Monte Carlo simulation with tensor network states

Abstract

We demonstrate that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with a large virtual bond dimension. We use the simple update rule introduced by Xiang et al. in Phys. Rev. Lett 101, 090603 (2008) to obtain the tensors describing the ground state wavefunction of the Antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of linear sizes up to L = 16 and virtual bond dimensions up to D = 16. The finite size magnetization errors are 0.003(2) and 0.013(2) at D = 16 for a system of size L = 8, 16 respectively. Finite D extrapolation provides exact finite size magnetization for L = 8, and reduces the magnetization error to 0.005(3) for L = 16, significantly improving the previous state of the art results.The efficient simulation of strongly correlated quantum many body systems presents one of the major open problems and challenges in condensed matter physics. A major step forward was made by Steven White 2 in the case of 1 dimensional quantum spin chains by introducing the density matrix renormalization group (DMRG), which soon became the method of choice for simulating 1 dimensional manybody systems at zero temperature. By reformulating DMRG as a variational method within the class of matrix product states (MPS) 3-5 , it has become clear how DMRG can be generalized to deal with systems in two dimensions 6,7 ; the quantum states of the corresponding variational class are known as projected entangled pair states (PEPS) and are part of the class called tensor product states which also includes the multiscale entanglement renormalization ansatz 8 and infinite PEPS 9 . More recently, it has also been demonstrated how the PEPS class can take into account fermionic anti-commutation relations 10-16 . Numerical algorithms based on these ansatze, such as variational minimization of the ground state energy and imaginary time evolution are also developing fast 1,7,9,17,18 , and a wide range of applications has been studied 19-27 .The computational complexity of algorithms based on the PEPS ansatz with virtual bond dimension D scales as D 12 for the finite PEPS algorithm with open boundary condition 17 , χ 3 D 4 for the infinite PEPS (iPEPS) algorithm 9 , χ 6 for the tensor entanglement renormalization (TERG) algorithm for square lattices 18 and χ 5 for honeycomb lattices 1,18 , where χ is the number of Schmidt coefficients kept in the various approximations. The large scaling power presents the main bottleneck in scaling up the number of variational parameters, which is necessary near second order phase transitions 28 . The common characteristics of all these algorithms is that the tensor network is always contracted over the physical indices, which effectively squares the computational cost of contracting the tensor network as compared to a tensor network corresponding to a classical spin system. As first shown in 29,30 for the case of matrix product states and str...

Gradient optimization of finite projected entangled pair states

“…The computational scaling for PBC is even higher with O(D 18 ). 21 To reduce the high scaling of the methods, Monte Carlo sampling techniques have been introduced by several groups 11,19,20 in which the contraction over physical indices is replaced by the MC sampling over the "spin configurations". In this algorithm, the energy is reexpressed as follows,…”

“…However, even though the MC sampling method has been used to calculate some physical quantities for a given PEPS, 11 it has not been applied to optimize PEPS wave functions themselves.…”

Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

“…The Monte Carlo sampling techniques have been proposed as a prescription to reduce the computational cost of tensor network contraction in the MPS 29 , PEPS 30 and MERA 31 . Another advantage of employing the Monte Carlo sampling into tensor network methods is that it is possible to choose various types of suitable reference basis beyond the real space basis.…”

Time evolution of projected entangled pair states in the single-layer picture

“…The first step in this direction was made by not calculating the environment at all but letting it evolve in the process of imaginary time evolution [18]. Such an approach allows to achieve very large bond dimensions [19] but nevertheless seems to require a good initial approximation and works best for translation invariant PEPS states.…”

“…The approach allows for high bond dimensions D = 10 or more, although the calculation of energy and other observables still requires the full calculation with the double layer tensor network structure. Alternatively, quantum monte carlo sampling can be used to calculate the energy and expectation values of tensor networks with a large bond dimension [19][20][21].…”