1,122 research outputs found
On Local Testability in the Non-Signaling Setting
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions.
We prove that low-degree testing in the non-signaling setting is possible, assuming that the locality of the non-signaling function exceeds a threshold. We additionally show that if the locality is below the threshold then the test fails spectacularly, in that there exists a non-signaling function which passes the test with probability 1 and yet is maximally far from being low-degree.
Along the way, we present general results about the local testability of linear codes in the non-signaling setting. These include formulating natural definitions that capture the condition that a non-signaling function "belongs" to a given code, and characterizing the sets of local constraints that imply membership in the code. We prove these results by formulating a logical inference system for linear constraints on non-signaling functions that is complete and sound
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
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
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