341 research outputs found
Entangled Games Are Hard to Approximate
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games
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 interactive proofs with short messages
This paper considers three variants of quantum interactive proof systems in
which short (meaning logarithmic-length) messages are exchanged between the
prover and verifier. The first variant is one in which the verifier sends a
short message to the prover, and the prover responds with an ordinary, or
polynomial-length, message; the second variant is one in which any number of
messages can be exchanged, but where the combined length of all the messages is
logarithmic; and the third variant is one in which the verifier sends
polynomially many random bits to the prover, who responds with a short quantum
message. We prove that in all of these cases the short messages can be
eliminated without changing the power of the model, so the first variant has
the expressive power of QMA and the second and third variants have the
expressive power of BQP. These facts are proved through the use of quantum
state tomography, along with the finite quantum de Finetti theorem for the
first variant.Comment: 15 pages, published versio
Quantum Multi-Prover Interactive Proof Systems with Limited Prior Entanglement
This paper gives the first formal treatment of a quantum analogue of
multi-prover interactive proof systems. It is proved that the class of
languages having quantum multi-prover interactive proof systems is necessarily
contained in NEXP, under the assumption that provers are allowed to share at
most polynomially many prior-entangled qubits. This implies that, in
particular, if provers do not share any prior entanglement with each other, the
class of languages having quantum multi-prover interactive proof systems is
equal to NEXP. Related to these, it is shown that, in the case a prover does
not have his private qubits, the class of languages having quantum
single-prover interactive proof systems is also equal to NEXP.Comment: LaTeX2e, 19 pages, 2 figures, title changed, some of the sections are
fully revised, journal version in Journal of Computer and System Science
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
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