arXiv subjects
Publications and source records attributed to F. Verstraete.
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.
We propose a new method for computing the ground state properties and the time evolution of infinite chains based on a transverse contraction of the tensor network. The method does not require finite size extrapolation and avoids explicit truncation of the bond dimension along the evolution. By folding the network in the time direction prior to contraction, time dependent expectation values and dynamic correlation functions can be computed after much longer evolution time than with any previous method. Moreover, the algorithm we propose can be used for the study of some non-invariant infinite chains, including impurity models.
We study the nature of the ground state of the frustrated J1-J2 model and the J1-J3 model using a variational algorithm based on projected entangled-pair states (PEPS). By investigating spin-spin correlation functions, we observe a separation in regions with long-range and short-range order. A direct comparison with exact diagonalizations in the subspace of short-range valence bond singlets reveals that the system is well described by states within this subset in the short-range order regions. We discuss the question whether the system forms a spin-liquid, a plaquette valence bond crystal or a columnar dimer crystal in these regions.
Anyons are particlelike excitations of strongly correlated phases of matter with fractional statistics, characterized by nontrivial changes in the wave function, generalizing Bose and Fermi statistics, when two of them are interchanged. This can be used to perform quantum computations [A. Yu. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003)]. We show how to simulate the creation and manipulation of Abelian and non- Abelian anyons in topological lattice models using trapped atoms in optical lattices. Our proposal, feasible with present technology, requires an ancilla particle which can undergo single-particle gates, be moved close to each constituent of the lattice and undergo a simple quantum gate, and be detected.
We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the {\em projected entangled-pair state} algorithm for finite lattice systems [F. Verstraete and J.I. Cirac, cond-mat/0407066] and the infinite {\em time-evolving block decimation} algorithm for infinite one-dimensional 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.
We consider the problem of creating a long-distance entangled state between two stations of a network, where neighboring nodes are connected by noisy quantum channels. We show that any two stations can share an entangled pair if the effective probability for the quantum errors is below a certain threshold, which is achieved by using local redundant encoding to preserve the global phase and network-based correction for the bit-flip errors. In contrast to the convensional quantum repeater schemes we are not limited by the memory coherence time, because all quantum operations only use one-way classical communication and can be done in one shot. Meanwhile, the overhead of local resources only increases logarithmically with the size of the network, making our proposal favorable to practical applications of long-distance quantum communication.
We present a unified framework for renormalization group methods, including Wilson's numerical renormalization group (NRG) and White's density-matrix renormalization group (DMRG), within the language of matrix product states. This allows improvements over Wilson's NRG for quantum impurity models, as we illustrate for the one-channel Kondo model. Moreover, we use a variational method for evaluating Green's functions. The proposed method is more flexible in its description of spectral properties at finite frequencies, opening the way to time-dependent, out-of-equilibrium impurity problems. It also substantially improves computational efficiency for one-channel impurity problems, suggesting potentially \emph{linear} scaling of complexity for $n$-channel problems.
We show how to construct relevant families of matrix product operators in one and higher dimensions. Those form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In particular, we construct translational invariant matrix product operators suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We illustrate how those tools can be exploited for constructing new algorithms for simulating quantum spin systems.
The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. This counterintuitive idea which has its roots in the nonextensive nature of black-hole entropy serves as a guiding principle in the search for the fundamental laws of Planck-scale physics. In this paper we show that a similar phenomenon emerges from the established laws of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of microscopic degrees of freedom, it may diverge as the inverse temperature in quantum systems. A rigorous relation between area laws and correlations is established and their explicit behavior is revealed for a large class of quantum many-body states beyond equilibrium systems.
We introduce a class of variational states to describe quantum many-body systems. This class generalizes matrix product states which underly the density-matrix renormalization group approach by combining them with weighted graph states. States within this class may (i) possess arbitrarily long-ranged two-point correlations, (ii) exhibit an arbitrary degree of block entanglement entropy up to a volume law, (iii) may be taken translationally invariant, while at the same time (iv) local properties and two-point correlations can be computed efficiently. This new variational class of states can be thought of as being prepared from matrix product states, followed by commuting unitaries on arbitrary constituents, hence truly generalizing both matrix product and weighted graph states. We use this class of states to formulate a renormalization algorithm with graph enhancement (RAGE) and present numerical examples demonstrating that improvements over density-matrix renormalization group simulations can be achieved in the simulation of ground states and quantum algorithms. Further generalizations, e.g., to higher spatial dimensions, are outlined.
We show that the existence of string order in a given quantum state is intimately related to the presence of a local symmetry by proving that both concepts are equivalent within the framework of finitely correlated states. Once this connection is established, we provide a complete characterization of local symmetries in these states. The results allow to understand in a straightforward way many of the properties of string order parameters, like their robustness/fragility under perturbations and their typical disappearance beyond strictly one-dimensional lattices. We propose and discuss an alternative definition, ideally suited for detecting phase transitions, and generalizations to two and more spatial dimensions.
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance. The proof relies on two new techniques that have been introduced in [quant-ph/0610027] and [quant-ph/0607216], respectively, and that are also well suited to prove the quantum generalisation of the Hoeffding bound, which is a modification of the Chernoff distance and specifies the optimal achievable asymptotic error rate in the context of asymmetric hypothesis testing. This has been done subsequently by Hayashi [quant-ph/0611013] and Nagaoka [quant-ph/0611289] for the special case where both hypotheses have full support. Moreover, quantum Stein's Lemma and quantum Sanov's theorem may be derived directly from the quantum Hoeffding bound combining it with a result obtained recently in [math/0703772]. The goal of this paper is to present the proofs of the above mentioned results in a unified way and in full generality (allowing hypothetic states with different supports). Additionally, we give an in-depth treatment of the properties of the quantum Chernoff distance. We argue that, although it is not a metric, it is a natural distance measure on the set of density operators, due to its clear operational meaning.
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 have studied the system of hard-core bosons on a 2-D optical lattice using a variational algorithm based on projected entangled-pair states (PEPS). We have investigated the ground state properties of the system as well as the responses of the system to sudden changes in the parameters. We have compared our results to mean field results based on a Gutzwiller ansatz.
We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the optimal strategy that minimizes the total probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because of the fact that it does not seem to share the undesirable features of other distance measures like the fidelity, the trace norm and the relative entropy.
We study the computational complexity of the N-representability problem in quantum chemistry. We show that this problem is QMA-complete, which is the quantum generalization of NP-complete. Our proof uses a simple mapping from spin systems to fermionic systems, as well as a convex optimization technique that reduces the problem of finding ground states to N-representability.
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 space-like separated regions is negligable, 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.
We introduce a variational method for the approximation of ground states of strongly interacting spin systems in arbitrary geometries and spatial dimensions. The approach is based on weighted graph states and superpositions thereof. These states allow for the efficient computation of all local observables (e.g. energy) and include states with diverging correlation length and unbounded multi-particle entanglement. As a demonstration we apply our approach to the Ising model on 1D, 2D and 3D square-lattices. We also present generalizations to higher spins and continuous-variable systems, which allows for the investigation of lattice field theories.