8 research outputs found
Quantum Merlin-Arthur and proofs without relative phase
We study a variant of QMA where quantum proofs have no relative phase (i.e.
non-negative amplitudes, up to a global phase). If only completeness is
modified, this class is equal to QMA [arXiv:1410.2882]; but if both
completeness and soundness are modified, the class (named QMA+ by Jeronimo and
Wu) can be much more powerful. We show that QMA+ with some constant gap is
equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One
interpretation is that Merlin's ability to "deceive" originates from relative
phase at least as much as from entanglement, since QMA(2) NEXP.Comment: 18 pages, 2 figure
Distributed Quantum Proofs for Replicated Data
This paper tackles the issue of checking that all copies of a large data set replicated at several nodes of a network are identical. The fact that the replicas may be located at distant nodes prevents the system from verifying their equality locally, i.e., by having each node consult only nodes in its vicinity. On the other hand, it remains possible to assign certificates to the nodes, so that verifying the consistency of the replicas can be achieved locally. However, we show that, as the replicated data is large, classical certification mechanisms, including distributed Merlin-Arthur protocols, cannot guarantee good completeness and soundness simultaneously, unless they use very large certificates. The main result of this paper is a distributed quantum Merlin-Arthur protocol enabling the nodes to collectively check the consistency of the replicas, based on small certificates, and in a single round of message exchange between neighbors, with short messages. In particular, the certificate-size is logarithmic in the size of the data set, which gives an exponential advantage over classical certification mechanisms. We propose yet another usage of a fundamental quantum primitive, called the SWAP test, in order to show our main result
Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement
Savitch's theorem states that NPSPACE computations can be simulated in
PSPACE. We initiate the study of a quantum analogue of NPSPACE, denoted
Streaming-QCMASPACE (SQCMASPACE), where an exponentially long classical proof
is streamed to a poly-space quantum verifier. Besides two main results, we also
show that a quantum analogue of Savitch's theorem is unlikely to hold, as
SQCMASPACE=NEXP. For completeness, we introduce Streaming-QMASPACE (SQMASPACE)
with an exponentially long streamed quantum proof, and show SQMASPACE=QMA_EXP
(quantum analogue of NEXP). Our first main result shows, in contrast to the
classical setting, the solution space of a quantum constraint satisfaction
problem (i.e. a local Hamiltonian) is always connected when exponentially long
proofs are permitted. For this, we show how to simulate any Lipschitz
continuous path on the unit hypersphere via a sequence of local unitary gates,
at the expense of blowing up the circuit size. This shows quantum
error-correcting codes can be unable to detect one codeword erroneously
evolving to another if the evolution happens sufficiently slowly, and answers
an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State
Connectivity problem. Our second main result is that any SQCMASPACE computation
can be embedded into "unentanglement", i.e. into a quantum constraint
satisfaction problem with unentangled provers. Formally, we show how to embed
SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux,
Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of
scaling the promise gap with the streamed proof size. As a corollary, we obtain
the first systematic construction for obtaining QMA(2)-type upper bounds on
arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap
scales exponentially with the number of bits of communication in the
interactive proof.Comment: 60 pages, 4 figure
Preuves interactives quantiques
Cette thèse est consacrée à la complexité basée sur le paradigme des preuves interactives.
Les classes ainsi définies ont toutes en commun qu’un ou plusieurs prouveurs,
infiniment puissants, tentent de convaincre un vérificateur, de puissance bornée, de
l’appartenance d’un mot à un langage. Nous abordons ici le modèle classique, où les
participants sont des machines de Turing, et le modèle quantique, où ceux-ci sont
des circuits quantiques. La revue de littérature que comprend cette thèse s’adresse
à un lecteur déjà familier avec la complexité et l’informatique quantique.
Cette thèse présente comme résultat la caractérisation de la classe NP par une
classe de preuves interactives quantiques de taille logarithmique.
Les différentes classes sont présentées dans un ordre permettant d’aborder aussi
facilement que possible les classes interactives. Le premier chapitre est consacré aux
classes de base de la complexité ; celles-ci seront utiles pour situer les classes subséquemment
présentées. Les chapitres deux et trois présentent respectivement les
classes à un et à plusieurs prouveurs. La présentation du résultat ci-haut mentionné
est l’objet du chapitre quatre.This thesis is devoted to complexity theory based on the interactive proof paradigm.
All classes defined in this way involve one or many infinitely powerful provers
attempting to convince a verifier of limited power that a string belongs to a certain
language. We will consider the classical model, in which the various participants
are Turing machines, as well as the quantum model, in which they are quantum
circuits. The literature review included in this thesis assume that the reader is
familiar with the basics of complexity theory and quantum computing.
This thesis presents the original result that the class NP can be characterized
by a class of quantum interactive proofs of logarithmic size.
The various classes are presented in an order that facilitates the treatment of
interactive classes. The first chapter is devoted to the basic complexity classes;
these will be useful points of comparison for classes presented subsequently. Chapters
two and three respectively present classes with one and many provers. The
presentation of the result mentioned above is the object of chapter four
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum