24,315 research outputs found
Codes and Protocols for Distilling , controlled-, and Toffoli Gates
We present several different codes and protocols to distill ,
controlled-, and Toffoli (or ) gates. One construction is based on
codes that generalize the triorthogonal codes, allowing any of these gates to
be induced at the logical level by transversal . We present a randomized
construction of generalized triorthogonal codes obtaining an asymptotic
distillation efficiency . We also present a Reed-Muller
based construction of these codes which obtains a worse but performs
well at small sizes. Additionally, we present protocols based on checking the
stabilizers of magic states at the logical level by transversal gates
applied to codes; these protocols generalize the protocols of 1703.07847.
Several examples, including a Reed-Muller code for -to-Toffoli distillation,
punctured Reed-Muller codes for -gate distillation, and some of the check
based protocols, require a lower ratio of input gates to output gates than
other known protocols at the given order of error correction for the given code
size. In particular, we find a T-gate to Toffoli gate code with
distance as well as triorthogonal codes with parameters
with very low prefactors in front of
the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal
codes, added comments on Clifford circuits for Reed-Muller states (v3) minor
chang
Complementary Sets, Generalized Reed-Muller Codes, and Power Control for OFDM
The use of error-correcting codes for tight control of the peak-to-mean
envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing
(OFDM) transmission is considered in this correspondence. By generalizing a
result by Paterson, it is shown that each q-phase (q is even) sequence of
length 2^m lies in a complementary set of size 2^{k+1}, where k is a
nonnegative integer that can be easily determined from the generalized Boolean
function associated with the sequence. For small k this result provides a
reasonably tight bound for the PMEPR of q-phase sequences of length 2^m. A new
2^h-ary generalization of the classical Reed-Muller code is then used together
with the result on complementary sets to derive flexible OFDM coding schemes
with low PMEPR. These codes include the codes developed by Davis and Jedwab as
a special case. In certain situations the codes in the present correspondence
are similar to Paterson's code constructions and often outperform them
On Algebraic Decoding of -ary Reed-Muller and Product-Reed-Solomon Codes
We consider a list decoding algorithm recently proposed by Pellikaan-Wu
\cite{PW2005} for -ary Reed-Muller codes of
length when . A simple and easily accessible
correctness proof is given which shows that this algorithm achieves a relative
error-correction radius of . This is
an improvement over the proof using one-point Algebraic-Geometric codes given
in \cite{PW2005}. The described algorithm can be adapted to decode
Product-Reed-Solomon codes.
We then propose a new low complexity recursive algebraic decoding algorithm
for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a
relative error correction radius of . This technique is then proved to outperform the Pellikaan-Wu
method in both complexity and error correction radius over a wide range of code
rates.Comment: 5 pages, 5 figures, to be presented at 2007 IEEE International
Symposium on Information Theory, Nice, France (ISIT 2007
A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut
We give improved pseudorandom generators (PRGs) for Lipschitz functions of
low-degree polynomials over the hypercube. These are functions of the form
psi(P(x)), where P is a low-degree polynomial and psi is a function with small
Lipschitz constant. PRGs for smooth functions of low-degree polynomials have
received a lot of attention recently and play an important role in constructing
PRGs for the natural class of polynomial threshold functions. In spite of the
recent progress, no nontrivial PRGs were known for fooling Lipschitz functions
of degree O(log n) polynomials even for constant error rate. In this work, we
give the first such generator obtaining a seed-length of (log
n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps.
Previous generators had an exponential dependence on the degree.
We use our PRG to get better integrality gap instances for sparsest cut, a
fundamental problem in graph theory with many applications in graph
optimization. We give an instance of uniform sparsest cut for which a powerful
semi-definite relaxation (SDP) first introduced by Goemans and Linial and
studied in the seminal work of Arora, Rao and Vazirani has an integrality gap
of exp(\Omega((log log n)^{1/2})). Understanding the performance of the
Goemans-Linial SDP for uniform sparsest cut is an important open problem in
approximation algorithms and metric embeddings and our work gives a
near-exponential improvement over previous lower bounds which achieved a gap of
\Omega(log log n)
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