6,103 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
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
NP-complete Problems and Physical Reality
Can NP-complete problems be solved efficiently in the physical universe? I
survey proposals including soap bubbles, protein folding, quantum computing,
quantum advice, quantum adiabatic algorithms, quantum-mechanical
nonlinearities, hidden variables, relativistic time dilation, analog computing,
Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and
"anthropic computing." The section on soap bubbles even includes some
"experimental" results. While I do not believe that any of the proposals will
let us solve NP-complete problems efficiently, I argue that by studying them,
we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction
Semantic Security and Indistinguishability in the Quantum World
At CRYPTO 2013, Boneh and Zhandry initiated the study of quantum-secure
encryption. They proposed first indistinguishability definitions for the
quantum world where the actual indistinguishability only holds for classical
messages, and they provide arguments why it might be hard to achieve a stronger
notion. In this work, we show that stronger notions are achievable, where the
indistinguishability holds for quantum superpositions of messages. We
investigate exhaustively the possibilities and subtle differences in defining
such a quantum indistinguishability notion for symmetric-key encryption
schemes. We justify our stronger definition by showing its equivalence to novel
quantum semantic-security notions that we introduce. Furthermore, we show that
our new security definitions cannot be achieved by a large class of ciphers --
those which are quasi-preserving the message length. On the other hand, we
provide a secure construction based on quantum-resistant pseudorandom
permutations; this construction can be used as a generic transformation for
turning a large class of encryption schemes into quantum indistinguishable and
hence quantum semantically secure ones. Moreover, our construction is the first
completely classical encryption scheme shown to be secure against an even
stronger notion of indistinguishability, which was previously known to be
achievable only by using quantum messages and arbitrary quantum encryption
circuits.Comment: 37 pages, 2 figure
On Perfect Completeness for QMA
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with
one-sided error, has been an open problem for years. This note helps to explain
why the problem is difficult, by using ideas from real analysis to give a
"quantum oracle" relative to which they are different. As a byproduct, we find
that there are facts about quantum complexity classes that are classically
relativizing but not quantumly relativizing, among them such "trivial"
containments as BQP in ZQEXP.Comment: 9 pages. To appear in Quantum Information & Computatio
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