



01174nas a2200133   4500008004100000245005700041210005700098260001500155490000600170520076400176100002500940700002400965856005100989       2009                        eng d00aLocality Bounds on Hamiltonians for Stabilizer Codes0 aLocality Bounds on Hamiltonians for Stabilizer Codes  c2009/09/220 v93 aIn this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. We
introduce various notions of k-locality of a stabilizer code, inherited from the associated stabilizer
group. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establish
bounds on the locality of any other Hamiltonian whose groundspace contains such a code, whether
or not its Pauli tensor summands commute. Our results provide insight into the cost of creating an
energy gap for passive error correction and for adiabatic quantum computing. The results simplify in
the cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-ordered
stabilizer codes arising from surface cellulations.
1 aBullock, Stephen, S.1 aO'Leary, Dianne, P.  uhttp://www.cs.umd.edu/~oleary/reprints/j91.pdf01525nas a2200145   4500008004100000245005200041210005200093260001400145490000700159520109700166100002401263700002301287700002501310856004401335       2006                        eng d00aParallelism for Quantum Computation with Qudits0 aParallelism for Quantum Computation with Qudits  c2006/9/280 v743 a  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.
1 aO'Leary, Dianne, P.1 aBrennen, Gavin, K.1 aBullock, Stephen, S.  uhttp://arxiv.org/abs/quant-ph/0603081v101765nas a2200145   4500008004100000245006400041210006300105260001400168490000700182520131400189100002501503700002401528700002301552856004401575       2005                        eng d00aAsymptotically Optimal Quantum Circuits for d-level Systems0 aAsymptotically Optimal Quantum Circuits for dlevel Systems  c2005/6/140 v943 a  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.
1 aBullock, Stephen, S.1 aO'Leary, Dianne, P.1 aBrennen, Gavin, K.  uhttp://arxiv.org/abs/quant-ph/0410116v201715nas a2200145   4500008004100000245004200041210004200083260001400125490000700139520130700146100002301453700002401476700002501500856004401525       2005                        eng d00aCriteria for Exact Qudit Universality0 aCriteria for Exact Qudit Universality  c2005/5/160 v713 a  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><j|+|j><k|) and H_{jk}^y =\hbar\Omega
(i|k><j|-i|j><k|). A coupling graph results taking nodes 0,... d-1 and edges
j<->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><d-1,d-1| is also allowed. We
discuss implementation in the eight dimensional ground electronic states of
^{87}Rb and construct an optimal gate sequence using Raman laser pulses.
1 aBrennen, Gavin, K.1 aO'Leary, Dianne, P.1 aBullock, Stephen, S.  uhttp://arxiv.org/abs/quant-ph/0407223v101321nas a2200157   4500008004100000022001400041245005800055210005800113260001500171300001000186490000700196520083300203100002401036700002501060856007801085       2005                        eng d  a1068-961300aQR Factorizations Using a Restricted Set of Rotations0 aQR Factorizations Using a Restricted Set of Rotations  c2005/07/11  a20-270 v213 aAny matrix A of dimension m × n (m ≥ n) can be reduced to upper triangular form
by multiplying by a sequence of mn − n(n + 1)/2 appropriately chosen rotation matrices. In this
work, we address the question of whether such a factorization exists when the set of allowed rotation
planes is restricted. We introduce the rotation graph as a tool to devise elimination orderings in
QR factorizations. Properties of this graph characterize sets of rotation planes that are sufficient (or
sufficient under permutation) and identify rotation planes to add to complete a deficient set. We
also devise a constructive way to determine all feasible rotation sequences for performing the QR
factorization using a restricted set of rotation planes. We present applications to quantum circuit
design and parallel QR factorization.1 aO'Leary, Dianne, P.1 aBullock, Stephen, S.  uhttp://www.emis.ams.org/journals/ETNA/vol.21.2005/pp20-27.dir/pp20-27.pdf01736nas a2200157   4500008004100000245005500041210005400096260001500150300001100165490000700176520127900183100002501462700002301487700002401510856004401534       2005                        eng d00aTime Reversal and n-qubit Canonical Decompositions0 aTime Reversal and nqubit Canonical Decompositions  c2005/01/01  a0621040 v463 a  For n an even number of qubits and v a unitary evolution, a matrix
decomposition v=k1 a k2 of the unitary group is explicitly computable and
allows for study of the dynamics of the concurrence entanglement monotone. The
side factors k1 and k2 of this Concurrence Canonical Decomposition (CCD) are
concurrence symmetries, so the dynamics reduce to consideration of the a
factor. In this work, we provide an explicit numerical algorithm computing v=k1
a k2 for n odd. Further, in the odd case we lift the monotone to a two-argument
function, allowing for a theory of concurrence dynamics in odd qubits. The
generalization may also be studied using the CCD, leading again to maximal
concurrence capacity for most unitaries. The key technique is to consider the
spin-flip as a time reversal symmetry operator in Wigner's axiomatization; the
original CCD derivation may be restated entirely in terms of this time
reversal. En route, we observe a Kramers' nondegeneracy: the existence of a
nondegenerate eigenstate of any time reversal symmetric n-qubit Hamiltonian
demands (i) n even and (ii) maximal concurrence of said eigenstate. We provide
examples of how to apply this work to study the kinematics and dynamics of
entanglement in spin chain Hamiltonians.
1 aBullock, Stephen, S.1 aBrennen, Gavin, K.1 aO'Leary, Dianne, P.  uhttp://arxiv.org/abs/quant-ph/0402051v2