| matrix product state | |
|---|---|
| Name | Matrix product state |
| Caption | Schematic of an MPS for a one-dimensional lattice |
| Type | Quantum many-body ansatz |
| Field | Quantum Physics |
| Introduced | 1990s |
| Notable examples | DMRG, AKLT model |
matrix product state
A matrix product state (MPS) is a class of variational wavefunctions that efficiently represent low-entanglement quantum states of one-dimensional lattice systems. MPS provide compact descriptions of ground states of local gapped Hamiltonians and underpin numerical methods such as the DMRG; they matter for both practical simulation of condensed matter models and foundational studies in quantum information science because they make entanglement structure explicit.
A matrix product state represents the amplitudes of an N-site quantum lattice as a product of site-dependent matrices. For a chain with local physical dimension d, an MPS expresses the coefficient of the product basis |s1...sN> as Tr[A1[s1] A2[s2] ... AN[sN, where each A_k[s_k] is a D_k−1 × D_k matrix and D_k are bond dimensions controlling representational power. The MPS form captures the area-law scaling of entanglement for one-dimensional gapped systems, connecting to results by M. B. Hastings and others on exponential decay of correlations. Physically, the bond indices can be viewed as virtual degrees of freedom that mediate entanglement between subsystems, making MPS a natural language for understanding low-energy sectors of models like the Heisenberg model, the Hubbard model, and the AKLT model.
Formally, an MPS is an element of the tensor product Hilbert space (C^d)^{⊗N} parameterized by a sequence of third-order tensors. Gauge freedom in the tensors allows conversion to canonical forms (left-, right-, and mixed-canonical) that expose orthonormality conditions and singular value spectra. The Schmidt decomposition across a bond relates directly to the singular values in the mixed-canonical form; truncation of small singular values implements optimal low-rank approximations in the 2-norm, analogous to SVD. The concept of injective and canonical MPS connects to classification results for one-dimensional SPT phases and to the construction of parent Hamiltonians. Rigorous statements about approximation accuracy link MPS bond dimension D to entanglement entropy S via bounds S ≤ log D and to approximation theorems due to F. Verstraete, I. Cirac, and collaborators.
MPS are the computational backbone of algorithms for one-dimensional quantum many-body problems. The DMRG algorithm variationally optimizes an MPS ansatz and remains the state-of-the-art for ground states of gapped chains; seminal contributors include S. R. White and later formalizations by S. Östlund and S. Rommer. Time evolution of MPS is performed with methods such as time-evolving block decimation (TEBD) by Guifr\'e Vidal and later time-dependent DMRG, employing Suzuki–Trotter decompositions or Krylov-space integrators. Optimization uses local tensor updates, canonicalization, and truncation controlled by discarded weights. Software ecosystems implementing MPS include ITensor, TeNPy, and ALPS, with many implementations used in condensed matter physics and quantum chemistry to treat systems like spin chains, ladders, and one-dimensional fermions.
MPS have enabled quantitative studies of quantum phase transitions, critical behavior (with adaptions to capture conformal scaling), and topological phases in one dimension. They serve in modeling low-dimensional materials, cold atom experiments in optical lattices (e.g., experiments by groups at Max Planck Institute for Quantum Optics and MIT), and simulation of quench dynamics relevant to nonequilibrium quantum systems. In quantum chemistry, matrix product state variants (matrix product operator representations) allow correlated electronic structure calculations for quasi-one-dimensional molecules; notable applications involve the DMET and active-space methods. MPS-based approaches have also supported theoretical studies of thermal states (purifications) and open quantum systems via coupling to Lindblad equation descriptions.
MPS are a special case of tensor network states and relate to higher-dimensional generalizations such as PEPS and MERA. They make entanglement entropy and the Schmidt spectrum explicit, providing tools to analyze entanglement scaling, area laws, and the role of entanglement in efficient classical simulation of quantum systems. Connections to quantum error correction and channel capacities arise via the structure of virtual indices and transfer matrices; transfer matrix spectra determine correlation lengths and mixing properties. MPS also contribute to studies in quantum information theory on state convertibility, matrix product operator representations of quantum channels, and resource theories for many-body states, linking to work by researchers at institutions like Perimeter Institute and IQC.
While MPS excel in one dimension, they face exponential cost when representing volume-law entangled states or generic high-dimensional systems. Extensions address these limits: PEPS and MERA for higher dimensions, Tree Tensor Networks (TTN) for hierarchical entanglement, and continuous MPS (cMPS) for field theories. Recent developments include scalable algorithms for long-range interactions, hybrid quantum–classical variational schemes integrating MPS with quantum computing hardware, and rigorous complexity results linking bond dimension to computational hardness. Socially relevant aspects include open-source tool development (e.g., ITensor community) that democratizes access to simulation tools for researchers globally, and methodological advances that enable more equitable participation in computational condensed matter and quantum information research. Active research hubs include CERN for numerical methods crossover, university groups at University of Vienna (Vidal, Cirac networks), University of Innsbruck, and national labs exploring applications to quantum simulation and materials.
Category:Quantum many-body theory Category:Tensor network states