Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment

We propose an environment recycling scheme to speed up a class of tensor network algorithms that produce an approximation to the ground state of a local Hamiltonian by simulating an evolution in imaginary time. Specifically, we consider the time-evolving block decimation (TEBD) algorithm applied to infinite systems in 1D and 2D, where the ground state is encoded, respectively, in a matrix product state (MPS) and in a projected entangled-pair state (PEPS). An important ingredient of the TEBD algorithm (and a main computational bottleneck, especially with PEPS in 2D) is the computation of the so-called environment, which is used to determine how to optimally truncate the bond indices of the tensor network so that their dimension is kept constant. In current algorithms, the environment is computed at each step of the imaginary time evolution, to account for the changes that the time evolution introduces in the many-body state represented by the tensor network. Our key insight is that close to convergence, most of the changes in the environment are due to a change in the choice of gauge in the bond indices of the tensor network, and not in the many-body state. Indeed, a consistent choice of gauge in the bond indices confirms that the environment is essentially the same over many time steps and can thus be re-used, leading to very substantial computational savings. We demonstrate the resulting approach in 1D and 2D by computing the ground state of the quantum Ising model in a transverse magnetic field.

Figure 1

Diagrammatic representations of an MPS corresponding to different boundary conditions: (a) The MPS with OBC. (b) The MPS with PBC. In these representations, a tensor is represented by a circle which has legs corresponding to the indices of the tensor. The links connecting tensors together represent the contraction over bond indices. The open legs of the tensors correspond to the physical indices, which are not shared between tensors in the network.

Figure 2

Diagrammatic representations of PEPSs that correspond to different lattice patterns: (a) a triangular lattice, (b) a hexagonal lattice, and (c) a square lattice.

Figure 3

(i)–(iii) Fundamental steps applied to transfer an iMPS $\{\Gamma ,\lambda \}$ into its canonical form. These steps are iterated until the Schmidt coefficients converge. (iv) Two iMPSs $\{\Gamma ,\lambda \}$ and $\{\tilde{\Gamma },\tilde{\lambda }\}$ are different from each other but they represent the same physical state $|\Psi \rangle$.

Figure 4

Diagrammatic representation of an infinite one-site translationally invariant iPEPS, which consists of tensors $\{{\lambda }_h,{\lambda }_v,\Gamma \}$.

Figure 5

Diagrammatic representation of constraints for the canonical iPEPS.

Figure 6

Diagrammatical illustration of how to transfer an iPEPS into its canonical form.

Figure 7

Diagrammatic illustration for constructing an iMPO. (a) The imaginary-time evolution operator is decomposed into a product of nearest-neighbor interaction terms denoted by $g_I$ and magnetic field terms named $g_F$. (b) Perform the SVD of the interaction operator $g_I=USV$. (c) The iMPO is obtained by contracting all the tensors $g_F,U,V,S$ together.

Figure 8

Plots of relative errors in the local expectation values regarding the number of times $i_{\mathrm{Re}}$ that the environment is calculated. The upper plot (a) shows the errors in the energy per link where $E_{\mathrm{exact}}\approx -1.306856$. The bottom plot (b) shows the errors in the local magnetization per site where $m_{z\text{exact}}=0$. The results for different $N_{\mathrm{Re}}$'s are obtained by simulating with the same initial iMPS of bond dimension $\chi =50,i_{\mathrm{Re}}$ is counted from one, the transverse magnetic field $h=1.05$, time step $\delta =0.05$. The $N_{\mathrm{Re}}\to \infty$ lines correspond to the cases where the dominant eigenvalue of the renormalized gate is found.

Figure 9

Diagrammatic illustration for building an iPEPO. (a) The imaginary-time evolution operator is decomposed into two terms: the interaction terms denoted by $g_I$ in horizontal and vertical directions and the on-site magnetic field terms $g_F$. (b) A PEPO will be created from the horizontal and vertical contributions and the on-site magnetic field terms. (c) Perform the SVD of the interaction operators $g_I=USV$. (d) PEPO obtained from contracting tensors $g_F,U,V,S$. Each PEPO is a tensor consisting of two physical indices of dimension $d$ and four virtual indices ofdimension $\kappa$. (e) An iPEPO.

Figure 10

Compute the environment around a specific link using the CTM method. (a) An infinite square lattice $\mathcal{L}\left(a\right)$ is decomposed into two contiguous parts $\mathcal{A}$ and $\mathcal{B}$. The environment $\mathcal{A}$ around $\mathcal{B}$ is characterized by corner tensors $\{C_1,C_2,C_3,C_4\}$ and edge tensors $\{T{a}_1,T{a}_2,T{a}_3,T{a}_4,T{b}_1,T{b}_2,T{b}_3,T{b}_4\}$. (b) The environment around a specific link is represented by six tensors $\{E_1,E_2,E_3,E_4,E_5,E_6\}$.

Figure 11

Expectation value for the local magnetization per site $m_z$ as a function of the transverse magnetic field $h$. The result is obtained by using a one-site translationally invariant iPEPS simulating an imaginary time evolution with the iTEBD algorithm, both with SU and FU. The results close to the criticality are shown in the inset.

Figure 12

Plots of relative errors in the local expectation values regarding the number of times $i_{\mathrm{Re}}$ that the environment is calculated. The upper plot (a) shows the errors in the energy energy per link where $E_0\approx -1.619581$. The bottom plot (b) shows the errors in the local magnetization per site where $m_{z_0}\approx 0.240717$. The results for different $N_{\mathrm{Re}}$'s are obtained by simulating with the same initial iPEPS of bond dimension $D=3,\chi =30,i_{\mathrm{Re}}$ is counted from one, the transverse magnetic field $h=3.05$ and time step $\delta =0.005$.

Figure 13

Plots of the elapsed time measured in terms of the number of environment recyclings $N_{\mathrm{Re}}$. The system is evolved with the same amount of time $T=12.05$ at time step $\delta =0.01$ and $h=3.01$. The plots are shown for different iPEPS bond dimensions where the exponential fitting curves are added to observe the abrupt decay in computational time when $N_{\mathrm{Re}}$ increases. The inset shows the relative errors in the energy per link obtained after $T$ compared between $N_{\mathrm{Re}}=1$ and the others in each case of bond dimension where $E_{1D=2}\approx -1.601426$ and $E_{1D=3}\approx -1.601809$.