71 research outputs found
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
Entanglement-Resistant Two-Prover Interactive Proof Systems and Non-Adaptive Private Information Retrieval Systems
We show that, for any language in NP, there is an entanglement-resistant
constant-bit two-prover interactive proof system with a constant completeness
vs. soundness gap. The previously proposed classical two-prover constant-bit
interactive proof systems are known not to be entanglement-resistant. This is
currently the strongest expressive power of any known constant-bit answer
multi-prover interactive proof system that achieves a constant gap. Our result
is based on an "oracularizing" property of certain private information
retrieval systems, which may be of independent interest.Comment: 8 page
Constant-Soundness Interactive Proofs for Local Hamiltonians
We give a quantum multiprover interactive proof
system for the local Hamiltonian problem in which there is a constant number of
provers, questions are classical of length polynomial in the number of qubits,
and answers are of constant length. The main novelty of our protocol is that
the gap between completeness and soundness is directly proportional to the
promise gap on the (normalized) ground state energy of the Hamiltonian. This
result can be interpreted as a concrete step towards a quantum PCP theorem
giving entangled-prover interactive proof systems for QMA-complete problems.
The key ingredient is a quantum version of the classical linearity test of
Blum, Luby, and Rubinfeld, where the function is
replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C),
the set of -dimensional Hermitian matrices that square to identity. The test
enforces that (i) each function is exactly linear,
\Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the
two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx
(-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page
Spatial Isolation Implies Zero Knowledge Even in a Quantum World
Zero knowledge plays a central role in cryptography and complexity. The
seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be
achieved unconditionally for any language in NEXP, as long as one is willing to
make a suitable physical assumption: if the provers are spatially isolated,
then they can be assumed to be playing independent strategies. Quantum
mechanics, however, tells us that this assumption is unrealistic, because
spatially-isolated provers could share a quantum entangled state and realize a
non-local correlated strategy. The MIP* model captures this setting. In this
work we study the following question: does spatial isolation still suffice to
unconditionally achieve zero knowledge even in the presence of quantum
entanglement? We answer this question in the affirmative: we prove that every
language in NEXP has a 2-prover zero knowledge interactive proof that is sound
against entangled provers; that is, NEXP \subseteq ZK-MIP*. Our proof consists
of constructing a zero knowledge interactive PCP with a strong algebraic
structure, and then lifting it to the MIP* model. This lifting relies on a new
framework that builds on recent advances in low-degree testing against
entangled strategies, and clearly separates classical and quantum tools. Our
main technical contribution consists of developing new algebraic techniques for
obtaining unconditional zero knowledge; this includes a zero knowledge variant
of the celebrated sumcheck protocol, a key building block in many probabilistic
proof systems. A core component of our sumcheck protocol is a new algebraic
commitment scheme, whose analysis relies on algebraic complexity theory.Comment: 55 pages. arXiv admin note: text overlap with arXiv:1704.0208
Spatial isolation implies zero knowledge even in a quantum world
Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP* model captures this setting. In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP*. Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP* model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory
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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