3,791 research outputs found
Matrix Product States Algorithms and Continuous Systems
A generic method to investigate many-body continuous-variable systems is
pedagogically presented. It is based on the notion of matrix product states
(so-called MPS) and the algorithms thereof. The method is quite versatile and
can be applied to a wide variety of situations. As a first test, we show how it
provides reliable results in the computation of fundamental properties of a
chain of quantum harmonic oscillators achieving off-critical and critical
relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use
it to study the ground state properties of the quantum rotor model in one
spatial dimension, a model that can be mapped to the Mott insulator limit of
the 1-dimensional Bose-Hubbard model. At the quantum critical point, the
central charge associated to the underlying conformal field theory can be
computed with good accuracy by measuring the finite-size corrections of the
ground state energy. Examples of MPS-computations both in the finite-size
regime and in the thermodynamic limit are given. The precision of our results
are found to be comparable to those previously encountered in the MPS studies
of, for instance, quantum spin chains. Finally, we present a spin-off
application: an iterative technique to efficiently get numerical solutions of
partial differential equations of many variables. We illustrate this technique
by solving Poisson-like equations with precisions of the order of 10^(-7).Comment: 22 pages, 14 figures, final versio
On the role of entanglement and correlations in mixed-state quantum computation
In a quantum computation with pure states, the generation of large amounts of
entanglement is known to be necessary for a speedup with respect to classical
computations. However, examples of quantum computations with mixed states are
known, such as the deterministic computation with one quantum qubit (DQC1)
model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which
entanglement is at most marginally present, and yet a computational speedup is
believed to occur. Correlations, and not entanglement, have been identified as
a necessary ingredient for mixed-state quantum computation speedups. Here we
show that correlations, as measured through the operator Schmidt rank, are
indeed present in large amounts in the DQC1 circuit. This provides evidence for
the preclusion of efficient classical simulation of DQC1 by means of a whole
class of classical simulation algorithms, thereby reinforcing the conjecture
that DQC1 leads to a genuine quantum computational speedup
Tensor Network Methods for Invariant Theory
Invariant theory is concerned with functions that do not change under the
action of a given group. Here we communicate an approach based on tensor
networks to represent polynomial local unitary invariants of quantum states.
This graphical approach provides an alternative to the polynomial equations
that describe invariants, which often contain a large number of terms with
coefficients raised to high powers. This approach also enables one to use known
methods from tensor network theory (such as the matrix product state
factorization) when studying polynomial invariants. As our main example, we
consider invariants of matrix product states. We generate a family of tensor
contractions resulting in a complete set of local unitary invariants that can
be used to express the R\'enyi entropies. We find that the graphical approach
to representing invariants can provide structural insight into the invariants
being contracted, as well as an alternative, and sometimes much simpler, means
to study polynomial invariants of quantum states. In addition, many tensor
network methods, such as matrix product states, contain excellent tools that
can be applied in the study of invariants.Comment: 21 page
Easy implementable algorithm for the geometric measure of entanglement
We present an easy implementable algorithm for approximating the geometric
measure of entanglement from above. The algorithm can be applied to any
multipartite mixed state. It involves only the solution of an eigenproblem and
finding a singular value decomposition, no further numerical techniques are
needed. To provide examples, the algorithm was applied to the isotropic states
of 3 qubits and the 3-qubit XX model with external magnetic field.Comment: 9 pages, 3 figure
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Entanglement, randomness and chaos
Entanglement is not only the most intriguing feature of quantum mechanics,
but also a key resource in quantum information science. The entanglement
content of random pure quantum states is almost maximal; such states find
applications in various quantum information protocols. The preparation of a
random state or, equivalently, the implementation of a random unitary operator,
requires a number of elementary one- and two-qubit gates that is exponential in
the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q.
On the other hand, pseudo-random states approximating to the desired accuracy
the entanglement properties of true random states may be generated efficiently,
that is, polynomially in n_q. In particular, quantum chaotic maps are efficient
generators of multipartite entanglement among the qubits, close to that
expected for random states. This review discusses several aspects of the
relationship between entanglement, randomness and chaos. In particular, I will
focus on the following items: (i) the robustness of the entanglement generated
by quantum chaotic maps when taking into account the unavoidable noise sources
affecting a quantum computer; (ii) the detection of the entanglement of
high-dimensional (mixtures of) random states, an issue also related to the
question of the emergence of classicality in coarse grained quantum chaotic
dynamics; (iii) the decoherence induced by the coupling of a system to a
chaotic environment, that is, by the entanglement established between the
system and the environment.Comment: Review paper, 40 pages, 7 figures, added reference
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