982 research outputs found
Quantum Computation Relative to Oracles
The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP, and their derivatives. This paper presents new relativized worlds in which (i) co-RP is not a subset of NQE, (ii) P=BQP and UP=EXP, (iii) P=EQP and RP=EXP, and (iv) EQP is not a subset of the union of Sigma{p}{2} and Pi{p}{2}. We also show a partial answer to the question of whether Almost-BQP=BQP
A Comparison of Quantum Oracles
A standard quantum oracle for a general function is
defined to act on two input states and return two outputs, with inputs
and () returning outputs and
. However, if is known to be a one-to-one function, a
simpler oracle, , which returns given , can also be
defined. We consider the relative strengths of these oracles. We define a
simple promise problem which minimal quantum oracles can solve exponentially
faster than classical oracles, via an algorithm which cannot be naively adapted
to standard quantum oracles. We show that can be constructed by invoking
and once each, while invocations of
and/or are required to construct .Comment: 4 pages, 1 figure; Final version, with an extended discussion of
oracle inverses. To appear in Phys Rev
Random Oracles in a Quantum World
The interest in post-quantum cryptography - classical systems that remain
secure in the presence of a quantum adversary - has generated elegant proposals
for new cryptosystems. Some of these systems are set in the random oracle model
and are proven secure relative to adversaries that have classical access to the
random oracle. We argue that to prove post-quantum security one needs to prove
security in the quantum-accessible random oracle model where the adversary can
query the random oracle with quantum states.
We begin by separating the classical and quantum-accessible random oracle
models by presenting a scheme that is secure when the adversary is given
classical access to the random oracle, but is insecure when the adversary can
make quantum oracle queries. We then set out to develop generic conditions
under which a classical random oracle proof implies security in the
quantum-accessible random oracle model. We introduce the concept of a
history-free reduction which is a category of classical random oracle
reductions that basically determine oracle answers independently of the history
of previous queries, and we prove that such reductions imply security in the
quantum model. We then show that certain post-quantum proposals, including ones
based on lattices, can be proven secure using history-free reductions and are
therefore post-quantum secure. We conclude with a rich set of open problems in
this area.Comment: 38 pages, v2: many substantial changes and extensions, merged with a
related paper by Boneh and Zhandr
Computation in generalised probabilistic theories
From the existence of an efficient quantum algorithm for factoring, it is
likely that quantum computation is intrinsically more powerful than classical
computation. At present, the best upper bound known for the power of quantum
computation is that BQP is in AWPP. This work investigates limits on
computational power that are imposed by physical principles. To this end, we
define a circuit-based model of computation in a class of operationally-defined
theories more general than quantum theory, and ask: what is the minimal set of
physical assumptions under which the above inclusion still holds? We show that
given only an assumption of tomographic locality (roughly, that multipartite
states can be characterised by local measurements), efficient computations are
contained in AWPP. This inclusion still holds even without assuming a basic
notion of causality (where the notion is, roughly, that probabilities for
outcomes cannot depend on future measurement choices). Following Aaronson, we
extend the computational model by allowing post-selection on measurement
outcomes. Aaronson showed that the corresponding quantum complexity class is
equal to PP. Given only the assumption of tomographic locality, the inclusion
in PP still holds for post-selected computation in general theories. Thus in a
world with post-selection, quantum theory is optimal for computation in the
space of all general theories. We then consider if relativised complexity
results can be obtained for general theories. It is not clear how to define a
sensible notion of an oracle in the general framework that reduces to the
standard notion in the quantum case. Nevertheless, it is possible to define
computation relative to a `classical oracle'. Then, we show there exists a
classical oracle relative to which efficient computation in any theory
satisfying the causality assumption and tomographic locality does not include
NP.Comment: 14+9 pages. Comments welcom
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