26,827 research outputs found
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Causal interpretation of stochastic differential equations
We give a causal interpretation of stochastic differential equations (SDEs)
by defining the postintervention SDE resulting from an intervention in an SDE.
We show that under Lipschitz conditions, the solution to the postintervention
SDE is equal to a uniform limit in probability of postintervention structural
equation models based on the Euler scheme of the original SDE, thus relating
our definition to mainstream causal concepts. We prove that when the driving
noise in the SDE is a L\'evy process, the postintervention distribution is
identifiable from the generator of the SDE
Measuring Shared Information and Coordinated Activity in Neuronal Networks
Most nervous systems encode information about stimuli in the responding
activity of large neuronal networks. This activity often manifests itself as
dynamically coordinated sequences of action potentials. Since multiple
electrode recordings are now a standard tool in neuroscience research, it is
important to have a measure of such network-wide behavioral coordination and
information sharing, applicable to multiple neural spike train data. We propose
a new statistic, informational coherence, which measures how much better one
unit can be predicted by knowing the dynamical state of another. We argue
informational coherence is a measure of association and shared information
which is superior to traditional pairwise measures of synchronization and
correlation. To find the dynamical states, we use a recently-introduced
algorithm which reconstructs effective state spaces from stochastic time
series. We then extend the pairwise measure to a multivariate analysis of the
network by estimating the network multi-information. We illustrate our method
by testing it on a detailed model of the transition from gamma to beta rhythms.Comment: 8 pages, 6 figure
Classical Cryptographic Protocols in a Quantum World
Cryptographic protocols, such as protocols for secure function evaluation
(SFE), have played a crucial role in the development of modern cryptography.
The extensive theory of these protocols, however, deals almost exclusively with
classical attackers. If we accept that quantum information processing is the
most realistic model of physically feasible computation, then we must ask: what
classical protocols remain secure against quantum attackers?
Our main contribution is showing the existence of classical two-party
protocols for the secure evaluation of any polynomial-time function under
reasonable computational assumptions (for example, it suffices that the
learning with errors problem be hard for quantum polynomial time). Our result
shows that the basic two-party feasibility picture from classical cryptography
remains unchanged in a quantum world.Comment: Full version of an old paper in Crypto'11. Invited to IJQI. This is
authors' copy with different formattin
Parameter identifiability in a class of random graph mixture models
We prove identifiability of parameters for a broad class of random graph
mixture models. These models are characterized by a partition of the set of
graph nodes into latent (unobservable) groups. The connectivities between nodes
are independent random variables when conditioned on the groups of the nodes
being connected. In the binary random graph case, in which edges are either
present or absent, these models are known as stochastic blockmodels and have
been widely used in the social sciences and, more recently, in biology. Their
generalizations to weighted random graphs, either in parametric or
non-parametric form, are also of interest in many areas. Despite a broad range
of applications, the parameter identifiability issue for such models is
involved, and previously has only been touched upon in the literature. We give
here a thorough investigation of this problem. Our work also has consequences
for parameter estimation. In particular, the estimation procedure proposed by
Frank and Harary for binary affiliation models is revisited in this article
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