79 research outputs found
Detection of Algebraic Manipulation in the Presence of Leakage
We investigate the problem of algebraic manipulation detection (AMD) over a communication channel that partially leaks information to an adversary. We assume the adversary is computationally unbounded and there is no shared key or correlated randomness between the sender and the receiver. We introduce leakage-resilient (LR)-AMD codes to detect algebraic manipulation in this model.
We consider two leakage models. The first model, called \emph{linear leakage}, requires the adversary\u27s uncertainty (entropy) about the message (or encoding randomness) to be a constant fraction of its length. This model can be seen as an extension of the original AMD study by Cramer et al. \cite{CDFPW08} to when some leakage to the adversary is allowed. We study \emph{randomized strong} and \emph{deterministic weak} constructions of linear (L)LR-AMD codes. We derive lower and upper bounds on the redundancy of these codes and show that known optimal (in rate) AMD code constructions can serve as optimal LLR-AMD codes. In the second model, called \emph{block leakage}, the message consists of a sequence of blocks and at least one block remains with uncertainty that is a constant fraction of the block length. We focus on deterministic block (B)LR-AMD codes. We observe that designing optimal such codes is more challenging: LLR-AMD constructions cannot function optimally under block leakage. We thus introduce a new optimal BLR-AMD code construction and prove its security in the model.
We show an application of LR-AMD codes to tampering detection over wiretap channels. We next show how to compose our BLR-AMD construction, with a few other keyless primitives, to provide both integrity and confidentiality in transmission of messages/keys over such channels. This is the best known solution in terms of randomness and code redundancy. We discuss our results and suggest directions for future research
From Classical to Quantum Shannon Theory
The aim of this book is to develop "from the ground up" many of the major,
exciting, pre- and post-millenium developments in the general area of study
known as quantum Shannon theory. As such, we spend a significant amount of time
on quantum mechanics for quantum information theory (Part II), we give a
careful study of the important unit protocols of teleportation, super-dense
coding, and entanglement distribution (Part III), and we develop many of the
tools necessary for understanding information transmission or compression (Part
IV). Parts V and VI are the culmination of this book, where all of the tools
developed come into play for understanding many of the important results in
quantum Shannon theory.Comment: v8: 774 pages, 301 exercises, 81 figures, several corrections; this
draft, pre-publication copy is available under a Creative Commons
Attribution-NonCommercial-ShareAlike license (see
http://creativecommons.org/licenses/by-nc-sa/3.0/), "Quantum Information
Theory, Second Edition" is available for purchase from Cambridge University
Pres
Approximate quantum error correction for generalized amplitude damping errors
We present analytic estimates of the performances of various approximate
quantum error correction schemes for the generalized amplitude damping (GAD)
qubit channel. Specifically, we consider both stabilizer and nonadditive
quantum codes. The performance of such error-correcting schemes is quantified
by means of the entanglement fidelity as a function of the damping probability
and the non-zero environmental temperature. The recovery scheme employed
throughout our work applies, in principle, to arbitrary quantum codes and is
the analogue of the perfect Knill-Laflamme recovery scheme adapted to the
approximate quantum error correction framework for the GAD error model. We also
analytically recover and/or clarify some previously known numerical results in
the limiting case of vanishing temperature of the environment, the well-known
traditional amplitude damping channel. In addition, our study suggests that
degenerate stabilizer codes and self-complementary nonadditive codes are
especially suitable for the error correction of the GAD noise model. Finally,
comparing the properly normalized entanglement fidelities of the best
performant stabilizer and nonadditive codes characterized by the same length,
we show that nonadditive codes outperform stabilizer codes not only in terms of
encoded dimension but also in terms of entanglement fidelity.Comment: 44 pages, 8 figures, improved v
Quantum channels and memory effects
Any physical process can be represented as a quantum channel mapping an
initial state to a final state. Hence it can be characterized from the point of
view of communication theory, i.e., in terms of its ability to transfer
information. Quantum information provides a theoretical framework and the
proper mathematical tools to accomplish this. In this context the notion of
codes and communication capacities have been introduced by generalizing them
from the classical Shannon theory of information transmission and error
correction. The underlying assumption of this approach is to consider the
channel not as acting on a single system, but on sequences of systems, which,
when properly initialized allow one to overcome the noisy effects induced by
the physical process under consideration. While most of the work produced so
far has been focused on the case in which a given channel transformation acts
identically and independently on the various elements of the sequence
(memoryless configuration in jargon), correlated error models appear to be a
more realistic way to approach the problem. A slightly different, yet
conceptually related, notion of correlated errors applies to a single quantum
system which evolves continuously in time under the influence of an external
disturbance which acts on it in a non-Markovian fashion. This leads to the
study of memory effects in quantum channels: a fertile ground where interesting
novel phenomena emerge at the intersection of quantum information theory and
other branches of physics. A survey is taken of the field of quantum channels
theory while also embracing these specific and complex settings.Comment: Review article, 61 pages, 26 figures; 400 references. Final version
of the manuscript, typos correcte
Ising Model on Locally Tree-like Graphs: Uniqueness of Solutions to Cavity Equations
In the study of Ising models on large locally tree-like graphs, in both
rigorous and non-rigorous methods one is often led to understanding the
so-called belief propagation distributional recursions and its fixed points. We
prove that there is at most one non-trivial fixed point for Ising models with
zero or certain random external fields. Previously this was only known for
sufficiently ``low-temperature'' models. Our main innovation is in applying
information-theoretic ideas of channel comparison leading to a new metric
(degradation index) between binary-input-symmetric (BMS) channels under which
the Belief Propagation (BP) operator is a strict contraction (albeit
non-multiplicative). A key ingredient of our proof is a strengthening of the
classical stringy tree lemma of (Evans-Kenyon-Peres-Schulman'00).
Our result simultaneously closes the following 6 conjectures in the
literature: 1) independence of robust reconstruction accuracy to leaf noise in
broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global
information for a labeled 2-community stochastic block model, or 2-SBM
(Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under
noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in
broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary
irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-P.'21); 6)
characterization of entropy (and mutual information) of community labels given
the graph in 2-SBM (ibid)
Characterization and implementation of robust quantum information processing
Quantum information processing has practical applications like exponential speed ups in optimisation problems or the simulation of complex quantum systems. However, well controlled quantum systems realised experimentally to process the information are sensitive to noise. The progress in leading experimental platforms like superconducting qubits or trapped ions has al-lowed the realisation of high-fidelity quantum processors known as Noisy Intermediate-Scale Quantum (NISQ) devices with roughly 50 qubits. NISQ devices are meant to be large enough to show, despite their imperfections, an advantage over classical processors in some computational tasks and pro-vide a rich playground to prove principles for future quantum algorithms and protocols. However, quantum processors need to be scaled up to imple-ment quantum algorithms that are relevant for practical applications. For this purpose, Quantum Error Correction (QEC) codes, which encode the information in multi-partite quantum states that are generally highly en-tangled, become crucial to eliminate the errors introduced by noise sources like qubit loss. Here we introduce a protocol to correct qubit loss, i.e., the impossibility to access the information encoded in a qubit, in the color code, a leading candidate for fault-tolerant quantum computation. We show that the achieved tolerance of 46(1)% to qubit loss is related to a novel percola-tion problem on three coupled lattices. Our work shows the high robustness of the color under our protocol and has practical importance for implemen-tations of fault-tolerant QEC. In our second line of research we propose and analyse local entanglement witnesses as efficient and platform-agnostic detectors of the entanglement between qubit subsystems, providing a de-scription of the entanglement structure in, in principle, arbitrarily large quantum systems. Since entanglement is a genuinely quantum property used as a resource in most quantum algorithms, local witnesses, which can be implemented with current technology, are of interest for current and future quantum processors
Polar Codes: Finite Length Implementation, Error Correlations and Multilevel Modulation
Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arıkan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arıkan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which
are approximate quantum error-correcting codes specified as the ground space of
a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum
error-correcting codes with sparse, low-weight stabilizer generators (i.e. each
stabilizer generator acts on a few qubits, and each qubit participates in a few
stabilizer generators). Our investigation is motivated by an important question
in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes
with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo
polylogarithmic corrections) is possible if we go beyond stabilizer codes: we
prove the existence of a family of approximate QLDPC
codes that encode logical qubits into physical
qubits with distance and approximation infidelity
. The code space is
stabilized by a set of 10-local noncommuting projectors, with each physical
qubit only participating in projectors. We
prove the existence of an efficient encoding map, and we show that arbitrary
Pauli errors can be locally detected by circuits of polylogarithmic depth.
Finally, we show that the spectral gap of the code Hamiltonian is
by analyzing a spacetime circuit-to-Hamiltonian
construction for a bitonic sorting network architecture that is spatially local
in dimensions.Comment: 51 pages, 13 figure
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