Tensor network methods with graph enhancement

We present applications of the renormalization algorithm with graph enhancement (RAGE). This analysis extends the algorithms and applications given for approaches based on matrix product states introduced in [Phys. Rev. A 79, 022317 (2009)] to other tensor-network states such as the tensor tree states (TTS) and projected entangled pair states. We investigate the suitability of the bare TTS to describe ground states, showing that the description of certain graph states and condensed-matter models improves. We investigate graph-enhanced tensor-network states, demonstrating that in some cases (disturbed graph states and for certain quantum circuits) the combination of weighted graph states with TTS can greatly improve the accuracy of the description of ground states and time-evolved states. We comment on delineating the boundary of the classically efficiently simulatable states of quantum many-body systems.

Figure 1

Subcubic tree representing a six-qubit quantum state.

Figure 2

Tensor network representation of MPS with open and closed-boundary conditions.

Figure 3

Tree graph state (top part) and specific arrangement (or labeling) of the qubits (bottom part) for an MPS description of the state. Qubits of the graph state and graph state interactions are shown in dark colors, while the MPS tree and points representing matrices are shown in light colors. One bipartitioning is shown. The entropy in this case, according to the rules described in the text, is 2 ebits.

Figure 4

Spin-glass, 16 sites. Depicted is the achieved accuracy when approximating the ground state with MPS and TTS. We find that for a small number of parameters used in the description (i.e., to signify the tensor entries), the subcubic tree describes a system better than the MPS. The situation is reversed for a large number of parameters.

Figure 5

Modified Ising 1D, open chain, length 24, transverse field $B=1$, ${\sigma }_z{\sigma }_z$ coupling with strength $1/r$. Depicted is the achieved accuracy when approximating the ground state with MPS and TTS. Although the MPS are not suited to carry long-range entanglement, they perform better than the subcubic tree, due to the fact that the tensor tree ansatz breaks the (translational) symmetries of the Hamiltonian explicitly.

Figure 6

The discrete Fourier transform is implemented by a sequence of Hadamard gates and controlled-phase gates with a rotation angle for a gate between qubits $i$ and $j$ of $\pi /2^{i-j}$. Time increases from left to right.

Figure 7

2D Ising model with $B=2$ on a $4\times 4$ periodic lattice. We compare the achieved accuracy within the RAGE approach and for MPS. The graph depicts the energy for different total numbers of complex parameters in comparison with the exact ground state as well as the first excited state. In Figs. 7, 8, and 9 the MPS is realized numerically as a flat TTS; that is, it is nonperiodic. This implies that tensors close to the leaves of the tree change the local bases only and are hence redundant.

Figure 8

2D Ising model on a $4\times 4$ periodic lattice. We compare the achieved accuracy using RAGE and MPS ($D=4$) with exact results. The total number of independent parameters is 384 for MPS and 476 for the RAGE state. Two-point correlations as function of $B$ are shown. The error bars stem from differences in the local two-point correlations due to the explicit breach of translational symmetry in the necessarily linear topology of the MPS ansatz.

Figure 9

2D Heisenberg model on a $6\times 6$ periodic lattice. We compare the achieved accuracy of bare tensor-network states with RAGE states. Also, values of the ground-state energy and the first excited energy from quantum Monte Carlo simulations are shown (Ref. 61). The number of computational steps for the evaluation of expectation values is, as shown in the main text, $O\left(N{D}^5\right)$ and $O\left(N{D}^6\right)$ for MPS and TTS, respectively; the combination of these states with WGS amounts to a change by a constant factor. The optimization, in turn, relies on a constant number of evaluations of expectation values per single optimization step. The number of such steps needed for satisfactory convergence is much higher for RAGE states than bare MPS and TTS as the state space is different.

Figure 10

Kitaev model on a periodic 2D lattice with $N=12$ with magnetic field $B$ in the $x$ direction. We compare the achieved accuracy for the ground state using MPS ($D=20$ and $D=10$) and RAGE states with fixed phases and underlying MPS of $D=1,2,3$, where $D=1$ corresponds to WGS. For $B=0$, the ground state is exactly described by a WGS. For comparison, the first excited state is plotted (E1). For larger $N$, similar results are found, although the exact treatment was no longer possible.

Figure 11

Comparison of MPS and RAGE with $D=2$ for the simulation of a random quantum circuit on $N=14$ qubits. An application of a random local phase gate followed by a random controlled-phase gate with random uniform phase in $[0,2\pi )$ constitutes one block. For given $k$ we apply this block $k$ times to a randomly chosen initial MPS state. Five hundred such runs are determined, and in each the fidelity with the exact state is computed. The average over 500 realizations is then plotted.

Figure 12

A bipartition of a TTS along an edge, leading to two simply connected substates. The Schmidt decomposition can be inferred directly from the tensors adjacent to the edge.

Figure 13

The graph corresponding to a two-dimensional PEPS shows a higher connectivity than the graph corresponding to an MPS.

Figure 14

Schematic representation of a PEPS contraction. The top layer corresponds to the ket PEPS and the lower layer corresponds to the bra PEPS. The dashed ellipse denotes the tensor structure which is contracted first, resulting in a single-layered rectangular tensor grid. The resulting rectangular grid is then contracted using MPS methods.