arXiv subjects
Publications and source records attributed to G. Vidal.
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 introduce the multi-scale entanglement renormalization ansatz (MERA), an efficient representation of certain quantum many-body states on a D-dimensional lattice. Equivalent to a quantum circuit with logarithmic depth and distinctive causal structure, the MERA allows for an exact evaluation of local expectation values. It is also the structure underlying entanglement renormalization, a coarse-graining scheme for quantum systems on a lattice that is focused on preserving entanglement.
We investigate for which resource states an efficient classical simulation of measurement based quantum computation is possible. We show that the Schmidt--rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt--rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement based quantum computation is possible for all states with logarithmically bounded Schmidt--rank width (with respect to the system size). For graph states where the Schmidt--rank width scales in this way, we efficiently construct the optimal tree tensor network descriptions, and provide several examples. We highlight parallels in the efficient description of complex systems in quantum information theory and graph theory.
An algorithm for the simulation of the evolution of slightly entangled quantum states has been recently proposed as a tool to study time-dependent phenomena in one-dimensional quantum systems. Its key feature is a time-evolving block-decimation (TEBD) procedure to identify and dynamically update the relevant, conveniently small subregion of the otherwise exponentially large Hilbert space. Potential applications of the TEBD algorithm are the simulation of time-dependent Hamiltonians, transport in quantum systems far from equilibrium and dissipative quantum mechanics. In this paper we translate the TEBD algorithm into the language of matrix product states in order to both highlight and exploit its resemblances to the widely used density-matrix renormalization-group (DMRG) algorithms. The TEBD algorithm being based on updating a matrix product state in time, it is very accessible to the DMRG community and it can be enhanced by using well-known DMRG techniques, for instance in the event of good quantum numbers. More importantly, we show how it can be simply incorporated into existing DMRG implementations to produce a remarkably effective and versatile ``adaptive time-dependent DMRG'' variant, that we also test and compare to previous proposals.
We explore entanglement loss along renormalization group trajectories as a basic quantum information property underlying their irreversibility. This analysis is carried out for the quantum Ising chain as a transverse magnetic field is changed. We consider the ground-state entanglement between a large block of spins and the rest of the chain. Entanglement loss is seen to follow from a rigid reordering, satisfying the majorization relation, of the eigenvalues of the reduced density matrix for the spin block. More generally, our results indicate that it may be possible to prove the irreversibility along RG trajectories from the properties of the vacuum only, without need to study the whole hamiltonian.
A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.
We present a numerical method to simulate the time evolution, according to a Hamiltonian made of local interactions, of quantum spin chains and systems alike. The efficiency of the scheme depends on the amount of the entanglement involved in the simulated evolution. Numerical analysis indicate that this method can be used, for instance, to efficiently compute time-dependent properties of low-energy dynamics of sufficiently regular but otherwise arbitrary one-dimensional quantum many-body systems.
We consider the implementation of two-qubit unitary transformations by means of CNOT gates and single-qubit unitary gates. We show, by means of an explicit quantum circuit, that together with local gates three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates.
Quantum phase transitions occur at zero temperature and involve the appearance of long-range correlations. These correlations are not due to thermal fluctuations but to the intricate structure of a strongly entangled ground state of the system. We present a microscopic computation of the scaling properties of the ground-state entanglement in several 1D spin chain models both near and at the quantum critical regimes. We quantify entanglement by using the entropy of the ground state when the system is traced down to $L$ spins. This entropy is seen to scale logarithmically with $L$, with a coefficient that corresponds to the central charge associated to the conformal theory that describes the universal properties of the quantum phase transition. Thus we show that entanglement, a key concept of quantum information science, obeys universal scaling laws as dictated by the representations of the conformal group and its classification motivated by string theory. This connection unveils a monotonicity law for ground-state entanglement along the renormalization group flow. We also identify a majorization rule possibly associated to conformal invariance and apply the present results to interpret the breakdown of density matrix renormalization group techniques near a critical point.
We compute the asymptotic entanglement capacity of the Ising interaction ZZ, the anisotropic Heisenberg interaction XX + YY, and more generally, any two-qubit Hamiltonian with canonical form K = a XX + b YY. We also describe an entanglement assisted classical communication protocol using the Hamiltonian K with rate equal to the asymptotic entanglement capacity.
A non-local unitary transformation of two qubits occurs when some Hamiltonian interaction couples them. Here we characterize the amount, as measured by time, of interaction required to perform two--qubit gates, when also arbitrarily fast, local unitary transformations can be applied on each qubit. The minimal required time of interaction, or interaction cost, defines an operational notion of the degree of non--locality of gates. We characterize a partial order structure based on this notion. We also investigate the interaction cost of several communication tasks, and determine which gates are able to accomplish them. This classifies two--qubit gates into four categories, differing in their capability to transmit classical, as well as quantum, bits of information.
We propose the use of quantum optical systems to perform universal simulation of quantum dynamics. Two specific implementations that require present technology are put forward for illustrative purposes. The first scheme consists of neutral atoms stored in optical lattices, while the second scheme consists of ions stored in an array of micro--traps. Each atom (ion) supports a two--level system, on which local unitary operations can be performed through a laser beam. A raw interaction between neighboring two--level systems is achieved by conditionally displacing the corresponding atoms (ions). Then, average Hamiltonian techniques are used to achieve evolutions in time according to a large class of Hamiltonians.
Consider a set of $N$ systems and an arbitrary interaction Hamiltonian $H$ that couples them. We investigate the use of local operations and classical communication (LOCC), together with the Hamiltonian $H$, to simulate a unitary evolution of the $N$ systems according to some other Hamiltonian $H'$. First, we show that the most general simulation using $H$ and LOCC can be also achieved, with the same time efficiency, by just interspersing the evolution of $H$ with local unitary manipulations of each system and a corresponding local ancilla (in a so-called LU+anc protocol). Thus, the ability to make local measurements and to communicate classical information does not help in non--local Hamiltonian simulation. Second, we show that both for the case of two $d$-level systems ($d>2$), or for that of a setting with more than two systems ($N>2$), LU+anc protocols are more powerful than LU protocols. Therefore local ancillas are a useful resource for non--local Hamiltonian simulation. Third, we use results of majorization theory to explicitly solve the problem of optimal simulation of two-qubit Hamiltonians using LU (equivalently, LU+anc, LO or LOCC).
We compute the entanglement cost of several families of bipartite mixed states, including arbitrary mixtures of two Bell states. This is achieved by developing a technique that allows us to ascertain the additivity of the entanglement of formation for any state supported on specific subspaces. As a side result, the proof of the irreversibility in asymptotic local manipulations of entanglement is extended to two-qubit systems.
We show that, in any open set of distillable states, all asymptotic entanglement measures $E(ρ)$ are continuous as a function of (a single copy of) $ρ$, even though they quantify the entanglement properties of $ρ^{\otimes N}$ is the large $N$ limit.
We show that Einstein--Podolsky--Rosen--Bohm (EPR) and Greenberger--Horne--Zeilinger--Mermin (GHZ) states can not generate, through local manipulation and in the asymptotic limit, all forms of three--partite pure--state entanglement in a reversible way. The techniques that we use suggest that there may be a connection between this result and the irreversibility that occurs in the asymptotic preparation and distillation of bipartite mixed states.
Motivated by experimental limitations commonly met in the design of solid state quantum computers, we study the problems of non-local Hamiltonian simulation and non-local gate synthesis when only homogeneous local unitaries are performed in order to tailor the available interaction. Homogeneous (i.e. identical for all subsystems) local manipulation implies a more refined classification of interaction Hamiltonians than the inhomogeneous case, as well as the loss of universality in Hamiltonian simulation. For the case of symmetric two-qubit interactions, we provide time-optimal protocols for both Hamiltonian simulation and gate synthesis.
We introduce the interaction cost of a non-local gate as the minimal time of interaction required to perform the gate when assisting the process with fast local unitaries. This cost, of interest both in the areas of quantum control and quantum information, depends on the specific interaction, and allows to compare in an operationally meaningful manner any two non-local gates. In the case of a two-qubit system, an analytical expression for the interaction cost of any unitary operation given any coupling Hamiltonian is obtained. One gate may be more time-consuming than another for any possible interaction. This defines a partial order structure in the set of non-local gates, that compares their degree of non-locality. We analytically characterize this partial order in a region of the set of two-qubit gates.