32 research outputs found
Low-degree testing for quantum states, and a quantum entangled games PCP for QMA
We show that given an explicit description of a multiplayer game, with a
classical verifier and a constant number of players, it is QMA-hard, under
randomized reductions, to distinguish between the cases when the players have a
strategy using entanglement that succeeds with probability 1 in the game, or
when no such strategy succeeds with probability larger than 1/2. This proves
the "games quantum PCP conjecture" of Fitzsimons and the second author
(ITCS'15), albeit under randomized reductions. The core component in our
reduction is a construction of a family of two-player games for testing
-qubit maximally entangled states. For any integer , we give a test
in which questions from the verifier are bits long, and answers are
bits long. We show that for any constant
, any strategy that succeeds with probability at least
in the test must use a state that is within distance
from a state that is locally equivalent to a maximally
entangled state on qubits, for some universal constant . The
construction is based on the classical plane-vs-point test for multivariate
low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test
to the quantum regime by executing independent copies of the test in the
generalized Pauli and bases over , where is a
sufficiently large prime power, and combine the two through a test for the
Pauli twisted commutation relations. Our main complexity-theoretic result is
obtained by combining this family of games with constructions of PCPs of
proximity introduced by Ben-Sasson et al. (CCC'05), and crucially relies on a
linear property of such PCPs. Another consequence of our results is a
deterministic reduction from the games quantum PCP conjecture to a suitable
formulation of the Hamiltonian quantum PCP conjecture.Comment: 59 pages. Game sized reduced from quasipolynomial to polynomial,
yielding improved complexity-theoretic result
Non-classicality as a computational resource
One of the main questions in the field of quantum computation is where the quantum computational speed-up comes from. Recent studies in the field of quantum foundations have suggested which are the features to be considered as inherently non-classical. One of the major contributions in this direction comes from a result known as Spekkens' toy theory, which is a model built to reproduce quantum theory as a classical phase-space-inspired theory with restrictions on what an observer can know about reality. The model reproduces many of the features of quantum mechanics, but it does not reproduce non-locality and contextuality. In this thesis we first complete Spekkens' toy theory with measurement update rules and a mathematical framework that generalises it to systems of any finite dimensions (prime and non-prime). We also extend the operational equivalence between the toy theory and stabilizer quantum mechanics to all odd dimensions via Gross' Wigner functions. We then use the toy theory to represent the non-contextual and classically simulatable part of the computation in state-injection schemes of quantum computation where contextuality is a resource. In the case of qubits, we show that the subtheories of quantum mechanics represented in the toy model can achieve the full stabilizer theory via state-injection and we associate different proofs of contextuality to different injection processes. Stepping back from Spekkens' toy theory, we conclude by focusing on single system protocols that compute non-linear functions (similarly to the popular CHSH game) which show quantum advantages even in absence of non-locality and contextuality (in its standard notions). We analyse their performances (formalised in Bell's and Tsirelson's bounds) in relation to Landauer's principle, which associates entropic costs to irreversible computations, and to a new notion of contextuality for sequences of transformations
Approximation, Proof Systems, and Correlations in a Quantum World
This thesis studies three topics in quantum computation and information: The
approximability of quantum problems, quantum proof systems, and non-classical
correlations in quantum systems.
In the first area, we demonstrate a polynomial-time (classical) approximation
algorithm for dense instances of the canonical QMA-complete quantum constraint
satisfaction problem, the local Hamiltonian problem. In the opposite direction,
we next introduce a quantum generalization of the polynomial-time hierarchy,
and define problems which we prove are not only complete for the second level
of this hierarchy, but are in fact hard to approximate.
In the second area, we study variants of the interesting and stubbornly open
question of whether a quantum proof system with multiple unentangled quantum
provers is equal in expressive power to a proof system with a single quantum
prover. Our results concern classes such as BellQMA(poly), and include a novel
proof of perfect parallel repetition for SepQMA(m) based on cone programming
duality.
In the third area, we study non-classical quantum correlations beyond
entanglement, often dubbed "non-classicality". Among our results are two novel
schemes for quantifying non-classicality: The first proposes the new paradigm
of exploiting local unitary operations to study non-classical correlations, and
the second introduces a protocol through which non-classical correlations in a
starting system can be "activated" into distillable entanglement with an
ancilla system.
An introduction to all required linear algebra and quantum mechanics is
included.Comment: PhD Thesis, 240 page
Computação quântica : autômatos, jogos e complexidade
Orientador: Arnaldo Vieira MouraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Desde seu surgimento, Teoria da Computação tem lidado com modelos computacionais de maneira matemática e abstrata. A noção de computação eficiente foi investigada usando esses modelos sem procurar entender as capacidades e limitações inerentes ao mundo físico. A Computação Quântica representa uma ruptura com esse paradigma. Enraizada nos postulados da Mecânica Quântica, ela é capaz de atribuir um sentido físico preciso à computação segundo nosso melhor entendimento da natureza. Esses postulados dão origem a propriedades fundamentalmente diferentes, uma em especial, chamada emaranhamento, é de importância central para computação e processamento de informação. O emaranhamento captura uma noção de correlação que é única a modelos quânticos. Essas correlações quânticas podem ser mais fortes do que qualquer correlação clássica estando dessa forma no coração de algumas capacidades quânticas que vão além do clássico. Nessa dissertação, nós investigamos o emaranhamento da perspectiva da complexidade computacional quântica. Mais precisamente, nós estudamos uma classe bem conhecida, definida em termos de verificação de provas, em que um verificador tem acesso à múltiplas provas não emaranhadas (QMA(k)). Assumir que as provas não contêm correlações quânticas parece ser uma hipótese não trivial, potencialmente fazendo com que essa classe seja maior do que aquela em que há apenas uma prova. Contudo, encontrar cotas de complexidade justas para QMA(k) permanece uma questão central sem resposta por mais de uma década. Nesse contexto, nossa contribuição é tripla. Primeiramente, estudamos classes relacionadas mostrando como alguns recursos computacionais podem afetar seu poder de forma a melhorar a compreensão a respeito da própria classe QMA(k). Em seguida, estabelecemos uma relação entre Probabilistically Checkable Proofs (PCP) clássicos e QMA(k). Isso nos permite recuperar resultados conhecidos de maneira unificada e simplificada. Para finalizar essa parte, mostramos que alguns caminhos para responder essa questão em aberto estão obstruídos por dificuldades computacionais. Em um segundo momento, voltamos nossa atenção para modelos restritos de computação quântica, mais especificamente, autômatos quânticos finitos. Um modelo conhecido como Two-way Quantum Classical Finite Automaton (2QCFA) é o objeto principal de nossa pesquisa. Seu estudo tem o intuito de revelar o poder computacional provido por memória quântica de dimensão finita. Nos estendemos esse autômato com a capacidade de colocar um número finito de marcadores na fita de entrada. Para qualquer número de marcadores, mostramos que essa extensão é mais poderosa do que seus análogos clássicos determinístico e probabilístico. Além de trazer avanços em duas linhas complementares de pesquisa, essa dissertação provê uma vasta exposição a ambos os campos: complexidade computacional e autômatosAbstract: Since its inception, Theoretical Computer Science has dealt with models of computation primarily in a very abstract and mathematical way. The notion of efficient computation was investigated using these models mainly without seeking to understand the inherent capabilities and limitations of the actual physical world. In this regard, Quantum Computing represents a rupture with respect to this paradigm. Rooted on the postulates of Quantum Mechanics, it is able to attribute a precise physical notion to computation as far as our understanding of nature goes. These postulates give rise to fundamentally different properties one of which, namely entanglement, is of central importance to computation and information processing tasks. Entanglement captures a notion of correlation unique to quantum models. This quantum correlation can be stronger than any classical one, thus being at the heart of some quantum super-classical capabilities. In this thesis, we investigate entanglement from the perspective of quantum computational complexity. More precisely, we study a well known complexity class, defined in terms of proof verification, in which a verifier has access to multiple unentangled quantum proofs (QMA(k)). Assuming the proofs do not exhibit quantum correlations seems to be a non-trivial hypothesis, potentially making this class larger than the one in which only a single proof is given. Notwithstanding, finding tight complexity bounds for QMA(k) has been a central open question in quantum complexity for over a decade. In this context, our contributions are threefold. Firstly, we study closely related classes showing how computational resources may affect its power in order to shed some light on \QMA(k) itself. Secondly, we establish a relationship between classical Probabilistically Checkable Proofs and QMA(k) allowing us to recover known results in unified and simplified way, besides exposing the interplay between them. Thirdly, we show that some paths to settle this open question are obstructed by computational hardness. In a second moment, we turn our attention to restricted models of quantum computation, more specifically, quantum finite automata. A model known as Two-way Quantum Classical Finite Automaton (2QCFA) is the main object of our inquiry. Its study is intended to reveal the computational power provided by finite dimensional quantum memory. We extend this automaton with the capability of placing a finite number of markers in the input tape. For any number of markers, we show that this extension is more powerful than its classical deterministic and probabilistic analogues. Besides bringing advances to these two complementary lines of inquiry, this thesis also provides a vast exposition to both subjects: computational complexity and automata theoryMestradoCiência da ComputaçãoMestre em Ciência da Computaçã
MIP*=RE
We show that the class MIP* of languages that can be decided by a classical
verifier interacting with multiple all-powerful quantum provers sharing
entanglement is equal to the class RE of recursively enumerable languages. Our
proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS
2018) and the classical low-individual degree test of (Ji, et al., 2020) by
integrating recent developments from (Natarajan and Wright, FOCS 2019) and
combining them with the recursive compression framework of (Fitzsimons et al.,
STOC 2019).
An immediate byproduct of our result is that there is an efficient reduction
from the Halting Problem to the problem of deciding whether a two-player
nonlocal game has entangled value or at most . Using a known
connection, undecidability of the entangled value implies a negative answer to
Tsirelson's problem: we show, by providing an explicit example, that the
closure of the set of quantum tensor product correlations is strictly
included in the set of quantum commuting correlations. Following work
of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our
results provide a refutation of Connes' embedding conjecture from the theory of
von Neumann algebras.Comment: 206 pages. v2: Updated to use arXiv:2009.12982. New appendi