| projected entangled pair states | |
|---|---|
| Name | Projected entangled pair states |
| Field | Quantum physics |
| Introduced | 2004 |
| Introduced by | Frank Verstraete and J. I. Cirac |
| Related | Tensor network states, Matrix product state, Density matrix renormalization group |
projected entangled pair states
Projected entangled pair states (PEPS) are a class of tensor network ansätze for quantum many-body wavefunctions that generalize matrix product states (MPS) to higher spatial dimensions. PEPS provide an efficient representation of low-entanglement states relevant to condensed matter physics and quantum information, enabling approximations of ground states of local Hamiltonians and analyses of entanglement structure in lattice systems.
PEPS were introduced in the context of variational methods for two-dimensional quantum lattice models and are closely tied to the study of quantum phases, topological order, and critical phenomena. They form part of the broader family of tensor network states used in many-body physics and share conceptual lineage with the density matrix renormalization group (DMRG) and matrix product operator (MPO) formalisms. PEPS connect to rigorous results in area law entanglement scaling and are used to study models such as the Heisenberg model, Hubbard model, and Kitaev model. Foundational contributors include Frank Verstraete, J. Ignacio Cirac, and Norbert Schuch who developed theoretical and algorithmic aspects of PEPS and their relation to topological order.
A PEPS is defined on a graph or lattice by assigning a rank-(z+1) tensor to each site (z the coordination number) and contracting virtual indices along edges to form a global wavefunction in the physical Hilbert space. The construction uses local projection maps from virtual entangled pair degrees of freedom to physical degrees of freedom, yielding a variational family parameterized by local tensors. PEPS generalize the MPS description of 1D systems and can encode symmetry-protected topological phases via tensor category structures or explicit symmetry constraints such as global SU(2) or U(1) invariance. Key mathematical issues include tensor symmetries, gauge freedom, canonical forms, and the scaling of the virtual bond dimension D, which controls expressivity versus computational cost.
PEPS have been applied to approximate ground states and low-energy excitations of strongly correlated systems, including the study of quantum phase transitions, fractional quantum Hall effect analogs on lattices, and models with intrinsic topological order such as resonating valence bond (RVB) states and the Toric code. In quantum information, PEPS provide constructive families for resource states in measurement-based quantum computation (MBQC), including cluster states and universal resource proposals studied by Robert Raussendorf and collaborators. PEPS have been used to analyze entanglement spectra, represent projective symmetry group ansätze for spin liquids, and to provide rigorous examples in discussions on the area law for entanglement entropy and the computational complexity of local Hamiltonian problems (e.g., QMA-hardness).
Numerical treatment of PEPS requires contracting two-dimensional tensor networks, an operation that is in general computationally hard and approximate. Algorithms include approximate contraction schemes such as boundary MPS / corner transfer matrix (CTM) methods developed in the spirit of Baxter's classical approaches, tensor renormalization group (TRG) and higher-order TRG (HOTRG), and variational optimization methods based on imaginary-time evolution and gradient techniques. Implementations leverage symmetries, sparse tensor formats, and Monte Carlo sampling over physical indices (variational Monte Carlo with PEPS). Important software and algorithmic benchmarks come from groups at institutions like Max Planck Institute for Quantum Optics, CERN-affiliated collaborations, and academic teams led by researchers such as Guifre Vidal, Frank Verstraete, and Norbert Schuch.
PEPS naturally encode the area law for entanglement entropy in gapped local systems on lattices, making them suitable variational families for ground states of local gapped Hamiltonians. Unlike MPS, PEPS can capture nontrivial topological entanglement entropy and long-range entanglement patterns relevant to topologically ordered phases. Scaling properties depend on the bond dimension D: correlation lengths and the ability to represent criticality grow with D, while computational cost typically scales polynomially or exponentially in D depending on the task. PEPS also expose connections to notions of locality and information propagation formalized by the Lieb–Robinson bound and have been used to study entanglement phase transitions and many-body localization phenomena.
PEPS are part of a landscape of tensor network architectures including multiscale entanglement renormalization ansatz (MERA), projected entangled simplex states (PESS), and continuous tensor networks for field theories such as continuous MPS (cMPS). Connections to category theory and tensor category models underlie constructions for chiral and symmetry-enriched phases. Generalizations introduce fermionic PEPS (fPEPS) to represent fermionic systems with proper antisymmetry, and hybrid approaches combine PEPS with Monte Carlo or neural-network quantum states to tackle larger bond dimensions. Comparative studies place PEPS alongside MERA for critical systems and MPS/DMRG for quasi-one-dimensional problems, with ongoing work to bridge gaps between these formalisms.
PEPS-inspired states guide the design and interpretation of experiments in ultracold atoms in optical lattices, Rydberg atom arrays, and superconducting qubit platforms aiming to realize many-body ground states with nontrivial entanglement. Preparation of PEPS-like resource states for MBQC remains an experimental goal, with challenges including state preparation fidelity, coherent control of many-body interactions, and mitigation of decoherence. Scalability is constrained by noise and the difficulty of performing full state tomography in large systems; proposals often use local observables, entanglement witnesses, or tomography of reduced density matrices. Collaborative efforts between experimental groups at institutions such as Harvard University, MIT, Institute for Quantum Information and Matter (IQIM), and national labs seek to translate PEPS theory into demonstrable quantum advantage and socially equitable access to quantum technologies, advocating for open scientific resources and inclusive research programs.
Category:Tensor network states Category:Quantum many-body theory