| matrix product states | |
|---|---|
| Name | Matrix product state |
| Caption | Diagrammatic representation of a one-dimensional matrix product state |
| Type | Tensor network state |
| Introduced | 1990s |
| Field | Quantum many-body physics |
| Notable | Density matrix renormalization group |
matrix product states
Matrix product states (MPS) are a class of variational quantum states that represent the wavefunction of one-dimensional quantum systems as a product of local tensors (matrices). MPS provide an efficient description for low-entanglement states that obey one-dimensional area law behavior and form the theoretical foundation of the Density matrix renormalization group (DMRG) and many tensor network algorithms. They matter in Quantum Physics because they capture ground states of gapped local Hamiltonians with polynomial resources and bridge quantum information theory with computational many-body methods.
Matrix product states were motivated by attempts to understand why DMRG, developed by Steven R. White in 1992, is so successful for one-dimensional lattices. MPS make explicit the structure exploited by DMRG: local tensors with a finite bond dimension encode correlations and entanglement that decay with distance in noncritical systems. Physically, MPS are well suited to describe ground states of short-range, gapped Hamiltonians such as the Heisenberg model, Hubbard model, and AKLT model, and provide compact representations for states produced by low-depth quantum circuits or adiabatic evolutions. The insight that many physically relevant states occupy only a small corner of Hilbert space underpins many numerical and analytical advances across condensed matter and quantum information.
An MPS on a chain of N sites with local dimension d is defined by a set of tensors A^{[i]}_{α_{i-1},α_i}(s_i) where s_i labels the local basis. The wavefunction amplitude is Ψ(s_1...s_N)=Tr[A^{[1]}(s_1)A^{[2]}(s_2)...A^{[N]}(s_N)]. The internal index size D is the bond dimension; D controls the number of variational parameters and the entanglement capacity. MPS admit gauge freedoms that allow canonical forms: left-canonical, right-canonical, and mixed-canonical gauges, obtained by successive Schmidt decompositions or singular value decompositions (SVD). The canonical form makes explicit the relation between singular values and bipartite Schmidt coefficients, facilitating truncation and error estimates. Formal results link MPS to finitely correlated states introduced by Fannes, Nachtergaele, and Werner and to matrix product operators (MPO) for representing density matrices and operators.
MPS satisfy an entanglement entropy upper bound S ≤ log D for any bipartition, hence they inherently obey one-dimensional area laws. For gapped local Hamiltonians, rigorous results (e.g., Hastings' area law proofs) show ground states have entanglement scaling compatible with efficient MPS approximation with polynomial D. At criticality, where entanglement grows logarithmically with subsystem size (as in conformal field theory predictions), an MPS requires bond dimension scaling polynomially or exponentially in system size to capture long-range correlations faithfully. The Schmidt spectrum of bipartitions in an MPS yields the entanglement spectrum, a quantity of interest in topological phases and symmetry-protected order, connecting to work by Haldane, Li and Haldane, and classification schemes based on projective representations of symmetry groups.
DMRG can be formulated as a variational optimization over MPS with open boundary conditions; sweeping updates optimize local tensors while preserving canonical gauges. Time evolution methods for MPS include the Time-Evolving Block Decimation (TEBD) introduced by G. Vidal, which applies Trotterized two-site gates and truncates by SVD, and variants using MPO-based Krylov or time-dependent variational principle (TDVP) integrators. Variational algorithms optimize expectation values ⟨Ψ|H|Ψ⟩ by local optimization (single-site or two-site updates) and exploit efficient contraction routines. MPS-based techniques integrate with Lanczos, Davidson solvers for excited states, and with techniques for finite-temperature simulation via purification or MPO evolution.
MPS generalize to higher-dimensional tensor networks. Projected entangled pair states (PEPS) extend the MPS construction to 2D lattices and capture area laws in higher dimensions but incur substantially higher computational cost for contraction. The multiscale entanglement renormalization ansatz (MERA) incorporates scale transformations to represent critical states with logarithmic entanglement scaling and relates to real-space renormalization group ideas by G. Vidal. Other extensions include tree tensor networks (TTN), string-bond states, and continuous MPS (cMPS) for quantum fields. These families preserve the philosophy of encoding entanglement locally while adapting to geometry or scale.
MPS underpin state-of-the-art simulations of spin chains, fermionic systems (via Jordan–Wigner transformations and fermionic MPS formalisms), and bosonic lattice models. They enable computation of correlation functions, spectral properties, dynamical response, and phase diagrams, including symmetry-protected topological phases and symmetry breaking. In quantum information, MPS clarify resource requirements for one-dimensional quantum simulation, certify parent Hamiltonians, and assist in designing matrix-product operator representations of quantum channels and noise. MPS have been used in studies related to AdS/CFT toy models, quantum state tomography via locally correlated ansätze, and benchmarking of near-term quantum computing devices.
Practical MPS algorithms are implemented in libraries such as ITensor, TeNPy, and ALPS and rely on efficient linear algebra: SVD, eigensolvers, and tensor contractions. Computational cost scales as O(N d D^3) or O(N d^2 D^3) depending on algorithmic choices; memory scales as O(N d D^2). Complexity limits arise from required bond dimension D to achieve given fidelity: for gapped 1D systems D can remain modest, while critical or highly excited states demand large D, making classical simulation hard and connecting to questions in quantum complexity theory (e.g., hardness of approximating ground states). Techniques to mitigate cost include exploiting symmetries (abelian and non-abelian), compression, parallelization, and hybrid classical–quantum schemes that use MPS as verification or compression layers for quantum hardware.
Category:Quantum many-body physics Category:Tensor network states