QIC Abstracts

 Vol.8 No.6/7 July 1, 2008

Research Articles: 
On the transport of atomic ions in linear and multidimensional ion trap arrays (pp0501-0578) PDF
D. Hucul, M. Yeo, W.K. Hensinger, J. Rabchuk, S. Olmschenk, and C. Monroe
Trapped atomic ions have become one of the most promising architectures for a quantum computer, and current effort is now devoted to the transport of trapped ions through complex segmented ion trap structures in order to scale up to much larger numbers of trapped ion qubits. This paper covers several important issues relevant to ion transport in any type of complex multidimensional rf (Paul) ion trap array. We develop a general theoretical framework for the application of time-dependent electric fields to shuttle laser-cooled ions along any desired trajectory, and describe a method for determining the effect of arbitrary shuttling schedules on the quantum state of trapped ion motion. In addition to the general case of linear shuttling over short distances, we introduce issues particular to the shuttling through multidimensional junctions, which are required for the arbitrary control of the positions of large arrays of trapped ions. This includes the transport of ions around a corner, through a cross or T junction, and the swapping of positions of multiple ions in a laser-cooled crystal. Where possible, we make connections to recent experimental results in a multidimensional T junction trap, where arbitrary 2-dimensional transport was realized.

On solving systems of random linear disequations (pp0579-0594) PDF
G. Ivanyos
An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.

Homological invariants of stabilizer states (pp0585-0621) PDF
K. Wirthmuller
We propose a new kind of invariant of multi-party stabilizer states with respect to local Clifford equivalence. These homological invariants are discrete entities defined in terms of the entanglement a state enjoys with respect to arbitrary groupings of the parties, and they may be thought of as reflecting entanglement in a qualitative way. We investigate basic properties of the invariants and link them with known results on the extraction of GHZ states.

Faked states attack using detector efficiency mismatch on SARG04, phase-time, DPSK, and Ekert protocols (pp0622-0635) PDF
V. Makarov and J. Skaar 
In quantum cryptosystems, variations in detector efficiency can be exploited to stage a successful attack. This happens when the efficiencies of Bob's two detectors are different functions of a control parameter accessible to Eve (e.g., timing of the incoming pulses). It has previously been shown that the Bennett-Brassard 1984 (BB84) protocol is vulnerable to this attack. In this paper, we show that several other protocols and encodings may also be vulnerable. We consider a faked states attack in the case of a partial efficiency mismatch on the Scarani-Acin-Ribordy-Gisin 2004 (SARG04) protocol, and derive the quantum bit error rate as a function of detector efficiencies. Additionally, it is shown how faked states can in principle be constructed for quantum cryptosystems that use a phase-time encoding, the differential phase shift keying (DPSK) and the Ekert protocols.

A fast quantum circuit for addition with few qubits (pp0636-0649) PDF
Y. Takahashi and N. Kunihiro
We show how to construct a fast quantum circuit for computing the sum of two $n$-bit binary numbers with few qubits. The constructed circuit uses $O(n/\log n)$ ancillary qubits and its depth and size are $O(\log n)$ and $O(n)$, respectively. The number of ancillary qubits is asymptotically less than that in Draper et al.'s quantum carry-lookahead adder, and the depth and size are asymptotically the same as those of Draper et al.'s. Moreover, we show that the circuit is useful for constructing an efficient quantum circuit for Shor's factoring algorithm.

PEPS as unique ground states of local Hamiltonians (pp0650-0663) PDF
D. Perez-Garcia, F. Verstraete, J.I. Cirac, and M.M. Wolf
In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.

On separability of graphs with some entangled edges (pp0664-0670) PDF
H. Rahiminia and M. Amini
In this paper we give a combinatorial proof of the entanglement of a graph whose number of vertices is a product of two primes, and has exactly one entangled edge, or all of whose entangled edges are emanating from one vertex. Thereby, we provide easier and more direct proofs of the conjecture by Braunstein, Ghosh, and Severini in {\it quant-ph}/0406165 v2 (2006) [2].

Matrix rearrangement approach for the entangling power with hybrid qudit systems (pp0671-0680) PDF
X.-M. Lu, X. Wang, Y. Yang, and J. Chen
We extend the former matrix rearrangement approach of the entangling power to the general cases, without requirement of same dimensions of the subsystems. The entangling power of a unitary operator is completely determined by its realignment and partial transposition. As applications, we calculate the entangling power for the Ising interaction and the isotropic Heisenberg interaction in the hybrid qudit $d_1 \times d_2$ systems.

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