QIC Abstracts

 Vol.11 No.1&2 January 1, 2011

Research Articles:

Macroscopic multispecies entanglement near quantum phase transitions (pp0001-0007)
V. Subrahmanyam
Multi-Species entanglement, defined for a many-particle system as the entanglement between different species of particles, is shown to exist in the thermodynamic limit of the system size going to infinity. This macroscopic entanglement, as it can exhibit singular behavior, is capable of tracking quantum phase transitions. The entanglement between up and down spins has been analytically calculated for the one-dimensional Ising model in a transverse magnetic field. As the coupling strength is varied, the first derivative of the entanglement shows a jump discontinuity and the second derivative diverges near the quantum critical point.

Surface code quantum error correction incorporating accurate error propagation (pp0008-0018)
Austin G. Fowler, David S. Wang, and Lloyd C. L. Hollenberg
The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.

Characterization of universal two-qubit Hamiltonians (pp0019-0039)
Andrew M. Childs, Debbie Leung, Laura Mancinska, and Maris Ozols
Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal, an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian $H$ can fail to be universal:
(1) H shares an eigenvector with the gate that swaps two qubits,
(2) H acts on the two qubits independently (in any of a certain family of bases), or
(3) H has zero trace
(with the third condition relevant only when the global phase of the unitary matters).
A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n \geq 3. We give some partial results on 3-universality.

Non-local box complexity and secure function evaluation (pp0040-0069)
Marc Kaplan, Sophie Laplante, Iordanis Kerenidis, and J\'er\'emie Roland
A non-local box is an abstract device into which Alice and Bob input bits $x$ and $y$ respectively and receive outputs $a$ and $b$, where $a,b$ are uniformly distributed and $a \oplus b = x \wedge y$. Such boxes have been central to the study of quantum or generalized non-locality, as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function $f$. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We show that non-local box complexity has interesting applications to classical cryptography, in particular to secure function evaluation, and study the question posed by Beimel and Malkin \cite{BM} of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function $f$. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 \nlbs, while no finite bound was previously known.

A new entanglement measure: D-concurrence (pp0070-0078)
Zhihao Ma, Weigang Yuan, Minli Bao, and Xiao-Dong Zhang
A new entanglement measure, which is called D-concurrence, is proposed. Then the upper bound for D-concurrence is obtained. In addition, D-concurrence has some special merits, such as it is an entanglement monotone, and is sub-additive.

Measurable lower bounds on concurrence (pp0079-0094)
Iman Sargolzahi, Sayyed Yahya Mirafzali, and Mohsen Sarbishaei
We derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the \textit{purely algebraic} bounds of concurrence [F. Mintert \textit{et al.} (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.

Quantum interpolation of polynomials (pp0095-0103)
Daniel M. Kane and Samuel A. Kutin
Can a quantum computer efficiently interpolate polynomials? We consider black-box algorithms that seek to learn information about a polynomial $f$ from input/output pairs $(x_i, f(x_i))$. We define a more general class of \emph{$(d,S)$-independent} function properties, where, outside of a set $S$ of exceptions, knowing $d$ input values does not help one predict the answer. There are essentially two strategies to computing such a function: query $d+1$ random input values, or search for one of the $|S|$ exceptions. We show that, up to constant factors, we cannot beat these two approaches.

A family of norms with applications in quantum information theory II (pp0104-0123)
Nathaniel Johnston and David W. Kribs
We consider the problem of computing the family of operator norms recently introduced. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory of semidefinite programming, including a new constructive proof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of the semidefinite program, including the case of Werner states and randomly generated states via the Bures measure, and approximate distributions of the norms are provided. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states.

Generation of cluster-type entangled coherent states using weak nonlinearities (pp0124-0141)
Nguyen B. An, Kisik Kim, and Jaewan Kim
We propose a scheme to generate a recently introduced type of entangled coherent states using realistic weak cross-Kerr nonlinearities and intense laser beams. An intense laser can be filtered to make a faint one to be used for production of a single photon which is necessary in our scheme. The optical devices used are conventional ones such as interferometer, mirrors, beam-splitters, phase-shifters and photo-detectors. We also provide a detailed analysis on the effects of possible imperfections and decoherence showing that our scheme is robust against such effects.

An efficient conversion of quantum circuits to a linear nearest neighbor architecture (pp0142-0166)
Yuichi Hirata, Masaki Nakanishi, Shigeru Yamashita and Yasuhiko Nakashima
Several promising implementations of quantum computation rely on a Linear Nearest Neighbor (LNN) architecture, which arranges quantum bits on a line, and allows neighbor interactions only. Therefore, several specific circuits have been designed on an LNN architecture. However, a general and efficient conversion method for an arbitrary circuit has not been established. Therefore, this paper gives an efficient conversion technique to convert quantum circuits to an LNN architecture. When a quantum circuit is converted to an LNN architecture, the objective is to reduce the size of the additional circuit added by the conversion and the time complexity of the conversion. The proposed method requires less additional circuitry and time complexity compared with naive techniques. To develop the method, we introduce two key theorems that may be interesting on their own. In addition, the proposed method also achieves less overhead than some known circuits designed from scratch on an LNN architecture.

Mathematical framework for detection and quantification of nonclassical correlation (pp0167-0180)
Akira SaiToh, Robabeh Rahimi, and Mikio Nakahara
Existing measures of bipartite nonclassical correlation that is typically characterized by nonvanishing nonlocalizable information under the zero-way CLOCC protocol are expensive in computational cost. We define and evaluate economical measures on the basis of a new class of maps, eigenvalue-preserving-but-not-completely-eigenvalue-preserving (EnCE) maps. The class is in analogy to the class of positive-but-not-completely-positive (PnCP) maps that have been commonly used in the entanglement theories. Linear and nonlinear EnCE maps are investigated. We also prove subadditivity of the measures in the form of logarithmic fidelity.

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