



@article {1568,
	title = {Locality Bounds on Hamiltonians for Stabilizer Codes},
	journal = {Quantum Information and Computation},
	volume = {9},
	year = {2009},
	month = {2009/09/22},
	abstract = {In 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.
},
	url = {http://www.cs.umd.edu/~oleary/reprints/j91.pdf},
	author = {Stephen S. Bullock and Dianne P. O{\textquoteright}Leary}
}
@article {1375,
	title = {Parallelism for Quantum Computation with Qudits},
	journal = {Physical Review A},
	volume = {74},
	year = {2006},
	month = {2006/9/28},
	abstract = {  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.
},
	doi = {10.1103/PhysRevA.74.032334},
	url = {http://arxiv.org/abs/quant-ph/0603081v1},
	author = {Dianne P. O{\textquoteright}Leary and Gavin K. Brennen and Stephen S. Bullock}
}
@article {1373,
	title = {Asymptotically Optimal Quantum Circuits for d-level Systems},
	journal = {Physical Review Letters},
	volume = {94},
	year = {2005},
	month = {2005/6/14},
	abstract = {  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{\textquoteright}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{\textquoteright}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.
},
	doi = {10.1103/PhysRevLett.94.230502},
	url = {http://arxiv.org/abs/quant-ph/0410116v2},
	author = {Stephen S. Bullock and Dianne P. O{\textquoteright}Leary and Gavin K. Brennen}
}
@article {1372,
	title = {Criteria for Exact Qudit Universality},
	journal = {Physical Review A},
	volume = {71},
	year = {2005},
	month = {2005/5/16},
	abstract = {  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.
},
	doi = {10.1103/PhysRevA.71.052318},
	url = {http://arxiv.org/abs/quant-ph/0407223v1},
	author = {Gavin K. Brennen and Dianne P. O{\textquoteright}Leary and Stephen S. Bullock}
}
@article {1567,
	title = {QR Factorizations Using a Restricted Set of Rotations},
	journal = {Electronic Transactions on Numerical Analysis},
	volume = {21},
	year = {2005},
	month = {2005/07/11},
	pages = {20-27},
	abstract = {Any matrix A of dimension m {\texttimes} 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.},
	issn = {1068-9613},
	url = {http://www.emis.ams.org/journals/ETNA/vol.21.2005/pp20-27.dir/pp20-27.pdf},
	author = {Dianne P. O{\textquoteright}Leary and Stephen S. Bullock}
}
@article {1371,
	title = {Time Reversal and n-qubit Canonical Decompositions},
	journal = {Journal of Mathematical Physics},
	volume = {46},
	year = {2005},
	month = {2005/01/01},
	pages = {062104},
	abstract = {  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{\textquoteright}s axiomatization; the
original CCD derivation may be restated entirely in terms of this time
reversal. En route, we observe a Kramers{\textquoteright} 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.
},
	doi = {10.1063/1.1900293},
	url = {http://arxiv.org/abs/quant-ph/0402051v2},
	author = {Stephen S. Bullock and Gavin K. Brennen and Dianne P. O{\textquoteright}Leary}
}
