PEPS

Official Resources

• Homepage: https://github.com/QuantumLiquids/PEPS
• Source Repository: https://github.com/QuantumLiquids/PEPS
• License: Open Source (Header-only core)

Overview

PEPS (Projected Entangled Pair States) by the QuantumLiquids group is a high-performance C++ library for simulating 2D strongly correlated electron systems. It implements the PEPS tensor network ansatz and uses a Variational Monte Carlo (VMC) approach to optimize the tensor elements. This combination allows for the rigorous study of 2D fermionic systems (fPEPS) that are challenging for traditional QMC due to the sign problem.

Scientific domain: 2D Quantum Lattice Models, Tensor Networks, Strongly Correlated Electrons Target user community: Researchers studying high-Tc superconductivity, topological order, and 2D magnetism

Theoretical Methods

• Projected Entangled Pair States (PEPS): A 2D generalization of MPS that naturally captures Area Law entanglement.
• Variational Monte Carlo (VMC): stochastic sampling to evaluate the contraction of the norm $\langle\psi|\psi\rangle$ and observables, avoiding the high cost of exact contraction.
• Stochastic Reconfiguration (SR): Optimization method (similar to Natural Gradient) to update variational parameters.
• Fermionic PEPS (fPEPS): Explicit handling of fermion statistics (swap gates).

Capabilities (CRITICAL)

• Fermionic Systems: Native support for fPEPS, crucial for the Hubbard and t-J models.
• Optimization: State-of-the-art VMC optimizers including Adam and Stochastic Reconfiguration.
• Custom Models: Plugin architecture for defining new Hamiltonians and lattice geometries.
• Observables: Built-in measurement toolkit for energy, correlation functions, and structure factors.
• Header-Only: Easy integration into other C++ projects.

Key Features

Performance:

• C++20: modern C++ implementation for efficiency and safety.
• Parallelism: MPI support for parallelizing the Monte Carlo sampling steps.
• Math Libraries: Links against high-performance BLAS (Intel MKL / OpenBLAS).

Flexibility:

• Lattices: Built-in support for Square and other common 2D lattices.
• Extensible: Plugin interfaces for Models and Updaters.

Inputs & Outputs

• Input formats:
  • C++ driver files or configuration scripts.
• Output data types:
  • Energy logs, optimized tensor parameters.
  • Measured observables (Spin/Charge correlations).

Interfaces & Ecosystem

• Dependencies: C++20 compiler, CMake, BLAS/LAPACK, MPI.
• Integration: Part of the QuantumLiquids methods suite (TensorToolkit).

Workflow and Usage

Users define the model (e.g., Heisenberg or Hubbard) and the PEPS ansatz (bond dimension D) in C++. The VMC optimizer is then run to minimize the energy.

// Example conceptual usage
auto model = HubbardModel(Lx, Ly, U);
auto peps = PEPSState(Lx, Ly, D);
VMCOptimizer optimizer(peps, model);
optimizer.run();

Performance Characteristics

• Cost: Contraction is #P-hard, but VMC sampling makes it manageable ($O(D^6)$ or similar depending on update scheme).
• Scalability: Excellent scaling with Monte Carlo samples via MPI.

Comparison with Other Codes

Verification & Sources

Primary sources:

1. GitHub Repository: https://github.com/QuantumLiquids/PEPS
2. Recent arXiv preprints on fPEPS using VMC/SR methods (e.g. from the authors).

Verification status: ✅ VERIFIED

• Source code: OPEN (GitHub)
• State: Active development.