| projected entangled pair states | |
|---|---|
| Name | Projected Entangled Pair States |
| Introduced | 2004 |
| Authors | Frank Verstraete; J. Ignacio Cirac; Guifrè Vidal (related development) |
| Field | Quantum many-body physics |
| Related | Tensor network, Matrix product state |
projected entangled pair states
Projected entangled pair states (PEPS) are a family of tensor network ansätze used to represent quantum many-body wavefunctions on lattices in two or higher dimensions. They provide a compact description of entanglement compatible with locality and are important for studying strongly correlated systems, quantum phases, and scalable simulation methods in Quantum physics.
Projected entangled pair states were introduced to generalize one-dimensional matrix product state (MPS) representations to higher-dimensional lattices while preserving a physically motivated structure of local tensors and virtual entangled pairs. The basic picture envisions placing maximally entangled virtual pairs on edges of a lattice and applying local linear maps (projections) at each site to produce a physical Hilbert space, giving an explicit construction of states that obey expected locality and entropic constraints. PEPS are motivated by the desire to capture ground states of local Hamiltonians such as the Heisenberg model and Hubbard model beyond one dimension, where exact diagonalization and perturbative methods fail.
Formally, a PEPS on a graph assigns a tensor A_i with one physical index and several virtual indices to each lattice site i. Virtual indices of neighboring tensors are contracted along edges to form the network; the bond dimension D of virtual indices controls the representational power. The resulting wavefunction is a multilinear map from virtual spaces to the global physical Hilbert space. The construction extends concepts from tensor network theory and relates to notions in quantum information theory such as entanglement entropy and Schmidt decompositions. Concrete examples include square-lattice PEPS used to approximate ground states of the two-dimensional Hubbard model and variational states for topologically ordered phases like the Kitaev toric code.
PEPS naturally encode states that satisfy an entanglement area law: the entanglement entropy of a region scales with its boundary rather than its volume, matching proven and conjectured behavior of gapped local Hamiltonian ground states in condensed matter. This makes PEPS particularly suitable for describing low-energy sectors of models such as the quantum Ising model and quantum spin liquids. PEPS can also represent states with nontrivial topological order and symmetry-protected topological phases; examples constructed from PEPS include resonating valence bond (RVB) states and string-net models related to the work of Xiao-Gang Wen and Michael Levin.
Practical use of PEPS requires algorithms for contracting tensor networks, optimizing tensor entries, and computing observables. Exact contraction is #P-hard in general, so approximate schemes are used: boundary MPS methods, corner transfer matrix renormalization, and tensor renormalization group approaches inspired by Leo P. Kadanoff and Kenneth G. Wilson's renormalization ideas. Variational energy minimization, imaginary-time evolution, and gradient-based optimizers are employed; numerical implementations often rely on careful exploitation of symmetries (e.g., SU(2) or lattice point-group symmetries) and use libraries developed in academic groups at institutions such as Max Planck Institute for Quantum Optics and University of Vienna. Benchmarks compare PEPS to other methods like quantum Monte Carlo (where sign problems are relevant) and density matrix renormalization group (DMRG) for finite-width systems.
PEPS have been applied to study phase diagrams of strongly correlated models including the t-J model, the Kagome lattice antiferromagnet, and frustrated spin systems where conventional Monte Carlo fails. In quantum information, PEPS provide constructive examples of resources for measurement-based quantum computation; cluster states and resource states can be represented as PEPS, linking to works at IQC (Institute for Quantum Computing) and Perimeter Institute. PEPS constructions also enable rigorous proofs of parent Hamiltonians and of properties like spectral gaps for engineered models. In quantum simulation, PEPS inform proposals for cold-atom and trapped-ion emulation of lattice Hamiltonians and guide variational quantum-classical hybrid algorithms.
PEPS are one member of a broader family of tensor networks. In one dimension they reduce to matrix product states which underpin the density matrix renormalization group (DMRG) method developed by Steven R. White. Multiscale entanglement renormalization ansatz (MERA) by Guifrè Vidal addresses scale invariance and criticality, offering complementary strengths. Projected entangled pair operators (PEPO) generalize PEPS to operators such as thermal density matrices. Other related constructions include tensor product states (TPS), string-bond states, and entanglement renormalization networks used across condensed matter and high-energy contexts, with cross-fertilization from groups at Harvard University, ETH Zurich, and University of California, Berkeley.
Key limitations of PEPS are computational cost (contraction scaling with bond dimension and system size) and the lack of universally efficient optimization procedures. Open problems include rigorous characterization of expressive power given finite bond dimension, algorithms for reliable contraction with controlled errors, and extensions to real-time dynamics and finite-temperature states. Another front is connecting PEPS descriptions to experimentally observable signatures in materials and cold-atom platforms; bridging theory and practice requires collaboration between theoretical groups and experimental laboratories such as CERN-adjacent quantum initiatives and national laboratories. Establishing scalable, stable software ecosystems and standards for reproducible PEPS computations remains a community challenge.
Category:Quantum many-body physics Category:Tensor network states