A class of quantum many-body states that can be efficiently simulated

TL;DR

The multiscale entanglement renormalization ansatz is introduced, a class of quantum many-body states on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that the expectation value of local observables can be computed exactly and efficiently.

Abstract

We introduce the multi-scale entanglement renormalization ansatz (MERA), an efficient representation of certain quantum many-body states on a D-dimensional lattice. Equivalent to a quantum circuit with logarithmic depth and distinctive causal structure, the MERA allows for an exact evaluation of local expectation values. It is also the structure underlying entanglement renormalization, a coarse-graining scheme for quantum systems on a lattice that is focused on preserving entanglement.

Figures (5)

• Figure 1: Quantum circuit ${\cal C}$ transforms state $\hbox{$| 0 \rangle$}^{\otimes N}$ into the $N$-site state $\hbox{$| \Psi \rangle$}$ of a 1D lattice ${\cal L}$. ${\cal C}$ contains $2N-1$ gates organized in $O(\log N)$ layers labeled by a discrete time $\theta$.
• Figure 2: A MERA ${\cal M}$ inherits the causal structure of quantum circuit ${\cal C}$: the causal cone ${\cal C}^{[s]}$ for site $s$ has bounded width. When reversing the arrow of time $\theta$, ${\cal M}$ implements entanglement renormalization transformations.
• Figure 3: Detail of a causal cone in a 2D MERA, implying that its width is bounded. $3\times 3$ sites (bottom left) are mapped into $3\times 3$ sites (top right) by layers of tensors that act along the $y$ and $x$ direction [some stages involve $3\times 4$ sites].
• Figure 4: Efficient computation of $\rho^{[s]}$. ($i$) Part of the causal cone ${\cal C}^{[s]}$ of $\rho^{[s]}$, $D=1$. ($ii$) How to compute $\sigma_{2k+1}$ from $\sigma_{2k-1}$ through simple tensor multiplications ($iii$) and ($iv$).
• Figure 5: Causal cones ${\cal C}^{[s_1s_2]}$ and ${\cal C}^{[B_l]}$ for a two-site density matrix $\rho^{[s_1 s_2]}$ (left) and an $l$-site density matrix $\rho^{[B_l]}$ (right).