26,575 research outputs found
Three-dimensional color code thresholds via statistical-mechanical mapping
Three-dimensional (3D) color codes have advantages for fault-tolerant quantum
computing, such as protected quantum gates with relatively low overhead and
robustness against imperfect measurement of error syndromes. Here we
investigate the storage threshold error rates for bit-flip and phase-flip noise
in the 3D color code on the body-centererd cubic lattice, assuming perfect
syndrome measurements. In particular, by exploiting a connection between error
correction and statistical mechanics, we estimate the threshold for 1D
string-like and 2D sheet-like logical operators to be and . We obtain these
results by using parallel tempering Monte Carlo simulations to study the
disorder-temperature phase diagrams of two new 3D statistical-mechanical
models: the 4- and 6-body random coupling Ising models.Comment: 4+7 pages, 6 figures, 1 tabl
Perfect Space–Time Block Codes
In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas
Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes
Color codes are topological stabilizer codes with unusual transversality
properties. Here I show that their group of transversal gates is optimal and
only depends on the spatial dimension, not the local geometry. I also introduce
a generalized, subsystem version of color codes. In 3D they allow the
transversal implementation of a universal set of gates by gauge fixing, while
error-detecting measurements involve only 4 or 6 qubits.Comment: 10 pages, 6 figures, as accepted in journa
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
We propose a family of exactly solvable toy models for the AdS/CFT
correspondence based on a novel construction of quantum error-correcting codes
with a tensor network structure. Our building block is a special type of tensor
with maximal entanglement along any bipartition, which gives rise to an
isometry from the bulk Hilbert space to the boundary Hilbert space. The entire
tensor network is an encoder for a quantum error-correcting code, where the
bulk and boundary degrees of freedom may be identified as logical and physical
degrees of freedom respectively. These models capture key features of
entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi
formula and the negativity of tripartite information are obeyed exactly in many
cases. That bulk logical operators can be represented on multiple boundary
regions mimics the Rindler-wedge reconstruction of boundary operators from bulk
operators, realizing explicitly the quantum error-correcting features of
AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and
bibliographic amendments and minor correction
Secure Compute-and-Forward in a Bidirectional Relay
We consider the basic bidirectional relaying problem, in which two users in a
wireless network wish to exchange messages through an intermediate relay node.
In the compute-and-forward strategy, the relay computes a function of the two
messages using the naturally-occurring sum of symbols simultaneously
transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the
computed function value is forwarded to the user nodes in an ensuing broadcast
phase. In this paper, we study the problem under an additional security
constraint, which requires that each user's message be kept secure from the
relay. We consider two types of security constraints: perfect secrecy, in which
the MAC channel output seen by the relay is independent of each user's message;
and strong secrecy, which is a form of asymptotic independence. We propose a
coding scheme based on nested lattices, the main feature of which is that given
a pair of nested lattices that satisfy certain "goodness" properties, we can
explicitly specify probability distributions for randomization at the encoders
to achieve the desired security criteria. In particular, our coding scheme
guarantees perfect or strong secrecy even in the absence of channel noise. The
noise in the channel only affects reliability of computation at the relay, and
for Gaussian noise, we derive achievable rates for reliable and secure
computation. We also present an application of our methods to the multi-hop
line network in which a source needs to transmit messages to a destination
through a series of intermediate relays.Comment: v1 is a much expanded and updated version of arXiv:1204.6350; v2 is a
minor revision to fix some notational issues; v3 is a much expanded and
updated version of v2, and contains results on both perfect secrecy and
strong secrecy; v3 is a revised manuscript submitted to the IEEE Transactions
on Information Theory in April 201
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