QIC Abstracts

 Vol.6 No.4&5 July 1, 2006   

Research Articles:
Electrode Configurations for fast separation of trapped ions (pp289-325)
         J.P.Home and A.M.Steane
We study the problem of designing electrode structures that allow pairs of ions to be brought together and separated rapidly in an array of linear Paul traps. We show that it is desirable for the electrode structure to produce a d.c. octupole moment with an a.c. radial quadrupole. For the case where electrical breakdown limits the voltages that can be applied, we show that the octupole is more demanding than the quadrupole when the characteristic distance scale of the structure is larger than 1 to 10 microns (for typical materials). We present a variety of approaches and optimizations of structures consisting of one to three layers of electrodes. The three-layer structures allow the fastest operation at given distance \rho from the trap centres to the nearest electrode surface, but when the total thickness w of the structure is constrained, leading to w < \rho, then two-layer structures may be preferable.

Implementation of continuous variable quantum cryptography in optical fibres using a go-&-return configuration (pp326-335)
         M. Legré, H. Zbinden, and N. Gisin 
We demonstrate an implementation of quantum key distribution with continuous variables based on a go-&-return configuration over distances up to 14km. This configuration leads to self-compensation of polarisation and phase fluctuations. We observe a high degree of stability of our set-up over many hours.

Quantum information processing with hyperentangled photon states (pp336-350)
         S. P. Walborn, M. P.  Almeida, P. H. Souto Ribeiro, and C. H. Monken 
We discuss quantum information processing with hyperentangled photon states - states entangled in multiple degrees of freedom. Using an additional entangled degree of freedom as an ancilla space, it has been shown that it is possible to perform efficient Bell-state measurements. We briefly review these results and present a novel deterministic quantum key distribution protocol based on Bell-state measurements of hyperentangled photons. In addition, we propose a scheme for a probabilistic controlled-not gate which operates with a 50% success probability. We also show that despite its probabilistic nature, the controlled-not gate can be used for an efficient, nonlocal demonstration of the Deutsch algorithm using two separate photons.

A logarithmic-depth quantum carry-lookahead adder (pp351-369)
         T.G. Draper, S.A. Kutin, E.M. Rains, and K.M. Svore
We present an efficient addition circuit, borrowing techniques from classical carry-lookahead arithmetic. Our quantum carry-lookahead (\QCLA) adder accepts two n-bit numbers and adds them in O(\log n) depth using O(n) ancillary qubits. We present both in-place and out-of-place versions, as well as versions that add modulo 2^n and modulo 2^n - 1. Previously, the linear-depth ripple-carry addition circuit has been the method of choice. Our work reduces the cost of addition dramatically with only a slight increase in the number of required qubits. The \QCLA\ adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor's algorithm.

Mixing of quantum walk on circulant bunkbeds (pp370-381)
         P. Lo, S. Rajaram, D. Schepens, D. Sullivan, C. Tamon, and J. Ward
This paper gives new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbeds. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G \cprod H of graphs G and H. Our results include the following:  (i) The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity; this extends a known fact about the cycles C_{n}.  (ii) Explicit analysis of the probability distribution of the quantum walk on the join of circulants; this explains why complete multipartite graphs are not average uniform mixing, using the fact K_{n} = K_{1} + K_{n-1} and K_{n,\ldots,n} = \overline{K}_{n} + \ldots + \overline{K}_{n}.  (iii) The quantum walk on the Cartesian product of a $m$-vertex path P_{m} and a circulant G, namely, P_{m} \cprod G, is average uniform mixing if G is; this highlights a difference between circulants and the hypercubes Q_{n} = P_{2} \cprod Q_{n-1}. Our proofs employ purely elementary arguments based on the spectra of the graphs.

Operator quantum error correction (pp382-399)
         D.W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky
This paper is an expanded and more detailed version of the work \cite{KLP04} in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques --- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method --- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems''.

The geometry and topology of entanglement: Conifold, Segre variety, and Hopf fibration (pp400-409)
         H. Heydari
We establish relations between conifold, Segre variety, Hopf fibration, and separable sets of pure two-qubit states. Moreover, we investigate the geometry and topology of separable sets of pure multi-qubit states based on the complex multi-projective Segre variety and higher order Hopf fibration. We show that the Segre variety and Hopf fibration give a unified geometrical and topological picture of multi-qubit states. We also construct entanglement monotones for multi-qubit states.

Generalized asymmetric quantum cloning machines (pp410-435)
         S. Iblisdir, A. Acin, and N. Gisin
We study machines that take N identical replicas of a pure qudit state as input and output a set of M_A clones of a given fidelity and another set of M_B clones of another fidelity. The trade-off between these two fidelities is investigated, and numerous examples of optimal N \to M_A+M_B cloning machines are exhibited using a generic method. A generalisation to more than two sets of clones is also discussed. Finally, an optical implementation of some such machines is proposed. This paper is an extended version of  Phys. Rev. A 72, 042328 (2005).

Efficient circuits for exact-universal computation with qudits (pp436-454)
         G.K. Brennen, S.S. Bullock, and D.P. O'Leary 
This paper concerns the efficient implementation of quantum circuits for qudits. We show that controlled two-qudit gates can be implemented without ancillas and prove that the gate library containing arbitrary local unitaries and one two-qudit gate, $\CINC$, is exact-universal. A recent paper [S.Bullock, D.O'Leary, and G.K. Brennen, Phys. Rev. Lett.  94, 230502 (2005)] describes quantum circuits for qudits which require O(d^n) two-qudit gates for state synthesis and O(d^{2n}) two-qudit gates for unitary synthesis, matching the respective lower bound complexities. In this work, we present the state-synthesis circuit in much greater detail and prove that it is correct.  Also, the (n-2)/(d-2) ancillas required in the original algorithm may be removed without changing the asymptotics. Further, we present a new algorithm for unitary synthesis, inspired by the QR matrix decomposition, which is also asymptotically optimal.

Concurrence and Schwarz inequality (pp455-460)
         H. Heydari and G. Björk
We show that generalized concurrence is closely related to, and can be derived from, the Schwarz inequality. This connection places concurrence in a geometrical and functional-analytical setting.

Communication complexity of remote state preparation with entanglement (pp461-464)
         R Jain
We consider the problem of remote state preparation recently studied in several papers. We study the communication complexity of this problem, in the presence of entanglement and in the scenario of single use of the channel.

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