47 research outputs found
Engineering Functional Quantum Algorithms
Suppose that a quantum circuit with K elementary gates is known for a unitary
matrix U, and assume that U^m is a scalar matrix for some positive integer m.
We show that a function of U can be realized on a quantum computer with at most
O(mK+m^2log m) elementary gates. The functions of U are realized by a generic
quantum circuit, which has a particularly simple structure. Among other
results, we obtain efficient circuits for the fractional Fourier transform.Comment: 4 pages, 2 figure
Efficient Quantum Tensor Product Expanders and k-designs
Quantum expanders are a quantum analogue of expanders, and k-tensor product
expanders are a generalisation to graphs that randomise k correlated walkers.
Here we give an efficient construction of constant-degree, constant-gap quantum
k-tensor product expanders. The key ingredients are an efficient classical
tensor product expander and the quantum Fourier transform. Our construction
works whenever k=O(n/log n), where n is the number of qubits. An immediate
corollary of this result is an efficient construction of an approximate unitary
k-design, which is a quantum analogue of an approximate k-wise independent
function, on n qubits for any k=O(n/log n). Previously, no efficient
constructions were known for k>2, while state designs, of which unitary designs
are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe
Temperature dependance of the tunneling density of states in sub-micron planar metal / oxide / graphene junctions
We present tunneling measurements of sub-micron metal/insulator/graphene
planar tunnel junctions up to room temperature. We observe a gate independent
gap, as previously observed only by low temperature STM[Y. Zhang et al., Nat.
Phys. 4, 627 (2008)]. No gap appears at temperatures above 150K, which is four
times smaller than the theoretically expected , from the accepted mean
field model[T. O. Wehling et al. Phys. Rev. Lett. 101, 216803 (2008)]. We show
that taking into account an additional vibrational effect of out-of-plane
phonon soft modes the gap may disappear from the measurements at temperatures
much lower than the calculated .Comment: http://link.aip.org/link/?APL/99/17210
Topological Superfluid in one-dimensional Ultracold Atomic System with Spin-Orbit Coupling
We propose a one-dimensional Hamiltonian which supports Majorana
fermions when -wave superfluid appears in the ultracold atomic
system and obtain the phase-separation diagrams both for the
time-reversal-invariant case and time-reversal-symmetry-breaking case. From the
phase-separation diagrams, we find that the single Majorana fermions exist in
the topological superfluid region, and we can reach this region by tuning the
chemical potential and spin-orbit coupling . Importantly, the
spin-orbit coupling has realized in ultracold atoms by the recent experimental
achievement of synthetic gauge field, therefore, our one-dimensional ultra-cold
atomic system described by is a promising platform to find the
mysterious Majorana fermions.Comment: 5 papers, 2 figure
Correlation inequalities for classical and quantum XY models
We review correlation inequalities of truncated functions for the classical
and quantum XY models. A consequence is that the critical temperature of the XY
model is necessarily smaller than that of the Ising model, in both the
classical and quantum cases. We also discuss an explicit lower bound on the
critical temperature of the quantum XY model.Comment: 13 pages. Submitted to the volume "Advances in Quantum Mechanics:
contemporary trends and open problems" of the INdAM-Springer series,
proceedings of the INdAM meeting "Contemporary Trends in the Mathematics of
Quantum Mechanics" (4-8 July 2016) organised by G. Dell'Antonio and A.
Michelangel
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
Experimentally feasible measures of distance between quantum operations
We present two measures of distance between quantum processes based on the
superfidelity, introduced recently to provide an upper bound for quantum
fidelity. We show that the introduced measures partially fulfill the
requirements for distance measure between quantum processes. We also argue that
they can be especially useful as diagnostic measures to get preliminary
knowledge about imperfections in an experimental setup. In particular we
provide quantum circuit which can be used to measure the superfidelity between
quantum processes.
As the behavior of the superfidelity between quantum processes is crucial for
the properties of the introduced measures, we study its behavior for several
families of quantum channels. We calculate superfidelity between arbitrary
one-qubit channels using affine parametrization and superfidelity between
generalized Pauli channels in arbitrary dimensions. Statistical behavior of the
proposed quantities for the ensembles of quantum operations in low dimensions
indicates that the proposed measures can be indeed used to distinguish quantum
processes.Comment: 9 pages, 4 figure
The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
The Quantum Reverse Shannon Theorem states that any quantum channel can be
simulated by an unlimited amount of shared entanglement and an amount of
classical communication equal to the channel's entanglement assisted classical
capacity. In this paper, we provide a new proof of this theorem, which has
previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof
has a clear structure being based on two recent information-theoretic results:
one-shot Quantum State Merging and the Post-Selection Technique for quantum
channels.Comment: 30 pages, 4 figures, published versio
Digital Quantum Simulation with Rydberg Atoms
We discuss in detail the implementation of an open-system quantum simulator
with Rydberg states of neutral atoms held in an optical lattice. Our scheme
allows one to realize both coherent as well as dissipative dynamics of complex
spin models involving many-body interactions and constraints. The central
building block of the simulation scheme is constituted by a mesoscopic Rydberg
gate that permits the entanglement of several atoms in an efficient, robust and
quick protocol. In addition, optical pumping on ancillary atoms provides the
dissipative ingredient for engineering the coupling between the system and a
tailored environment. As an illustration, we discuss how the simulator enables
the simulation of coherent evolution of quantum spin models such as the
two-dimensional Heisenberg model and Kitaev's toric code, which involves
four-body spin interactions. We moreover show that in principle also the
simulation of lattice fermions can be achieved. As an example for controlled
dissipative dynamics, we discuss ground state cooling of frustration-free spin
Hamiltonians.Comment: submitted to special issue "Quantum Information with Neutral
Particles" of "Quantum Information Processing
Quantitative Treatment of Decoherence
We outline different approaches to define and quantify decoherence. We argue
that a measure based on a properly defined norm of deviation of the density
matrix is appropriate for quantifying decoherence in quantum registers. For a
semiconductor double quantum dot qubit, evaluation of this measure is reviewed.
For a general class of decoherence processes, including those occurring in
semiconductor qubits, we argue that this measure is additive: It scales
linearly with the number of qubits.Comment: Revised version, 26 pages, in LaTeX, 3 EPS figure