147 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 âroundedâ 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 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
The Hilbertian Tensor Norm and Entangled Two-Prover Games
We study tensor norms over Banach spaces and their relations to quantum
information theory, in particular their connection with two-prover games. We
consider a version of the Hilbertian tensor norm and its dual
that allow us to consider games with arbitrary output alphabet
sizes. We establish direct-product theorems and prove a generalized
Grothendieck inequality for these tensor norms. Furthermore, we investigate the
connection between the Hilbertian tensor norm and the set of quantum
probability distributions, and show two applications to quantum information
theory: firstly, we give an alternative proof of the perfect parallel
repetition theorem for entangled XOR games; and secondly, we prove a new upper
bound on the ratio between the entangled and the classical value of two-prover
games.Comment: 33 pages, some of the results have been obtained independently in
arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6
rewritten, v3: completely rewritten in order to improve readability; title
changed; references added; published versio
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
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
Perfect zero knowledge for quantum multiprover interactive proofs
In this work we consider the interplay between multiprover interactive
proofs, quantum entanglement, and zero knowledge proofs - notions that are
central pillars of complexity theory, quantum information and cryptography. In
particular, we study the relationship between the complexity class MIP, the
set of languages decidable by multiprover interactive proofs with quantumly
entangled provers, and the class PZKMIP, which is the set of languages
decidable by MIP protocols that furthermore possess the perfect zero
knowledge property.
Our main result is that the two classes are equal, i.e., MIP
PZKMIP. This result provides a quantum analogue of the celebrated result of
Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP
PZKMIP (in other words, all classical multiprover interactive protocols can be
made zero knowledge). We prove our result by showing that every MIP
protocol can be efficiently transformed into an equivalent zero knowledge
MIP protocol in a manner that preserves the completeness-soundness gap.
Combining our transformation with previous results by Slofstra (Forum of
Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we
obtain the corollary that all co-recursively enumerable languages (which
include undecidable problems as well as all decidable problems) have zero
knowledge MIP protocols with vanishing promise gap
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