This is a partly

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, cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite onedimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)]. 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 analysing its second order quantum phase transition. PACS numbers:Strongly interacting quantum many-body systems are of central importance in several areas of science and technology, including condensed matter and high-energy physics, quantum chemistry, quantum computation and nanotechnology. From a theoretical perspective, the study of such systems often poses a great computational challenge. Despite the existence of well-stablished numerical techniques, such as exact diagonalization, quantum monte carlo [1], the density matrix renormalization group [2] or series expansion [3] to mention some, a large class of two-dimensional lattice models involving frustrated spins or fermions remain unsolved.Fresh ideas from quantum information have recently led to a series of new simulation algorithms based on an efficient representation of the lattice many-body wave-function through a tensor network. This is a network made of small tensors interconnected according to a pattern that reproduces the structure of entanglement in the system. Thus, a matrix product state (MPS) [4], a tensor network already implicit in the density matrix renormalization group, is used in the time-evolving block decimation (TEBD) algorithm to simulate time evolution in one-dimensional lattice systems [5], whereas a tensor product state [6] or projected entangled-pair state (PEPS) [7] is the basis to simulate two-dimensional lattice systems. In turn, the multi-scale entanglement renormalization ansatz accuratedly describes critical and topologically ordered systems [8].In this work we explain how to modify the PEPS algorithm of Ref.[7] to simulate two-dimensional lattice systems in the thermodynamic limit. By addressing an infinite system directly, the infinite PEPS (iPEPS) algorithm can analyse bulk properties without dealing with boundary effects or finite-size corrections. This is achieved by generalizing, to two dimensions, the basic ideas underlying the infinite TEBD (iTEBD) [9]. Namely, we exploit translational invariance (i) to obtain a very compact PEPS description with only two independent tensors and (ii) to simulate time evolution by just updating these two tensors. We describe the essential new ingredients of the iPEPS algorithm, which is based on numerically solving a transfer matrix problem with an MPS. We then use it to compute the ground state of the quantum Ising model with...

The infinite time-evolving block decimation algorithm ͓G. Vidal, Phys. Rev. Lett. 98, 070201 ͑2007͔͒ allows to simulate unitary evolution and to compute the ground state of one-dimensional ͑1D͒ quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a ͑translationally invariant͒ tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open ͑i.e., master equation͒ dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional ͑2D͒ classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.

An extension of the projected entangled-pair states (PEPS) algorithm to infinite systems, known as the iPEPS algorithm, was recently proposed to compute the ground state of quantum systems on an infinite two-dimensional lattice. Here we investigate a modification of the iPEPS algorithm, where the environment is computed using the corner transfer matrix renormalization group (CTMRG) method, instead of using one-dimensional transfer matrix methods as in the original proposal. We describe a variant of the CTMRG that addresses different directions of the lattice independently, and use it combined with imaginary time evolution to compute the ground state of the two-dimensional quantum Ising model. Near criticality, the modified iPEPS algorithm is seen to provide a better estimation of the order parameter and correlators.

We explain how to implement, in the context of projected entangled-pair states ͑PEPSs͒, the general procedure of fermionization of a tensor network introduced in P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 ͑2009͒. The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the t-J model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.

Tensor network states and methods have erupted in recent years. Originally developed in the context of condensed matter physics and based on renormalization group ideas, tensor networks lived a revival thanks to quantum information theory and the understanding of entanglement in quantum many-body systems. Moreover, it has been not-so-long realized that tensor network states play a key role in other scientific disciplines, such as quantum gravity and artificial intelligence. In this context, here we provide an overview of basic concepts and key developments in the field. In particular, we briefly discuss the most important tensor network structures and algorithms, together with a sketch on advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, AdS/CFT, artificial intelligence, the 2d Hubbard model, 2d quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems, and many-body localization.

We analyze the entanglement entropy in the Lipkin-Meshkov-Glick model, which describes mutually interacting spins half embedded in a magnetic field. This entropy displays a singularity at the critical point that we study as a function of the interaction anisotropy, the magnetic field, and the system size. Results emerging from our analysis are surprisingly similar to those found for the one-dimensional XY chain.PACS numbers: 03.65. Ud,03.67.Mn,73.43.Nq Within the last few years, entanglement properties of spin systems have attracted much attention. As initially shown in one-dimensional (1D) spin chains [1,2,3,4], observables measuring this genuine quantum mechanical feature are strongly affected by the existence of a quantum phase transition. For instance, the so-called concurrence [5] that quantifies the two-spin entanglement displays some nontrivial universal scaling properties. Similarly, the von Neumann entropy which rather characterizes the bipartite entanglement between any two subsystems scales logarithmically with the typical size L of these subsystems at the quantum critical point, with a prefactor given by the central charge of the corresponding theory [3,4,6,7]. Note that the role played by the boundaries in these conformal invariant systems has been only recently elucidated [8]. Apart from 1D systems, very few models have been studied so far [9,10,11,12,13], either due to the absence of exact solution or to a difficult numerical treatment. In this context, the LipkinMeshkov-Glick (LMG) model [14,15,16] discussed here has drawn much attention since it allows for very efficient numerical treatment as well as analytical calculations. Introduced by Lipkin, Meshkov and Glick in Nuclear Physics, this model has been the subject of intensive studies during the last two decades because of its relevance for quantum tunneling of bosons between two levels. It is thus of prime interest to describe in particular the Josephson effect in two-mode Bose-Einstein condensates [17,18]. The entanglement properties of this model have been already discussed through the concurrence, which exhibits a cusp-like behavior at the critical point [19,20,21,22] as well as interesting dynamical properties [23]. Note that similar results have also been obtained in the Dicke model [24,25] which can be mapped onto the LMG model in some cases [26], or in the reduced BCS model [27].In this letter, we analyze the von Neumann entropy computed from the ground state of the LMG model. We show that, at the critical point, it behaves logarithmically with the size of the blocks L used in the bipartite decomposition of the density matrix with a prefactor that depends on the anisotropy parameter tuning the universality class. We also discuss the dependence of the entropy with the magnetic field and stress the analogy with 1D systems.The LMG model is defined by the Hamiltonianwhere σ k α is the Pauli matrix at position k in the direction α, and N the total number of spins. This Hamiltonian describes a set of spins half located at the vertices of ...

We discuss how quantum computation can be applied to financial problems, providing an overview of current approaches and potential prospects. We review quantum optimization algorithms, and expose how quantum annealers can be used to optimize portfolios, find arbitrage opportunities, and perform credit scoring. We also discuss deep-learning in finance, and suggestions to improve these methods through quantum machine learning. Finally, we consider quantum amplitude estimation, and how it can result in a quantum speed-up for Monte Carlo sampling. This has direct applications to many current financial methods, including pricing of derivatives and risk analysis. Perspectives are also discussed.