694 research outputs found

### Classical versus quantum errors in quantum computation of dynamical systems

We analyze the stability of a quantum algorithm simulating the quantum
dynamics of a system with different regimes, ranging from global chaos to
integrability. We compare, in these different regimes, the behavior of the
fidelity of quantum motion when the system's parameters are perturbed or when
there are unitary errors in the quantum gates implementing the quantum
algorithm. While the first kind of errors has a classical limit, the second one
has no classical analogue. It is shown that, whereas in the first case
(``classical errors'') the decay of fidelity is very sensitive to the dynamical
regime, in the second case (``quantum errors'') it is almost independent of the
dynamical behavior of the simulated system. Therefore, the rich variety of
behaviors found in the study of the stability of quantum motion under
``classical'' perturbations has no correspondence in the fidelity of quantum
computation under its natural perturbations. In particular, in this latter case
it is not possible to recover the semiclassical regime in which the fidelity
decays with a rate given by the classical Lyapunov exponent.Comment: 8 pages, 7 figure

### Robust and efficient generator of almost maximal multipartite entanglement

Quantum chaotic maps can efficiently generate pseudo-random states carrying
almost maximal multipartite entanglement, as characterized by the probability
distribution of bipartite entanglement between all possible bipartitions of the
system. We show that such multipartite entanglement is robust, in the sense
that, when realistic noise is considered, distillable entanglement of
bipartitions remains almost maximal up to a noise strength that drops only
polynomially with the number of qubits.Comment: 4 pages, 4 figures. Published versio

### Separability in Riemannian Manifolds

An outline of the basic Riemannian structures underlying the separation of
variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.Comment: This paper was submitted in 2004 to the Royal Society and accepted
for publication in a special volume dedicated to the 'State of the Art of the
Separation of Variables'. However, this volume was never published due to
death of the Editor, V. Kutznetso

### 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

### Optimal purification of a generic n-qudit state

We propose a quantum algorithm for the purification of a generic mixed state
$\rho$ of a $n$-qudit system by using an ancillary $n$-qudit system. The
algorithm is optimal in that (i) the number of ancillary qudits cannot be
reduced, (ii) the number of parameters which determine the purification state
$|\Psi>$ exactly equals the number of degrees of freedom of $\rho$, and (iii)
$|\Psi>$ is easily determined from the density matrix $\rho$. Moreover, we
introduce a quantum circuit in which the quantum gates are unitary
transformations acting on a $2n$-qudit system. These transformations are
determined by parameters that can be tuned to generate, once the ancillary
qudits are disregarded, any given mixed $n$-qudit state.Comment: 8 pages, 9 figures, remarks adde

### A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems

Two effective methods for writing the dynamical equations for non-holonomic
systems are illustrated. They are based on the two types of representation of
the constraints: by parametric equations or by implicit equations. They can be
applied to linear as well as to non-linear constraints. Only the basic notions
of vector calculus on Euclidean 3-space and on tangent bundles are needed.
Elementary examples are illustrated.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA

### An Error Model for the Cirac-Zoller CNOT gate

In the framework of ion-trap quantum computing, we develop a characterization
of experimentally realistic imperfections which may affect the Cirac-Zoller
implementation of the CNOT gate. The CNOT operation is performed by applying a
protocol of five laser pulses of appropriate frequency and polarization. The
laser-pulse protocol exploits auxiliary levels, and its imperfect
implementation leads to unitary as well as non-unitary errors affecting the
CNOT operation. We provide a characterization of such imperfections, which are
physically realistic and have never been considered before to the best of our
knowledge. Our characterization shows that imperfect laser pulses unavoidably
cause a leak of information from the states which alone should be transformed
by the ideal gate, into the ancillary states exploited by the experimental
implementation.Comment: 10 pages, 1 figure. Accepted as a contributed oral communication in
the QuantumComm 2009 International Conference on Quantum Communication and
Quantum Networking, Vico Equense, Italy, October 26-30, 200

### Computing the distance between quantum channels: Usefulness of the Fano representation

The diamond norm measures the distance between two quantum channels. From an
operational vewpoint, this norm measures how well we can distinguish between
two channels by applying them to input states of arbitrarily large dimensions.
In this paper, we show that the diamond norm can be conveniently and in a
physically transparent way computed by means of a Monte-Carlo algorithm based
on the Fano representation of quantum states and quantum operations. The
effectiveness of this algorithm is illustrated for several single-qubit quantum
channels.Comment: 8 pages, 7 figure

### How complex is the quantum motion?

In classical mechanics the complexity of a dynamical system is characterized
by the rate of local exponential instability which effaces the memory of
initial conditions and leads to practical irreversibility. In striking
contrast, quantum mechanics appears to exhibit strong memory of the initial
state. Here we introduce a notion of complexity for a quantum system and relate
it to its stability and reversibility properties.Comment: 4 pages, 3 figures, new figure adde

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