QIC Abstracts

 Vol.9 No.3&4 March 1, 2009

Research Articles: 
Integrated optical approach to  trapped ion quantum computation (pp0181-0202)
J. Kim and C. Kim
Recent experimental progress in quantum information processing with trapped ions have demonstrated most of the fundamental elements required to realize a scalable quantum computer. The next set of challenges lie in realization of a large number of qubits and the means to prepare, manipulate and measure them, leading to error-protected qubits and fault tolerant architectures. The integration of qubits necessarily require integrated optical approach as most of these operations involve interaction with photons. In this paper, we discuss integrated optics technologies and concrete optical designs needed for the physical realization of scalable quantum computer.

Scalable, efficient ion-photon coupling with phase Fresnel lenses for large-scale quantum computing (pp0203-0214)
E.W. Streed, B.G. Norton, J.J. Chapman, and D. Kielpinski
Efficient ion-photon coupling is an important component for large-scale ion-trap quantum computing. We propose that arrays of phase Fresnel lenses (PFLs) are a favorable optical coupling technology to match with multi-zone ion traps. Both are scalable technologies based on conventional micro-fabrication techniques. The large numerical apertures (NAs) possible with PFLs can reduce the readout time for ion qubits. PFLs also provide good coherent ion-photon coupling by matching a large fraction of an ion's emission pattern to a single optical propagation mode (TEM$_{00}$). To this end we have optically characterized a large numerical aperture phase Fresnel lens (NA=0.64) designed for use at 369.5 nm, the principal fluorescence detection transition for Yb$^+$ ions. A diffraction-limited spot $w_0=350\pm15$ nm ($1/e^2$ waist) with mode quality $M^2= 1.08\pm0.05$ was measured with this PFL. From this we estimate the minimum expected free space coherent ion-photon coupling to be 0.64\%, which is twice the best previous experimental measurement using a conventional multi-element lens. We also evaluate two techniques for improving the entanglement fidelity between the ion state and photon polarization with large numerical aperture lenses.

Efficient quantum algorithm for identifying hidden polynomials (pp0215-0230)
T. Decker, J. Draisma, and P. Wocjan
We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$-variate polynomial functions of total degree $n\geq 2$.  The problem of identifying hidden $m$-variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$.

Graph embedding using quantum hitting time (pp0231-0254)
D. Emms, R. Wilson, and E. Hancock 
In this paper, we explore analytically and experimentally a quasi-quantum analogue of the hitting time of the continuous-time quantum walk on a graph. For the classical random walk, the hitting time has been shown to be robust to errors in edge weight structure and to lead to spectral clustering algorithms with improved performance. Our analysis shows that the quasi-quantum analogue of the hitting time of the continuous-time quantum walk can be determined via integrals of the Laplacian spectrum, calculated using Gauss-Laguerre quadrature. We analyse the quantum hitting times with reference to their classical counterpart. Specifically, we explore the graph embeddings that preserve hitting time. Experimentally, we show that the quantum hitting times can be used to emphasise cluster-structure.

Communication complexities of symmetric XOR functions (pp0255-0263)
Z.-Q. Zhang and Y.-Y. Shi
We call $F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}$ a symmetric XOR function if for a function $S:\{0, 1, ..., n\}\to\{0, 1\}$, $F(x, y)=S(|x\oplus y|)$, for any $x, y\in\{0, 1\}^n$, where $|x\oplus y|$ is the Hamming weight of the bit-wise XOR of $x$ and $y$. We show that for any such function, (a) the deterministic communication complexity is always $\Theta(n)$ except for four simple functions that have a constant complexity, and (b) up to a polylog factor, both the error-bounded randomized complexity and quantum communication with entanglement complexity are $\Theta(r_0+r_1)$, where $r_0$ and $r_1$ are the minimum integers such that $r_0, r_1\leq n/2$ and $S(k)=S(k+2)$ for all $k\in[r_0, n-r_1)$.

Estimating Jones and HOMFLY polynomials with one clean qubit (pp0264-0289)
S.P. Jordan and P. Wocjan
The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class\cite{Shor_Jordan}. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at \emph{any} root of unity.

High fidelity universal set of quantum gates using non-adiabatic rapid passage (pp0290-0316)
R. Li, M. Hoover, and F. Gaitan
Numerical simulation results are presented which suggest that a class of non-adiabatic rapid passage sweeps first realized experimentally in 1991 should be capable of implementing a universal set of quantum gates \uniset\ that operate with high fidelity. The gates constituting \uniset\ are the Hadamard and NOT gates, together with variants of the phase, $\pi /8$, and controlled-phase gates. The universality of \uniset\ is established by showing that it can construct the universal set consisting of Hadamard, phase, $\pi /8$, and controlled-NOT gates. Sweep parameter values are provided which simulations indicate will produce the different gates in \uniset , and for which the gate error probability $P_{e}$ satisfies: (i)~$P_{e}<10^{-4}$ for the one-qubit gates; and (ii)~$P_{e}<1.27\times 10^{-3}$ for the modified controlled-phase gate. The sweeps in this class are non-composite and generate controllable quantum interference effects that allow the gates in \uniset\ to operate non-adiabatically while maintaining high fidelity. These interference effects have been observed using NMR, and it has previously been shown how these rapid passage sweeps can be applied to atomic systems using electric fields. Here we show how these sweeps can be applied to both superconducting  charge and flux qubit systems. The simulations suggest that the universal set of gates \uniset\ produced by these rapid passage sweeps shows promise as possible elements of a fault-tolerant scheme for quantum computing.

Non-Markovian decoherence dynamics of entangled coherent states (pp0317-0335)
J.-H. An, M. Feng, and W. M. Zhang
We microscopically model the decoherence dynamics of entangled coherent states of two optical modes under the influence of vacuum fluctuation. We derive an exact master equation with time-dependent coefficients reflecting the memory effect of the environment, by using the Feynman-Vernon influence functional theory in the coherent-state representation. Under the Markov approximation, our master equation recovers the widely used Lindblad equation in quantum optics. We then investigate the non-Markovian entanglement dynamics of the two-mode entangled coherent states under vacuum fluctuation. Compared with the results in Markov limit, it shows that the non-Markovian effect enhances the disentanglement to the initially entangled coherent state. Our analysis also shows that the decoherence behaviors of the entangled coherent states depend on the symmetrical properties of the entangled coherent states as well as the couplings between the optical fields and the environment.

Classical and quantum tensor product expanders (pp0336-0360)
M.B. Hastings and A.W. Harrow
We introduce the concept of quantum tensor product expanders. These generalize the concept of quantum expanders, which are quantum maps that are efficient randomizers and use only a small number of Kraus operators. Quantum tensor product expanders act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.

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