56 research outputs found
The problem with the SURF scheme
There is a serious problem with one of the assumptions made in the security
proof of the SURF scheme. This problem turns out to be easy in the regime of
parameters needed for the SURF scheme to work.
We give afterwards the old version of the paper for the reader's convenience.Comment: Warning : we found a serious problem in the security proof of the
SURF scheme. We explain this problem here and give the old version of the
paper afterward
Linear programming bounds for quantum amplitude damping codes
Given that approximate quantum error-correcting (AQEC) codes have a
potentially better performance than perfect quantum error correction codes, it
is pertinent to quantify their performance. While quantum weight enumerators
establish some of the best upper bounds on the minimum distance of quantum
error-correcting codes, these bounds do not directly apply to AQEC codes.
Herein, we introduce quantum weight enumerators for amplitude damping (AD)
errors and work within the framework of approximate quantum error correction.
In particular, we introduce an auxiliary exact weight enumerator that is
intrinsic to a code space and moreover, we establish a linear relationship
between the quantum weight enumerators for AD errors and this auxiliary exact
weight enumerator. This allows us to establish a linear program that is
infeasible only when AQEC AD codes with corresponding parameters do not exist.
To illustrate our linear program, we numerically rule out the existence of
three-qubit AD codes that are capable of correcting an arbitrary AD error.Comment: 5 page
Curves of every genus with many points, I: Abelian and toric families
Let N_q(g) denote the maximal number of F_q-rational points on any curve of
genus g over the finite field F_q. Ihara (for square q) and Serre (for general
q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their
proofs they constructed curves with many points in infinitely many genera;
however, their sequences of genera are somewhat sparse. In this paper, we prove
that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian
covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we
use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} >
0; we also show that these results are the best possible that can be proved
with these families of curves.Comment: LaTeX, 20 page
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