Field | Quantum physics |
| Introduced | 1990s |
| Notable people | Steven R. White, Frank Verstraete, Guifré Vidal, Norbert Schuch, J. Ignacio Cirac |
| Related | Density matrix renormalization group, Tensor network, Matrix product operator, Projected entangled pair state |
matrix product state
A matrix product state (MPS) is a class of variational wavefunctions that efficiently represent low-entanglement quantum many-body states, especially in one spatial dimension. MPS are central to numerical methods in condensed matter physics and quantum information theory because they capture the area-law entanglement scaling of gapped one-dimensional systems and provide the theoretical foundation for the Density matrix renormalization group (DMRG) and related algorithms. Their structured form enables scalable simulation, analytic insight into phases of matter, and connections to rigorous results in quantum many-body theory.
An MPS represents the amplitude of a quantum state on an L-site lattice as a product of site-dependent tensors (matrices) contracted along a virtual bond index. For a chain with local physical dimension d, an MPS takes the form |ψ⟩ = Σ_{s1...sL} Tr[A^{s1}_1 A^{s2}_2 ... A^{sL}_L] |s1...sL⟩ where each A^{si}_i is a D×D matrix and D is the bond dimension. The bond dimension controls the amount of entanglement the ansatz can represent and determines the computational cost. MPS generalize earlier ideas from the Density matrix renormalization group and are closely tied to notions of tensor factorization in linear algebra.
Mathematically, an MPS is a particular instance of a tensor network with a one-dimensional topology. Using singular value decomposition (SVD), any MPS can be brought into left-, right-, or mixed-canonical form, exposing the Schmidt decompositions across bipartitions. Canonical forms yield orthonormality conditions for the tensors and make computation of expectation values and entanglement spectra stable and efficient. Gauge transformations relate different tensor representations of the same physical state; fixing a gauge by canonicalization is standard in implementations in libraries developed at institutions such as Max Planck Institute for Physics groups and quantum software projects. Key theoretical results connect canonical MPS to completely positive maps and matrix product operators (MPOs), enabling proofs about correlation decay and parent Hamiltonians.
MPS occupy the simplest nontrivial class of tensor networks and exemplify the area-law paradigm: ground states of one-dimensional gapped local Hamiltonians obey an entanglement entropy bound that MPS with modest D can saturate. MPS naturally encode the Schmidt decomposition between blocks and provide direct access to the entanglement spectrum, introduced in work by Haldane and others in topological contexts. The relation between MPS and entanglement renormalization techniques by Guifré Vidal clarifies how entanglement is organized across scales, and comparisons to multiscale entanglement renormalization ansatz (MERA) highlight tradeoffs in representing critical versus gapped systems.
MPS underpin powerful numerical algorithms. The Density matrix renormalization group algorithm, developed by Steven R. White in 1992, can be formulated entirely in MPS language and is the premier tool for one-dimensional quantum lattice models. Time-evolution of MPS is performed with methods like the time-evolving block decimation (TEBD) introduced by Guifré Vidal and time-dependent DMRG variants; these apply Suzuki–Trotter decompositions or variational time evolution to approximate dynamics. Optimization and compression techniques include variational ground-state minimization, single-site and two-site updates, and MPO compression; they are implemented in community codes from research groups at University of Innsbruck, Harvard University, Caltech, and companies such as Google Quantum AI exploring simulators. Convergence properties, truncation errors, and entanglement growth set practical limits for simulations of quenches and finite-temperature algorithms like purification.
MPS are widely used to study quantum spin chains (e.g., Heisenberg model, Transverse field Ising model), fermionic systems via Jordan–Wigner transformations, and symmetry-protected topological phases such as the Haldane phase. They provide constructive realizations of valence-bond solid states, exemplified by the AKLT model, and are instrumental in characterizing order parameters, correlation functions, and excitations. In quantum information, MPS encode matrix product operators for quantum channels and are applied to tensor network descriptions of error-correcting codes, measurement-based quantum computation models, and as resources in experimental platforms at IBM Quantum and academic quantum optics labs. Analytical constructions yield parent Hamiltonians and rigorous classifications of one-dimensional phases under symmetries by groups like Nobel Laureates-adjacent theorists and institutional collaborations.
MPS generalize to higher-dimensional tensor networks, notably Projected entangled pair states (PEPS) for two dimensions and multiscale entanglement renormalization ansatz (MERA) for scale-invariant problems. While PEPS retain the local tensor structure, contracting PEPS is computationally costly and often relies on approximate schemes developed in research at institutions such as École Normale Supérieure and University of Amsterdam. MERA provides an alternative ansatz with explicit renormalization-group interpretations. These extensions inherit conceptual lessons from MPS about entanglement area laws but face increased computational complexity and different approximation regimes.
The representational power of MPS is characterized by bond dimension D: polynomial D suffices for gapped 1D ground states obeying area laws, while critical systems or highly excited states may require D that grows with system size. Results in complexity theory link tensor network contraction problems to #P-hard instances in higher dimensions and show that certain tasks for MPS can be performed in polynomial time in D and L, whereas others become intractable as D grows. Research by theoretical groups including Perimeter Institute researchers and university departments studies expressiveness, simulation hardness, and bounds on approximability, informing best practices for numerical studies and rigorous statements about quantum simulability.
Category:Quantum many-body theory Category:Tensor network states