Multiscale Entanglement Renormalization Ansatz Research Articles

Class of Quantum Many-Body States That Can Be Efficiently Simulated

We introduce the multiscale entanglement renormalization ansatz, 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. The multiscale entanglement renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel coarse-graining scheme for many-body quantum systems on a lattice.

Multiscale Entanglement Renormalization Ansatz in Two Dimensions: Quantum Ising Model

We propose a symmetric version of the multiscale entanglement renormalization ansatz in two spatial dimensions (2D) and use this ansatz to find an unknown ground state of a 2D quantum system. Results in the simple 2D quantum Ising model on the 8x8 square lattice are found to be very accurate even with the smallest nontrivial truncation parameter.

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 $L\ensuremath{\approx}{10}^{6}$ quantum spins. The cost of a simulation, which scales as $L\text{ }{\text{log}}_{2}(L)$, is reduced to ${\text{log}}_{2}(L)$ 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.

Entanglement Renormalization and Topological Order

The multiscale entanglement renormalization ansatz (MERA) is argued to provide a natural description for topological states of matter. The case of Kitaev's toric code is analyzed in detail and shown to possess a remarkably simple MERA description leading to distillation of the topological degrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are also shown to be a fixed point of the renormalization group flow associated with entanglement renormalization. All of these results generalize to arbitrary quantum double models.