207 research outputs found
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
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
Generalized Quantum Arthur-Merlin Games
This paper investigates the role of interaction and coins in public-coin
quantum interactive proof systems (also called quantum Arthur-Merlin games).
While prior works focused on classical public coins even in the quantum
setting, the present work introduces a generalized version of quantum
Arthur-Merlin games where the public coins can be quantum as well: the verifier
can send not only random bits, but also halves of EPR pairs. First, it is
proved that the class of two-turn quantum Arthur-Merlin games with quantum
public coins, denoted qq-QAM in this paper, does not change by adding a
constant number of turns of classical interactions prior to the communications
of the qq-QAM proof systems. This can be viewed as a quantum analogue of the
celebrated collapse theorem for AM due to Babai. To prove this collapse
theorem, this paper provides a natural complete problem for qq-QAM: deciding
whether the output of a given quantum circuit is close to a totally mixed
state. This complete problem is on the very line of the previous studies
investigating the hardness of checking the properties related to quantum
circuits, and is of independent interest. It is further proved that the class
qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems
with perfect completeness gives new bounds for standard well-studied classes of
two-turn interactive proof systems. Finally, the collapse theorem above is
extended to comprehensively classify the role of interaction and public coins
in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the
class of problems having an m-turn quantum Arthur-Merlin proof system is either
equal to PSPACE or equal to the class of problems having a two-turn quantum
Arthur-Merlin game of a specific type, which provides a complete set of quantum
analogues of Babai's collapse theorem.Comment: 31 pages + cover page, the proof of Lemma 27 (Lemma 24 in v1) is
corrected, and a new completeness result is adde
On the power of quantum, one round, two prover interactive proof systems
We analyze quantum two prover one round interactive proof systems, in which
noninteracting provers can share unlimited entanglement. The maximum acceptance
probability is characterized as a superoperator norm. We get some partial
results about the superoperator norm, and in particular we analyze the "rank
one" case.Comment: 12 pages, no figure
Quantum interactive proofs and the complexity of separability testing
We identify a formal connection between physical problems related to the
detection of separable (unentangled) quantum states and complexity classes in
theoretical computer science. In particular, we show that to nearly every
quantum interactive proof complexity class (including BQP, QMA, QMA(2), and
QSZK), there corresponds a natural separability testing problem that is
complete for that class. Of particular interest is the fact that the problem of
determining whether an isometry can be made to produce a separable state is
either QMA-complete or QMA(2)-complete, depending upon whether the distance
between quantum states is measured by the one-way LOCC norm or the trace norm.
We obtain strong hardness results by proving that for each n-qubit maximally
entangled state there exists a fixed one-way LOCC measurement that
distinguishes it from any separable state with error probability that decays
exponentially in n.Comment: v2: 43 pages, 5 figures, completely rewritten and in Theory of
Computing (ToC) journal forma
Quantum Interactive Proofs with Competing Provers
This paper studies quantum refereed games, which are quantum interactive
proof systems with two competing provers: one that tries to convince the
verifier to accept and the other that tries to convince the verifier to reject.
We prove that every language having an ordinary quantum interactive proof
system also has a quantum refereed game in which the verifier exchanges just
one round of messages with each prover. A key part of our proof is the fact
that there exists a single quantum measurement that reliably distinguishes
between mixed states chosen arbitrarily from disjoint convex sets having large
minimal trace distance from one another. We also show how to reduce the
probability of error for some classes of quantum refereed games.Comment: 13 pages, to appear in STACS 200
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