Invariance under translation is exploited to efficiently simulate one-dimensional quantum lattice systems in the limit of an infinite lattice. Both the computation of the ground state and the simulation of time evolution are considered.
We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the projected entangled-pair state algorithm for finite lattice systems [F. Verstraete and J. I. Cirac, arxiv:cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite one-dimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)10.1103/PhysRevLett.98.070201]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analyzing its second order quantum phase transition.
We develop an efficient algorithm to find a matrix product state representation of the ground-state wavefunctions for translationally invariant finite-size periodic lattice systems in one spatial dimension. This is based on the observation that the efficient computation of the ground-state energy per site only needs to retain a certain number of the largest eigenvalues of the transfer matrix for a matrix product state, without any sacrifice of accuracy. The computational cost is independent of the system's size, and scales as χ 3 with χ being the truncation dimension. The algorithm is tested for the critical quantum Ising model in a transverse field on a finite-size lattice, with the size as large as 4800 for the truncation dimension 200.
We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.
Discrete translational symmetry plays a fundamental role in condensed matter physics and lattice gauge theories, enabling the analysis of systems that would otherwise be intractable. Despite this, many open problems remain. Quantum simulation promises to offer new insights, but progress is often limited by device connectivity constraints, which lead to prohibitively long computation times. We extend the use of spatial symmetry from the systems to be simulated to the quantum circuits simulating them. One application is that it becomes possible to efficiently and optimally alleviate device connectivity constraints algorithmically. This leads to reductions in quantum computational time by several orders of magnitude even for moderate system sizes, making such simulations feasible, with even greater relative gains for larger systems. This substantially enhances the capabilities of quantum computers in the simulation of condensed matter systems and lattice gauge theories, even before hardware improvements. Our Letter forms the basis for using spatial symmetry of quantum circuits in other areas of quantum computation, such as in the design and implementation of quantum error correcting codes.
We propose and test a scheme for entanglement renormalization capable of addressing large two-dimensional quantum lattice systems. In a translationally invariant system, the cost of simulations grows only as the logarithm of the lattice size; at a quantum critical point, the simulation cost becomes independent of the lattice size and infinite systems can be analyzed. We demonstrate the performance of the scheme by investigating the low energy properties of the 2D quantum Ising model on a square lattice of linear size L={6, 9, 18, 54, infinity} with periodic boundary conditions. We compute the ground state and evaluate local observables and two-point correlators. We also produce accurate estimates of the critical magnetic field and critical exponent beta. A calculation of the energy gap shows that it scales as 1/L at the critical point.