61 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
List decoding of noisy Reed-Muller-like codes
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are
two fundamental error-correcting codes which arise in communication as well as
in probabilistically-checkable proofs and learning. In this paper, we take the
first steps toward extending the quick randomized decoding tools of RM(1) into
the realm of quadratic binary and, equivalently, Z_4 codes. Our main
algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin
and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and
RM(2). That is, given signal s of length N, we find a list that is a superset
of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times
the norm of s, in time polynomial in k and log(N). We also give a new and
simple formulation of a known Kerdock code as a subcode of the Hankel code. As
a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm
for finding a sparse Kerdock approximation. That is, for k small compared with
1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k
log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at
most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such
approximation
Kerdock Codes Determine Unitary 2-Designs
The binary non-linear Kerdock codes are Gray images of â€_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=ââ-1 by these â€_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits
Equiangular lines, mutually unbiased bases, and spin models
We use difference sets to construct interesting sets of lines in complex
space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in
C^k when k-1 is a prime power. Using semiregular relative difference sets with
parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k.
We show how to construct these difference sets from commutative semifields and
that several known maximal sets of mutually unbiased bases can be obtained in
this way, resolving a conjecture about the monomiality of maximal sets. We also
relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in
arxiv.org:1104.337
A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
The basic methods of constructing the sets of mutually unbiased bases in the
Hilbert space of an arbitrary finite dimension are discussed and an emerging
link between them is outlined. It is shown that these methods employ a wide
range of important mathematical concepts like, e.g., Fourier transforms, Galois
fields and rings, finite and related projective geometries, and entanglement,
to mention a few. Some applications of the theory to quantum information tasks
are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two
more references adde
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
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