898 research outputs found
Clustered Error Correction of Codeword-Stabilized Quantum Codes
Codeword stabilized (CWS) codes are a general class of quantum codes that
includes stabilizer codes and many families of non-additive codes with good
parameters. For such a non-additive code correcting all t-qubit errors, we
propose an algorithm that employs a single measurement to test all errors
located on a given set of t qubits. Compared with exhaustive error screening,
this reduces the total number of measurements required for error recovery by a
factor of about 3^t.Comment: 4 pages, 2 figures, revtex4; number of editorial changes in v
Structured Error Recovery for Codeword-Stabilized Quantum Codes
Codeword stabilized (CWS) codes are, in general, non-additive quantum codes
that can correct errors by an exhaustive search of different error patterns,
similar to the way that we decode classical non-linear codes. For an n-qubit
quantum code correcting errors on up to t qubits, this brute-force approach
consecutively tests different errors of weight t or less, and employs a
separate n-qubit measurement in each test. In this paper, we suggest an error
grouping technique that allows to simultaneously test large groups of errors in
a single measurement. This structured error recovery technique exponentially
reduces the number of measurements by about 3^t times. While it still leaves
exponentially many measurements for a generic CWS code, the technique is
equivalent to syndrome-based recovery for the special case of additive CWS
codes.Comment: 13 pages, 9 eps figure
Order 3 Symmetry in the Clifford Hierarchy
We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop
MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the
dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop
vectors splits into three Clifford orbits. These vectors form configurations
with so-called Zauner subspaces, eigenspaces of order 3 elements of the
Clifford group highly relevant to the SIC problem. We identify Alltop vectors
as the magic states that appear in the context of fault-tolerant universal
quantum computing, wherein the appearance of distinct Clifford orbits implies a
surprising inequivalence between some magic states.Comment: 20 pages, 2 figures. Published versio
Unitary reflection groups for quantum fault tolerance
This paper explores the representation of quantum computing in terms of
unitary reflections (unitary transformations that leave invariant a hyperplane
of a vector space). The symmetries of qubit systems are found to be supported
by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive
reflection groups, introduced (but not named) in a recent paper [Planat M and
Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The
automorphisms of multiple qubit systems are found to relate to some Clifford
operations once the corresponding group of reflections is identified. For a
short list, one may point out the Coxeter systems of type and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups and groups No 9 and 31 in
the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford
groups (the Bell groups) are generated by the Hadamard gate, the phase
gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it
New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum
computing, the lattice approach and the geometry of smooth cubic surfaces are
discussed.Comment: new version for the Journal of Computational and Theoretical
Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices
for Quantum Applications
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups
We establish braided tensor equivalences among module categories over the
twisted quantum double of a finite group defined by an extension of a group H
by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also
prove that the canonical ribbon structure of the module category of any twisted
quantum double of a finite group is preserved by braided tensor equivalences.
We give two main applications: first, if G is an extra-special 2-group of width
at least 2, we show that the quantum double of G twisted by a 3-cocycle w is
gauge equivalent to a twisted quantum double of an elementary abelian 2-group
if, and only if, w^2 is trivial; second, we discuss the gauge equivalence
classes of twisted quantum doubles of groups of order 8, and classify the
braided tensor equivalence classes of these quasi-triangular quasi-bialgebras.
It turns out that there are exactly 20 such equivalence classes.Comment: 27 pages, LateX, a few of typos in v2 correcte
Anyon computers with smaller groups
Anyons obtained from a finite gauge theory have a computational power that
depends on the symmetry group. The relationship between group structure and
computational power is discussed in this paper. In particular, it is shown that
anyons based on finite groups that are solvable but not nilpotent are capable
of universal quantum computation. This extends previously published results to
groups that are smaller, and therefore more practical. Additionally, a new
universal gate-set is built out of an operation called a probabilistic
projection, and a quasi-universal leakage correction scheme is discussed.Comment: 28 pages, REVTeX 4 (minor corrections in v2
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