This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

The strongest adversary in quantum information science is decoherence, which arises owing to the coupling of a system with its environment 1 . The induced dissipation tends to destroy and wash out the interesting quantum effects that give rise to the power of quantum computation 2 , cryptography 2 and simulation 3 . Whereas such a statement is true for many forms of dissipation, we show here that dissipation can also have exactly the opposite effect: it can be a fully fledged resource for universal quantum computation without any coherent dynamics needed to complement it. The coupling to the environment drives the system to a steady state where the outcome of the computation is encoded. In a similar vein, we show that dissipation can be used to engineer a large variety of strongly correlated states in steady state, including all stabilizer codes, matrix product states 4 , and their generalization to higher dimensions 5 .The situation we have in mind is shown in Fig. 1. A quantum system composed of N particles (such as qubits) is organized in space according to a particular geometry (in the figure, a onedimensional lattice). Neighbouring systems are coupled to some local environments, which are dissipative in nature and tend to drive the system to a steady state. Our idea is to engineer those couplings, so that the environments drive the system to a desired final state. The coupling to the environment will be static, so that the desired state is obtained after some time without having to actively control the system. Note that the role of the environments is to dissipate (or, more precisely, evacuate) the entropy of the system, and by choosing the couplings appropriately we can use this effect to drive our system.We will show first how to design the interactions with the environment to implement universal quantum computation. This new method, which we refer to as dissipative quantum computation (DQC), defies some of the standard criteria for quantum computation because it requires neither state preparation, nor unitary dynamics 6 . However, it is nevertheless as powerful as standard quantum computation. Then we will show that dissipation can be engineered 7 to prepare ground states of frustration-free Hamiltonians. Those include matrix product states 4,8,9 (MPSs) and projected entangled pair states 5,9 (PEPSs), such as graph states 10 and Kitaev 11 and Levin-Wen 12 topological codes. Both DQC and dissipative state engineering (DSE) are robust in the sense that, given the dissipative nature of the process, the system is driven towards its steady state independent of the initial state and hence of eventual perturbations along the way.Here, we will concentrate first on DQC, showing how given any quantum circuit one can construct a locally acting master equation for which the steady state is unique, encodes the outcome of the circuit and is reached in polynomial time (with respect to the one corresponding to the circuit). Then we will show how to construct dissipative processes that drive the system to the ground stat...

This work gives a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.

We show how to simulate numerically the evolution of 1D quantum systems under dissipation as well as in thermal equilibrium. The method applies to both finite and inhomogeneous systems, and it is based on two ideas: (a) a representation for density operators which extends that of matrix product states to mixed states; (b) an algorithm to approximate the evolution (in real or imaginary time) of matrix product states which is variational.

We consider a single copy of a pure four-partite state of qubits and investigate its behaviour under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four-qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to GHZ-like entanglement. The other ones contain essentially 2- or 3-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of 4 qubits, giving rise to a seven parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single copy distillation protocol

The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block. The principle of causality forms one of the pillars of modern physics. It dictates that there is a finite speed at which information can propagate. Because of the existence of a light cone, relativistic quantum field theories automatically exhibit that property. The situation is, however, not so clear in nonrelativistic quantum mechanics, where a strict notion of a light cone does not exist. It has indeed been noticed that local operations can in principle be used to send information over arbitrary distances in arbitrary small times [1]. The seminal work of Lieb and Robinson [2] and recent generalizations due to Hastings [3] and Nachtergaele and Sims [4], however, show that the situation in not so bad: if evolution is governed by local Hamiltonians, then nonrelativistic quantum mechanics gives rise to an effective light cone with exponentially decaying tails. Because of this exponential attenuation, we will show how a quantitative version of causality emerges where the amount of information that can be exchanged is exponentially small within space-time regions not connected by a light cone.A related question is how fast correlations can be created between two widely separated regions in space. Note that in this case, the principle of causality does not prohibit the buildup of correlations faster than the speed of light, as correlations as such cannot be used to signal information; this is precisely the argument used to show that the existence of entanglement does not violate causality. Again using the Lieb-Robinson bound, we will show that there is a finite velocity at which correlations can be distributed. This automatically implies that the time it takes to distribute entanglement between two nodes in a spin-network scales as the distance between the nodes, solving an open question raised in [5]. Note that we assume that all classical communication is also described by local Hamiltonian evolution, as otherwise it is possible to distribute entanglement over arbitrary distances in a single unit of time by making use of the concept of quantum teleportation [6,7]. Similar techniques can be used to prove lower bounds on the time it takes to create exotic quantum states exhibiting topological quantum order [8,9] by local Hamiltonian evolution: a time propo...

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real-and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example. DOI: 10.1103/PhysRevLett.107.070601 PACS numbers: 05.10.Cc, 02.70.Àc, 03.67.Àa, 75.40.Gb The density-matrix renormalization group (DMRG) is arguably the most powerful tool available for the study of one-dimensional strongly interacting quantum lattice systems [1]. The DMRG-now understood as an application of the variational principle to matrix product states (MPSs) [2]-was originally conceived as a method to calculate ground-state properties. However, there has been a recent explosion of activity, spurred by insights from quantum information theory, in developing powerful extensions allowing the study of finite-temperature properties and nonequilibrium physics via time evolution [3]. The simulation of nonequilibrium properties with the DMRG was first attempted in Ref.[4], but modern implementations are based on the time-evolving block decimation algorithm (TEBD) [5] or the variational matrix product state approach [6].At the core of a TEBD algorithm lies the Lie-Trotter decomposition for the propagator expðidtĤÞ, which splits it into a product of local unitaries. This product can then be dealt with in a parallelized and efficient way: When applied to an MPS, one obtains another MPS with a larger bond dimension. To proceed, one then truncates the MPS description by discarding irrelevant variational parameters. This is such a flexible idea that it has allowed even the study of the dynamics of infinite translation-invariant lattice systems via the infinite TEBD [7]. Despite its success, the TEBD has some drawbacks: (i) The truncation step may not be optimal; (ii) conservation laws, e.g., energy conservation, may be broken; and (iii) symmetries, e.g., translation invariance, are broken (although translation invariance by two-site shifts is retained for nearestneighbor Hamiltonians). The problem is that when the Lie-Trotter step is applied to the state-stored as an MPS-it leaves the variational manifold and a representative from the manifold must be found that best approximates the new time-evolved state. There are a variety of ways to do this based on diverse distance measures for quantum states, but implementations become awkward when symmetries are brought into account.In this Letter, we introduce a new algorithm to solve the aforementioned problems-intrinsic to the TEBD-without an appreciable increase in computational cost. The resulting imaginary-time algorithm quickly converges towards the globally best uniform MPS (uMPS) approximation for translational-invariant ground states of strongly corr...

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...