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Acta Physica Sinica

Optimization of the projected entangled pair state algorithm for quantum systems

李生好 LI Sheng-hao, Xiaobing Wu, Huang Chong-Fu, Honglei Wang

Published Jan 1, 2014• 1 citation
Publisher

Abstract

In the numerical calculation, the projected entangled pair state (PEPS) algorithm is the most important tensor network algorithm for two-dimensional strongly correlated electron quantum lattice system. In this paper, the optimization of PEPS for two-dimensional quantum system is discussed. An optimization connection between how to update the PEPS tensor and how to calculate the physical observable is investigated, for the tensor network algorithm based on the PEPS representation, which can greatly improve the utilization of computing resources. In this case, optimized PEPS algorithm, as a powerful tool, can be used to study quantum phase transitions and quantum critical phenomena in the thermodynamic limit of the two-dimensional strongly correlated electron quantum lattice system. Of course, optimization of PEPS algorithm program has many other applications, such as adding U(1) and SO(2) symmetry in PEPS algorithm, etc.

DOI
10.7498/aps.63.140501

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In the numerical calculation, the projected entangled pair state (PEPS) algorithm is the most important tensor network algorithm for two-dimensional strongly correlated electron quantum lattice system. In this paper, the optimization of PEPS for two-dimensional quantum system is discussed. An optimization connection between how to update the PEPS tensor and how to calculate the physical observable is investigated, for the tensor network algorithm based on the PEPS representation, which can greatly improve the utilization of computing resources. In this case, optimized PEPS algorithm, as a powerful tool, can be used to study quantum phase transitions and quantum critical phenomena in the thermodynamic limit of the two-dimensional strongly correlated electron quantum lattice system. Of course, optimization of PEPS algorithm program has many other applications, such as adding U(1) and SO(2) symmetry in PEPS algorithm, etc.
Projected entangled pair states (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. However, due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction and discuss how purifications can be used in the latter. On the other hand, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to $21\ifmmode\times\else\texttimes\fi{}21$ sites.
The Tensor Network (TN) algorithm for strongly related electronic quantum lattice systems has become the uppermost algorithm for studying strongly related electronic quantum lattice systems in numerical computation. Focusing on the TN algorithm based on 2-D spin systems, this paper discusses the theoretical principles of the Projected Entangled Pair States (PEPS) algorithm and its implementation process. It also explores the infinite PEPS (iPEPS) algorithm for simulating the spin-1/2 infinite 2-D square lattice quantum XYX model. The critical phenomena of the two-dimensional XYX quantum system are analyzed from the perspectives of ground-state fidelity and order parameters, and the quantum critical point is identified. It is anticipated that with the further development of TN algorithms, the single-site fidelity theory of quantum phase transitions (QPTs) can be applied to various quantum many-body systems in condensed matter physics, thereby deepening our understanding of the extremely rich states of matter in the quantum world. Research indicates that quantum information theory provides a universal method for the study of QPTs, which can identify several ground-state fidelities and order parameters through the ground-state wave function of the system, several ground state fidelity and local order parameters can be identified to fully describe some quantum phase transition points, greatly enriching the traditional Landau theory of symmetry breaking QPTs.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
Projected entangled pair states (PEPS) are a class of quantum many-body states that generalize matrix product states for one-dimensional systems to higher dimensions. In recent years, PEPS have advanced understanding of strongly correlated systems, especially in two dimensions, e.g., quantum spin liquids. Typically described by tensor networks on regular lattices (e.g., square, cubic), PEPS have also been adapted for irregular graphs; however, the computational cost becomes prohibitive for dense graphs with large vertex degrees. In this paper, we present a PEPS algorithm to simulate low-energy states and dynamics defined on arbitrary, fluctuating, and densely connected graphs. We introduce a cut-off, κ ∈ N , to constrain the vertex degree of the PEPS to a set but tunable value, which is enforced in the optimization by applying a simple edge-deletion rule, allowing the geometry of the PEPS to change and adapt dynamically to the system's correlation structure. We benchmark our flexible PEPS algorithm with simulations of classical spin glasses and quantum annealing on densely connected graphs with hundreds of spins, and also study the impact of tuning κ when simulating a uniform quantum spin model on a regular (square) lattice. Our paper opens the way to apply tensor network algorithms to arbitrary, even fluctuating, background geometries.
We adapt and optimize the projected-pair-entangled-state (PEPS) algorithm on finite lattices (fPEPS) for two-dimensional Hubbard models and apply the algorithm to the Hubbard model with nearest-neighbor hopping on a square lattice. In particular, we formulate the PEPS algorithm using projected entangled pair operators, incorporate SU(2) symmetry in all tensor indices, and optimize the PEPS using both iterative-diagonalization-based local bond optimization and gradient-based optimization of the PEPS. We discuss the performance and convergence of the algorithm for the Hubbard model on lattice sizes of up to $8\ifmmode\times\else\texttimes\fi{}8$ for PEPS states with U(1) symmetric bond dimensions of up to D=8D=8D=8 and SU(2) symmetric bond dimensions of up to D=6D=6D=6. Finally, we comment on the relative and overall efficiency of schemes for optimizing fPEPS.
Whether two-dimensional tensor networks, such as the so-called projected entangled pair states (PEPS), have the potential to describe a genuine quantum critical point or quantum critical phases separating two ordered phases is of much interest for the understanding of frustrated magnets and strongly correlated systems. Here, the authors identify a specific family of SU(2)-invariant PEPS, which provides excellent variational energies for the J1−J2{J}_{1}-{J}_{2}J1​−J2​ frustrated Heisenberg model at large frustration. Using a full-update infinite-PEPS scheme directly in the thermodynamic limit, the authors provide evidence of diverging correlation lengths in both the singlet and triplet channels. It is argued that such PEPS give a qualitative description of the critical point or critical phase of the J1−J2{J}_{1}-{J}_{2}J1​−J2​ model.