[Paper Review] Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions

Frank Verstraete, J. I. Cirac|arXiv (Cornell University)|Jul 2, 2004

Quantum many-body systems503 citations

TL;DR

This paper introduces Projected Entangled-Pair States (PEPS) as a natural extension of matrix product states to two and higher dimensions, enabling efficient variational simulation of quantum many-body systems. The authors develop a scalable algorithm for calculating correlation functions and applying it to ground state and imaginary time evolution, achieving high accuracy even with small bond dimensions D, as demonstrated on 2D Heisenberg and frustrated spin systems.

ABSTRACT

We describe quantum many--body systems in terms of projected entangled--pair states, which naturally extend matrix product states to two and more dimensions. We present an algorithm to determine correlation functions in an efficient way. We use this result to build powerful numerical simulation techniques to describe the ground state, finite temperature, and evolution of spin systems in two and higher dimensions.

Motivation & Objective

• To develop a scalable numerical method for simulating quantum many-body systems in two and higher dimensions, where traditional methods like DMRG and Monte Carlo face limitations.
• To generalize matrix product states (MPS) to higher dimensions using entangled pairs of auxiliary systems, leading to the concept of Projected Entangled-Pair States (PEPS).
• To enable efficient computation of correlation functions in PEPS, which is essential for variational methods and time evolution simulations.
• To demonstrate the effectiveness of PEPS-based algorithms in finding ground states and simulating real or imaginary time evolution of spin systems, including frustrated models.
• To establish a framework applicable to various geometries, finite temperatures, and open quantum systems with dissipation.

Proposed method

• Propose PEPS as a generalization of MPS, where each physical spin is formed by projecting entangled pairs of auxiliary systems with bond dimension D.
• Represent the quantum state as a tensor network with local tensors (projectors) mapping auxiliary degrees of freedom to physical spins, with entanglement between neighboring sites.
• Develop an efficient algorithm to compute correlation functions in PEPS by contracting tensor networks, enabling variational optimization.
• Implement a time-evolution algorithm using the Trotter decomposition, dividing time steps into substeps where only one interaction is active at a time.
• Reduce the bond dimension after each substep via optimal approximation, iteratively optimizing projectors row by row to maintain low computational cost.
• Use polynomial-time contraction techniques to handle the computational complexity, ensuring scalability with system size N and bond dimension D.

Experimental results

Research questions

• RQ1 Can matrix product states be generalized to two and higher dimensions to enable scalable simulation of quantum many-body systems?
• RQ2 Can efficient algorithms for computing correlation functions be developed within the PEPS framework to support variational methods?
• RQ3 Can PEPS-based algorithms accurately simulate ground states and imaginary time evolution of 2D spin systems, including frustrated models?
• RQ4 How does the accuracy of the PEPS approximation scale with increasing bond dimension D for systems like the 2D Heisenberg model?
• RQ5 Can the PEPS formalism be extended to finite-temperature states, real-time evolution, and systems with dissipation?

Key findings

• The PEPS formalism successfully extends DMRG-like variational methods to two and higher dimensions, enabling accurate simulations of strongly correlated systems.
• For a 4×4 Heisenberg antiferromagnet, the variational energy with D=3 is within 0.004% of the exact ground state energy, showing rapid convergence with D.
• Even with D=2, the PEPS algorithm achieves excellent agreement with exact results, with energy differences decreasing from 35% (D=1) to 2% (D=2).
• The algorithm converges quickly to the ground state, and the energy gap for the frustrated Heisenberg model is captured with increasing accuracy as D increases.
• Simulations on a 10×10 lattice show that the method scales well, with D=3 providing a significant improvement over D=2 in ground state energy estimation.
• The method is robust for both antiferromagnetic and frustrated interactions, demonstrating its generality and potential for broader applications.