%0 Journal Article
%J Nature Physics
%D 2008
%T Anyonic interferometry and protected memories in atomic spin lattices
%A Liang Jiang
%A Gavin K. Brennen
%A Alexey V. Gorshkov
%A Klemens Hammerer
%A Mohammad Hafezi
%A Eugene Demler
%A Mikhail D. Lukin
%A Peter Zoller
%X Strongly correlated quantum systems can exhibit exotic behavior called topological order which is characterized by non-local correlations that depend on the system topology. Such systems can exhibit remarkable phenomena such as quasi-particles with anyonic statistics and have been proposed as candidates for naturally fault-tolerant quantum computation. Despite these remarkable properties, anyons have never been observed in nature directly. Here we describe how to unambiguously detect and characterize such states in recently proposed spin lattice realizations using ultra-cold atoms or molecules trapped in an optical lattice. We propose an experimentally feasible technique to access non-local degrees of freedom by performing global operations on trapped spins mediated by an optical cavity mode. We show how to reliably read and write topologically protected quantum memory using an atomic or photonic qubit. Furthermore, our technique can be used to probe statistics and dynamics of anyonic excitations.
%B Nature Physics
%V 4
%P 482 - 488
%8 2008/4/20
%G eng
%U http://arxiv.org/abs/0711.1365v1
%N 6
%! Nat Phys
%R 10.1038/nphys943
%0 Journal Article
%J Physical Review A
%D 2006
%T Parallelism for Quantum Computation with Qudits
%A Dianne P. O'Leary
%A Gavin K. Brennen
%A Stephen S. Bullock
%X Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the atomic alkalies $^{87}$Rb and $^{133}$Cs. Second, we provide a protocol for implementing parallelized non-local two-qudit gates using the assistance of entangled qubit pairs. Because the entangled qubits can be prepared non-deterministically, this offers the possibility of high fidelity two-qudit gates.
%B Physical Review A
%V 74
%8 2006/9/28
%G eng
%U http://arxiv.org/abs/quant-ph/0603081v1
%N 3
%! Phys. Rev. A
%R 10.1103/PhysRevA.74.032334
%0 Journal Article
%J Physical Review Letters
%D 2005
%T Asymptotically Optimal Quantum Circuits for d-level Systems
%A Stephen S. Bullock
%A Dianne P. O'Leary
%A Gavin K. Brennen
%X As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.
%B Physical Review Letters
%V 94
%8 2005/6/14
%G eng
%U http://arxiv.org/abs/quant-ph/0410116v2
%N 23
%! Phys. Rev. Lett.
%R 10.1103/PhysRevLett.94.230502
%0 Journal Article
%J Physical Review A
%D 2005
%T Criteria for Exact Qudit Universality
%A Gavin K. Brennen
%A Dianne P. O'Leary
%A Stephen S. Bullock
%X We describe criteria for implementation of quantum computation in qudits. A qudit is a d-dimensional system whose Hilbert space is spanned by states |0>, |1>,... |d-1>. An important earlier work of Mathukrishnan and Stroud [1] describes how to exactly simulate an arbitrary unitary on multiple qudits using a 2d-1 parameter family of single qudit and two qudit gates. Their technique is based on the spectral decomposition of unitaries. Here we generalize this argument to show that exact universality follows given a discrete set of single qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to the QR-matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of H_{jk}^x=\hbar\Omega (|k>k iff H_{jk}^{x,y} are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian H=-\hbar\Omega |d-1,d-1>