7 research outputs found
Noisy Interactive Quantum Communication
We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for noiseless qudit channels (where d is arbitrary), our main result is a simulation method that fails with probability less than 2â»á¶żâœâżá”⟠and uses a qudit channel n(1 + Î(âΔ)) times, of which Δ fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the âΔ term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Surprisingly, this outperforms the best known overhead of 1 + O(â(Δ log log 1/Δ)) in the corresponding
classical model, which is also conjectured to be optimal [Haeupler, FOCSâ14]. Our work also improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCSâ14] for small Δ
Adaptive Protocols for Interactive Communication
How much adversarial noise can protocols for interactive communication
tolerate? This question was examined by Braverman and Rao (IEEE Trans. Inf.
Theory, 2014) for the case of "robust" protocols, where each party sends
messages only in fixed and predetermined rounds. We consider a new class of
non-robust protocols for Interactive Communication, which we call adaptive
protocols. Such protocols adapt structurally to the noise induced by the
channel in the sense that both the order of speaking, and the length of the
protocol may vary depending on observed noise.
We define models that capture adaptive protocols and study upper and lower
bounds on the permissible noise rate in these models. When the length of the
protocol may adaptively change according to the noise, we demonstrate a
protocol that tolerates noise rates up to . When the order of speaking may
adaptively change as well, we demonstrate a protocol that tolerates noise rates
up to . Hence, adaptivity circumvents an impossibility result of on
the fraction of tolerable noise (Braverman and Rao, 2014).Comment: Content is similar to previous version yet with an improved
presentatio
Coding for interactive communication correcting insertions and deletions
We consider the question of interactive communication, in which two remote
parties perform a computation while their communication channel is
(adversarially) noisy. We extend here the discussion into a more general and
stronger class of noise, namely, we allow the channel to perform insertions and
deletions of symbols. These types of errors may bring the parties "out of
sync", so that there is no consensus regarding the current round of the
protocol.
In this more general noise model, we obtain the first interactive coding
scheme that has a constant rate and resists noise rates of up to
. To this end we develop a novel primitive we name edit
distance tree code. The edit distance tree code is designed to replace the
Hamming distance constraints in Schulman's tree codes (STOC 93), with a
stronger edit distance requirement. However, the straightforward generalization
of tree codes to edit distance does not seem to yield a primitive that suffices
for communication in the presence of synchronization problems. Giving the
"right" definition of edit distance tree codes is a main conceptual
contribution of this work
Interactive quantum information theory
La théorie de l'information quantique s'est développée à une vitesse fulgurante au cours des vingt derniÚres années, avec des analogues et extensions des théorÚmes de codage de source et de codage sur canal bruité pour la communication unidirectionnelle. Pour la communication interactive, un analogue quantique de la complexité de la communication a été développé, pour lequel les protocoles quantiques peuvent performer exponentiellement mieux que les meilleurs protocoles classiques pour certaines tùches classiques. Cependant, l'information quantique est beaucoup plus sensible au bruit que l'information classique. Il est donc impératif d'utiliser les ressources quantiques à leur plein potentiel.
Dans cette thĂšse, nous Ă©tudions les protocoles quantiques interactifs du point de vue de la thĂ©orie de l'information et Ă©tudions les analogues du codage de source et du codage sur canal bruitĂ©. Le cadre considĂ©rĂ© est celui de la complexitĂ© de la communication: Alice et Bob veulent faire un calcul quantique biparti tout en minimisant la quantitĂ© de communication Ă©changĂ©e, sans Ă©gard au coĂ»t des calculs locaux. Nos rĂ©sultats sont sĂ©parĂ©s en trois chapitres distincts, qui sont organisĂ©s de sorte Ă ce que chacun puisse ĂȘtre lu indĂ©pendamment.
Ătant donnĂ© le rĂŽle central qu'elle occupe dans le contexte de la compression interactive, un chapitre est dĂ©diĂ© Ă l'Ă©tude de la tĂąche de la redistribution d'Ă©tat quantique. Nous prouvons des bornes infĂ©rieures sur les coĂ»ts de communication nĂ©cessaires dans un contexte interactif. Nous prouvons Ă©galement des bornes atteignables avec un seul message, dans un contexte d'usage unique.
Dans un chapitre subséquent, nous définissons une nouvelle notion de complexité de l'information quantique. Celle-ci caractérise la quantité d'information, plutÎt que de communication, qu'Alice et Bob doivent échanger pour calculer une tùche bipartie. Nous prouvons beaucoup de propriétés structurelles pour cette quantité, et nous lui donnons une interprétation opérationnelle en tant que complexité de la communication quantique amortie. Dans le cas particulier d'entrées classiques, nous donnons une autre caractérisation permettant de quantifier le coût encouru par un protocole quantique qui oublie de l'information classique. Deux applications sont présentées: le premier résultat général de somme directe pour la complexité de la communication quantique à plus d'une ronde, ainsi qu'une borne optimale, à un terme polylogarithmique prÚs, pour la complexité de la communication quantique avec un nombre de rondes limité pour la fonction « ensembles disjoints ».
Dans un chapitre final, nous initions l'Ă©tude de la capacitĂ© interactive quantique pour les canaux bruitĂ©s. Ătant donnĂ© que les techniques pour distribuer de l'intrication sont bien Ă©tudiĂ©es, nous nous concentrons sur un modĂšle avec intrication prĂ©alable parfaite et communication classique bruitĂ©e. Nous dĂ©montrons que dans le cadre plus ardu des erreurs adversarielles, nous pouvons tolĂ©rer un taux d'erreur maximal de une demie moins epsilon, avec epsilon plus grand que zĂ©ro arbitrairement petit, et ce avec un taux de communication positif. Il s'ensuit que les canaux avec bruit alĂ©atoire ayant une capacitĂ© positive pour la transmission unidirectionnelle ont une capacitĂ© positive pour la communication interactive quantique.
Nous concluons avec une discussion de nos résultats et des directions futures pour ce programme de recherche sur une théorie de l'information quantique interactive.Quantum information theory has developed tremendously over the past two decades, with analogues and extensions of the source coding and channel coding theorems for unidirectional communication. Meanwhile, for interactive communication, a quantum analogue of communication complexity has been developed, for which quantum protocols can provide exponential savings over the best possible classical protocols for some classical tasks. However, quantum information is much more sensitive to noise than classical information. It is therefore essential to make the best use possible of quantum resources.
In this thesis, we take an information-theoretic point of view on interactive quantum
protocols and study the interactive analogues of source compression and
noisy channel coding.
The setting we consider is that of quantum communication complexity:
Alice and Bob want to perform some joint quantum computation while
minimizing the required amount of communication.
Local computation is deemed free.
Our results are split
into three distinct chapters, and these are organized in such a way that each can
be read independently.
Given its central role in the context of interactive compression, we devote a chapter
to the task of quantum state redistribution. In particular, we prove lower
bounds on its communication cost that are robust in the context of interactive communication.
We also prove one-shot, one-message achievability bounds.
In a subsequent chapter, we define a new, fully quantum notion of information
cost for interactive protocols and a corresponding notion of information complexity for bipartite tasks.
It characterizes how much quantum information, rather than quantum
communication, Alice and Bob must exchange in order to implement a given bipartite task.
We prove many structural properties for these quantities, and provide an operational interpretation
for quantum information complexity as the amortized quantum communication complexity.
In the special case of classical inputs, we provide an alternate characterization of information
cost that provides an answer to the following question about quantum protocols:
what is the cost of forgetting classical information?
Two applications are presented: the first general multi-round direct-sum theorem for quantum protocols,
and a tight lower bound, up to polylogarithmic terms, for the bounded-round quantum communication complexity
of the disjointness function.
In a final chapter, we initiate the study of the interactive quantum capacity of noisy channels. Since techniques to distribute
entanglement are well-studied, we focus on a model with perfect pre-shared entanglement and noisy classical communication.
We show that even in the harder setting of adversarial errors, we can tolerate a provably maximal error rate of one half minus epsilon, for an arbitrarily small epsilon greater than zero, at positive communication rates. It then follows that random noise channels with positive capacity for unidirectional transmission also have positive interactive quantum capacity.
We conclude with a discussion of our results and further research directions in interactive quantum information theory