Simulation of time evolution with multiscale entanglement renormalization ansatz

We describe an algorithm to simulate time evolution using the multiscale entanglement renormalization ansatz and test it by studying a critical Ising chain with periodic boundary conditions and with up to $L\approx {10}^6$ quantum spins. The cost of a simulation, which scales as $L{\text{log}}_2\left(L\right)$, is reduced to ${\text{log}}_2\left(L\right)$ when the system is invariant under translations. By simulating an evolution in imaginary time, we compute the ground state of the system. The errors in the ground-state energy display no evident dependence on the system size. The algorithm can be extended to lattice systems in higher spatial dimensions.

Figure 1

(Color online) (a) MERA tensor structure with periodic boundary conditions and $\ell =3$. (b) Schematic representation of the fidelity (3). Black lower (red upper) tensors represent the tensors inside the causal cone of $|\psi ⟩$ $\left(⟨\tilde{\psi }|\right)$. Blue tensors (at sides) are outside the causal cone, thus they are contracted for free (c). (d)–(e) Schematic representation of the expressions (4, 5).

Figure 2

(Color online) Convergence of the computed ground-state energy $E\left(L\right)$ for $L=2^{\ell },\ell =8,10,\dots ,20$ to the exact ground-state energy $E_{ex}\left(L\right)$ with periodic boundary conditions [20].

Figure 3

(Color online) Energy $E\left(L\right)$ as a function of the system size $\left(L=2^{\ell },\ell =8,\dots ,20\right)$ for the critical Ising model (blue circles), $m=4$, $dt=0.1$, and $T_f=800$. Red full line represents the exact thermodynamical limit $E_{\infty }^{ex}=-4/\pi$. Inset: Difference of the computed energy with respect to thermodynamical limit $\Delta E=E\left(L\right)-E_{\infty }^{ex}$. The red line represents the exact scaling.

Figure 4

(Color online) (a) Error as a function of the system size $L$ of the computed energy with respect to the exact one $\delta E=E\left(L\right)-E_{ex}\left(L\right)$, $m=4$, $dt=0.1$, and $T_f=800$ [20]. (b) Error $\delta E$ as a function of $m$ for $L=32$, $dt=0.1$, and $T_f=800$. The dashed green line is an exponential fit.