182 research outputs found
Kerdock Codes Determine Unitary 2-Designs
The non-linear binary Kerdock codes are known to be Gray images of certain
extended cyclic codes of length over . We show that
exponentiating these -valued codewords by produces stabilizer states, that are quantum states obtained using
only Clifford unitaries. These states are also the common eigenvectors of
commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the
Pauli group. We use this quantum description to simplify the derivation of the
classical weight distribution of Kerdock codes. Next, we organize the
stabilizer states to form mutually unbiased bases and prove that
automorphisms of the Kerdock code permute their corresponding MCS, thereby
forming a subgroup of the Clifford group. When represented as symplectic
matrices, this subgroup is isomorphic to the projective special linear group
PSL(). We show that this automorphism group acts transitively on the Pauli
matrices, which implies that the ensemble is Pauli mixing and hence forms a
unitary -design. The Kerdock design described here was originally discovered
by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is
new which simplifies its description and translation to circuits significantly.
Sampling from the design is straightforward, the translation to circuits uses
only Clifford gates, and the process does not require ancillary qubits.
Finally, we also develop algorithms for optimizing the synthesis of unitary
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to
2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is
included in the arXiv packag
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
Characterisation of a family of neighbour transitive codes
We consider codes of length over an alphabet of size as subsets of
the vertex set of the Hamming graph . A code for which there
exists an automorphism group that acts transitively on the
code and on its set of neighbours is said to be neighbour transitive, and were
introduced by the authors as a group theoretic analogue to the assumption that
single errors are equally likely over a noisy channel. Examples of neighbour
transitive codes include the Hamming codes, various Golay codes, certain
Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and
frequency permutation arrays, which have connections with powerline
communication, and also completely transitive codes, a subfamily of completely
regular codes, which themselves have attracted a lot of interest. It is known
that for any neighbour transitive code with minimum distance at least 3 there
exists a subgroup of that has a -transitive action on the alphabet over
which the code is defined. Therefore, by Burnside's theorem, this action is of
almost simple or affine type. If the action is of almost simple type, we say
the code is alphabet almost simple neighbour transitive. In this paper we
characterise a family of neighbour transitive codes, in particular, the
alphabet almost simple neighbour transitive codes with minimum distance at
least , and for which the group has a non-trivial intersection with the
base group of . If is such a code, we show that, up to
equivalence, there exists a subcode that can be completely described,
and that either , or is a neighbour transitive frequency
permutation array and is the disjoint union of -translates of .
We also prove that any finite group can be identified in a natural way with a
neighbour transitive code.Comment: 30 Page
Pairwise transitive 2-designs
We classify the pairwise transitive 2-designs, that is, 2-designs such that a
group of automorphisms is transitive on the following five sets of ordered
pairs: point-pairs, incident point-block pairs, non-incident point-block pairs,
intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall
into two classes: the symmetric ones and the quasisymmetric ones. The symmetric
examples include the symmetric designs from projective geometry, the 11-point
biplane, the Higman-Sims design, and designs of points and quadratic forms on
symplectic spaces. The quasisymmetric examples arise from affine geometry and
the point-line geometry of projective spaces, as well as several sporadic
examples.Comment: 28 pages, updated after review proces
Doubly even self-orthogonal codes from quasi-symmetric designs
In this paper, we give a construction of doubly even self-orthogonal codes
from quasi-symmetric designs. Further, we study orbit matrices of
quasi-symmetric designs and give a construction of doubly even self-orthogonal
codes from orbit matrices of quasi-symmetric designs of Blokhuis-Haemers type.Comment: 13 page
- …