Projected Entangled Pair States (PEPS)

• PEPS are a class of tensor network states that efficiently represent quantum many-body wavefunctions, naturally encoding area-law entanglement, topological order, and symmetry.
• They are constructed from rank-(z+1) tensors with virtual degrees of freedom whose contraction mirrors lattice connectivity and embeds both physical and virtual symmetries.
• Algorithmic advances in PEPS enable approximate contraction and systematic quantum phase classification while addressing computational complexity challenges in high-dimensional systems.

Projected Entangled Pair States (PEPS) are tensor network states that enable the systematic and scalable representation of quantum many-body wavefunctions in two and higher spatial dimensions. PEPS generalize matrix product states (MPS) from one dimension, providing an efficient description of states that obey area laws of entanglement, and naturally encode a variety of physical phenomena including symmetry breaking, topological order, criticality, and gauge invariance.

1. Mathematical Formalism and Symmetry Principles

A PEPS is constructed by assigning a rank-$(z + 1)$ tensor $A^{s}_{\alpha_1 ... \alpha_z}$ to each lattice site, where %%%%2%%%% is the lattice coordination number, $s$ indexes the physical Hilbert space, and $\alpha_j$ label the virtual degrees of freedom connecting neighboring sites. The physical many-body wavefunction is given as a network contraction over all virtual indices: $$|\Psi\rangle = \sum_{ \{ s_j \} } \mathrm{tr} \left( \bigotimes_j A^{s_j}_j \right) | s_1 s_2 ... s_N \rangle.$$ The contraction structure (i.e., the pattern of virtual connections) reflects the connectivity of the underlying lattice or, more generally, interaction graph (Patra et al., 30 Jul 2024).

Symmetry properties of PEPS can be embedded both at the physical and virtual levels. For a finite group $G$ acting as a symmetry, PEPS enable the virtual implementation of $G$ via representations carried by the virtual bonds. For more advanced constructions, such as those classifying topological phases, PEPS can be twisted by a 3-cocycle $\omega\in Z^3(G,\mathrm{U}(1))$, producing a $(G, \omega)$-injective structure in which the virtual symmetry is effected by a matrix product operator (MPO) carrying additional $\omega$-dependent phase factors (Buerschaper, 2013).

2. Classification of Quantum Phases via Twisted Injectivity

A central application of PEPS is the characterization of two-dimensional gapped quantum phases, including the emergence of intrinsic topological order. In the $(G, \omega)$-injective PEPS framework (Buerschaper, 2013), the local tensor satisfies

$$A = A \cdot \mathcal{P}^{\omega},\quad \mathcal{P}^{\omega} = \frac{1}{|G|} \sum_{g \in G} V^{\omega}(g),$$

where $V^{\omega}(g)$ is an MPO representation of $G$ twisted by the cocycle $\omega$ and $\mathcal{P}^{\omega}$ acts as a projection onto the twisted-symmetric subspace. (Injectivity in this subspace ensures the viability of a unique ground state for a gapped parent Hamiltonian.)

The parent Hamiltonian constructed from such PEPS takes the form

$$H = \sum_{v} h_{v},\quad \mathrm{ker}(h_{v}) = \mathcal{L}_v,$$

where $\mathcal{L}_v$ is the local subspace defined by the PEPS tensors. At the so-called isometric point ($A = \mathcal{P}^{\omega}$), the Hamiltonian becomes a sum of commuting projectors, is frustration-free, and gapped.

The universality class of these Hamiltonians is determined entirely by the cohomology class $[\omega]$ of the 3-cocycle, directly linking PEPS descriptions to Dijkgraaf-Witten topological quantum field theories (TQFT). Specifically, the set of topological invariants such as ground state degeneracy on a torus and braiding data are in one-to-one correspondence with the Dijkgraaf-Witten TQFT of $(G,\omega)$. PEPS tensors with twists $\omega$ and $\omega'$ from the same cohomology class generate phase-equivalent states, connected by local unitaries.

3. Chiral Topological Order and the Limitations of Local Parent Hamiltonians

The original PEPS framework naturally described nonchiral topological phases. Extensions to chiral topological order—states with non-trivial Chern number—have been achieved with fermionic PEPS, e.g., Gaussian fermionic PEPS (GFPEPS) (Wahl et al., 2013, Yang et al., 2014). These GFPEPS are constructed from entangled virtual Majorana modes, and the mapping from virtual to physical space is encoded in a block-structured Gaussian map, yielding ground states with Chern numbers $C\ne 0$.

A key result is that injective GFPEPS—i.e., those whose local virtual-to-physical map is full rank after a block—cannot realize both strictly local, gapped, frustration-free parent Hamiltonians and nontrivial topology. For topological (Chern) phases, such GFPEPS either correspond to gapless local Hamiltonians or gapped Hamiltonians with only algebraically decaying (power-law) interactions. This constraint is fundamentally related to the impossibility of constructing exponentially localized Wannier functions in chiral topological bands.

Nonetheless, numerical studies demonstrate that GFPEPS with moderate bond dimension can approximate physical observables (e.g., Hall conductivity, von Neumann entropy) of topological insulators to exponential accuracy, making them a practical variational tool for strongly correlated chiral phases (Wahl et al., 2013, Yang et al., 2014).

4. Algorithmic Advances and Computational Complexity

The contraction of a PEPS network, required to evaluate normalization and expectation values, is generally a computationally hard (#P-hard) problem. Approximate contraction algorithms are therefore essential. The cluster update (CU) framework (Lubasch et al., 2013) introduces variable cluster size $\delta$, interpolating between the simple update (minimal environment, low cost, limited accuracy) and full contraction (maximal environment, high cost, optimal accuracy). Contraction error decays exponentially with cluster size and is governed by the system’s physical correlation length (Lubasch et al., 2014): $\epsilon(\delta) \propto e^{-\delta/\zeta},$ where $\zeta$ is the correlation length.

Algorithmic improvements for finite PEPS include the use of purification-inspired contractions (enforcing positivity), reduced tensor updates, and gauge-fixing for numerical stability (Lubasch et al., 2014). Monte Carlo sampling of single-layer tensor contractions, combined with stochastic gradient optimization, extends feasible bond dimensions and enables calculations in highly entangled regimes, such as the frustrated $J_1$–$J_2$ model (Liu et al., 2016).

From a complexity-theoretic perspective, even basic questions such as the presence of a nonzero PEPS wavefunction, the enforcement of symmetries, and parent Hamiltonian gap problems are undecidable in full generality for arbitrary PEPS (Scarpa et al., 2018). This sets a fundamental barrier for systematic classification or canonicalization; practical approaches either restrict to tractable subclasses or employ approximate methods.

Beyond complexity theory, random tensor network methods reveal a computational complexity phase transition as bond dimension increases (Gonzalez-Garcia et al., 2023). Simulating global wavefunction amplitudes becomes intractable beyond a critical bond dimension, but the norm and local observables arising in physically relevant systems can be approximated efficiently at any finite bond dimension for area-law entangled states.

5. Extensions: Continuous Symmetries, Gauge Invariance, and Flexible Geometry

PEPS frameworks have been extended to continuous virtual symmetries, including SU(2), leading to parent Hamiltonians with nontrivial ground space degeneracies and logarithmic corrections to entanglement entropy. In such models, the ground state degeneracy on the torus can scale extensively, and the entanglement entropy acquires a logarithmic correction: $$S_0(\rho_Q) = L\log 2 - \frac{3}{2}\log (L/2) + \text{const} + O(1/L)$$ where $L$ is the boundary perimeter (Dreyer et al., 2018). These properties suggest a critical or exotic phase rather than a conventional gapped topological phase.

For gauge-invariant PEPS, local (gauge) symmetries are implemented by coupling “matter” tensors at vertices and “gauge field” tensors at edges (Blanik et al., 24 Oct 2024, Zohar et al., 2015). Injectivity and unitality conditions for edge tensors ensure that local gauge transformations are “pushed” onto virtual indices as unique invertible representations, enforcing the correct gauge structure on the virtual level: $$\widehat{U}_v(g) |\Psi(A,B)\rangle = |\Psi(A,B)\rangle,$$ with $\widehat{U}_v(g)$ acting on both matter and gauge field Hilbert spaces as a combination of left and right group representations. This explicit intertwiner structure allows for efficient gauge-invariant variational simulations relevant to nonperturbative lattice gauge theory.

Recent developments include:

• PEPS on arbitrary, possibly fluctuating, graphs via a flexible graph structure constrained by a tunable maximal vertex degree $\kappa$, with redundant edges dynamically pruned based on their bond entanglement entropy (Patra et al., 30 Jul 2024).
• Efficient PEPS methods for periodic quantum systems by superposing open-boundary PEPS under translation operators to restore periodicity and translation invariance while maintaining computational efficiency ($O(D^6)$ scaling) (Dong et al., 22 Jul 2024).
• Dual-isometric PEPS with two independent isometric constraints, allowing efficient contraction of local observables by mapping the problem to an effective 1D channel, while retaining a rich set of tunable parameters and the capacity to interpolate between topological and trivial phases (Yu et al., 25 Apr 2024).

6. Physical Applications: Order, Excitations, and Machine Learning

PEPS frameworks support the paper of symmetry breaking and order parameters through the structure of their transfer operators. For models with $\mathbb{Z}_2$ symmetry, long-range order is identified with the near-degeneracy of leading eigenvalues of the transfer matrix. The nature of spontaneous symmetry breaking under perturbations is encoded in the fixed point structure (r$_\uparrow$, r$_\downarrow$) formed from linear combinations of even and odd symmetry sectors. Notably, constructing the entanglement Hamiltonian from these symmetry-broken fixed points restores local character in gapped phases (Rispler et al., 2015).

Excitation spectra can be accessed variationally by locally substituting one tensor in an infinite PEPS and superposing the configuration over all lattice sites with a given momentum, reducing the excitation calculation to a generalized eigenvalue problem (Vanderstraeten et al., 2018).

PEPS have proven effective beyond quantum many-body physics, for example in generative modeling of datasets with intrinsic 2D structure (images). Here, the natural inductive bias of PEPS for two-dimensional correlations leads to improved generalization and sampling over one-dimensional MPS—with sampling and learning enabled by efficient single-layer contraction and direct sampling algorithms (Vieijra et al., 2022).

7. Outlook and Open Problems

Ongoing research seeks to further bridge PEPS theory, algorithmic scalability, and physical modeling. Open challenges include:

• Rigorous characterization of possible phases for PEPS with continuous symmetries and the connection to exotic or critical phases.
• Efficient, accurate contraction for large-bond-dimension PEPS and complex geometries, especially for systems exhibiting long correlation lengths or criticality.
• Extension of gauge-invariant PEPS to higher-rank gauge groups, inclusion of matter coupling, and application to real-time and finite-density lattice gauge simulations.
• Mitigating the analytical and numerical undecidability barriers identified for general PEPS by exploiting physically motivated subclasses or hybrid classical-quantum methods.
• Deepening the integration of PEPS with machine learning and quantum computational paradigms, including their use as resource states for measurement-based quantum computing.

Research from the past decade has definitively established PEPS as a fundamental tool for the analytical and numerical exploration of high-dimensional quantum many-body systems, offering a systematic, variationally controlled approach to ground states, topological order, criticality, and computational modeling. Significant open questions remain regarding their fundamental limitations and ultimate classification capabilities, especially in the context of undecidability results and new classes of exotic quantum order.