19,719 research outputs found
Generalised state spaces and non-locality in fault tolerant quantum computing schemes
We develop connections between generalised notions of entanglement and
quantum computational devices where the measurements available are restricted,
either because they are noisy and/or because by design they are only along
Pauli directions. By considering restricted measurements one can (by
considering the dual positive operators) construct single particle state spaces
that are different to the usual quantum state space. This leads to a modified
notion of entanglement that can be very different to the quantum version (for
example, Bell states can become separable). We use this approach to develop
alternative methods of classical simulation that have strong connections to the
study of non-local correlations: we construct noisy quantum computers that
admit operations outside the Clifford set and can generate some forms of
multiparty quantum entanglement, but are otherwise classical in that they can
be efficiently simulated classically and cannot generate non-local statistics.
Although the approach provides new regimes of noisy quantum evolution that can
be efficiently simulated classically, it does not appear to lead to significant
reductions of existing upper bounds to fault tolerance thresholds for common
noise models.Comment: V2: 18 sides, 7 figures. Corrected two erroneous claims and one
erroneous argumen
Digital quantum simulation of lattice gauge theories in three spatial dimensions
In the present work, we propose a scheme for digital formulation of lattice
gauge theories with dynamical fermions in 3+1 dimensions. All interactions are
obtained as a stroboscopic sequence of two-body interactions with an auxiliary
system. This enables quantum simulations of lattice gauge theories where the
magnetic four-body interactions arising in two and more spatial dimensions are
obtained without the use of perturbation theory, thus resulting in stronger
interactions compared with analogue approaches. The simulation scheme is
applicable to lattice gauge theories with either compact or finite gauge
groups. The required bounds on the digitization errors in lattice gauge
theories, due to the sequential nature of the stroboscopic time evolution, are
provided. Furthermore, an implementation of a lattice gauge theory with a
non-abelian gauge group, the dihedral group , is proposed employing the
aforementioned simulation scheme using ultracold atoms in optical lattices.Comment: 38 pages, 5 figure
Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control
We derive expressions for the invariant length element and measure for the
simple compact Lie group SU(4) in a coordinate system particularly suitable for
treating entanglement in quantum information processing. Using this metric, we
compute the invariant volume of the space of two-qubit perfect entanglers. We
find that this volume corresponds to more than 84% of the total invariant
volume of the space of two-qubit gates. This same metric is also used to
determine the effective target sizes that selected gates will present in any
quantum-control procedure designed to implement them.Comment: 27 pages, 5 figure
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
A geometric theory of non-local two-qubit operations
We study non-local two-qubit operations from a geometric perspective. By
applying a Cartan decomposition to su(4), we find that the geometric structure
of non-local gates is a 3-Torus. We derive the invariants for local
transformations, and connect these local invariants to the coordinates of the
3-Torus. Since different points on the 3-Torus may correspond to the same local
equivalence class, we use the Weyl group theory to reduce the symmetry. We show
that the local equivalence classes of two-qubit gates are in one-to-one
correspondence with the points in a tetrahedron except on the base. We then
study the properties of perfect entanglers, that is, the two-qubit operations
that can generate maximally entangled states from some initially separable
states. We provide criteria to determine whether a given two-qubit gate is a
perfect entangler and establish a geometric description of perfect entanglers
by making use of the tetrahedral representation of non-local gates. We find
that exactly half the non-local gates are perfect entanglers. We also
investigate the non-local operations generated by a given Hamiltonian. We first
study the gates that can be directly generated by a Hamiltonian. Then we
explicitly construct a quantum circuit that contains at most three non-local
gates generated by a two-body interaction Hamiltonian, together with at most
four local gates generated by single qubit terms. We prove that such a quantum
circuit can simulate any arbitrary two-qubit gate exactly, and hence it
provides an efficient implementation of universal quantum computation and
simulation.Comment: 22 pages, 6 figure
Foliated fracton order in the checkerboard model
In this work, we show that the checkerboard model exhibits the phenomenon of
foliated fracton order. We introduce a renormalization group transformation for
the model that utilizes toric code bilayers as an entanglement resource, and
show how to extend the model to general three-dimensional manifolds.
Furthermore, we use universal properties distilled from the structure of
fractional excitations and ground-state entanglement to characterize the
foliated fracton phase and find that it is the same as two copies of the X-cube
model. Indeed, we demonstrate that the checkerboard model can be transformed
into two copies of the X-cube model via an adiabatic deformation.Comment: 8 pages, 9 figure
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