5 research outputs found
On the uselessness of quantum queries
Given a prior probability distribution over a set of possible oracle
functions, we define a number of queries to be useless for determining some
property of the function if the probability that the function has the property
is unchanged after the oracle responds to the queries. A familiar example is
the parity of a uniformly random Boolean-valued function over ,
for which classical queries are useless. We prove that if classical
queries are useless for some oracle problem, then quantum queries are also
useless. For such problems, which include classical threshold secret sharing
schemes, our result also gives a new way to obtain a lower bound on the quantum
query complexity, even in cases where neither the function nor the property to
be determined is Boolean
Optimal quantum algorithm for polynomial interpolation
We consider the number of quantum queries required to determine the
coefficients of a degree-d polynomial over GF(q). A lower bound shown
independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2
quantum queries are needed to solve this problem with bounded error, whereas an
algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We
show that the lower bound is achievable: d/2+1/2 quantum queries suffice to
determine the polynomial with bounded error. Furthermore, we show that d/2+1
queries suffice to achieve probability approaching 1 for large q. These upper
bounds improve results of Boneh and Zhandry on the insecurity of cryptographic
protocols against quantum attacks. We also show that our algorithm's success
probability as a function of the number of queries is precisely optimal.
Furthermore, the algorithm can be implemented with gate complexity poly(log q)
with negligible decrease in the success probability. We end with a conjecture
about the quantum query complexity of multivariate polynomial interpolation.Comment: 17 pages, minor improvements, added conjecture about multivariate
interpolatio
Oracles and query lower bounds in generalised probabilistic theories
We investigate the connection between interference and computational power
within the operationally defined framework of generalised probabilistic
theories. To compare the computational abilities of different theories within
this framework we show that any theory satisfying three natural physical
principles possess a well-defined oracle model. Indeed, we prove a subroutine
theorem for oracles in such theories which is a necessary condition for the
oracle to be well-defined. The three principles are: causality (roughly, no
signalling from the future), purification (each mixed state arises as the
marginal of a pure state of a larger system), and strong symmetry existence of
non-trivial reversible transformations). Sorkin has defined a hierarchy of
conceivable interference behaviours, where the order in the hierarchy
corresponds to the number of paths that have an irreducible interaction in a
multi-slit experiment. Given our oracle model, we show that if a classical
computer requires at least n queries to solve a learning problem, then the
corresponding lower bound in theories lying at the kth level of Sorkin's
hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum
oracle needed to solve certain problems are not optimal in the space of all
generalised probabilistic theories, although it is not yet known whether the
optimal bounds are achievable in general. Hence searches for higher-order
interference are not only foundationally motivated, but constitute a search for
a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue
"Foundational Aspects of Quantum Information" in Foundations of Physic
Uselessness for an Oracle Model with Internal Randomness
We consider a generalization of the standard oracle model in which the oracle
acts on the target with a permutation selected according to internal random
coins. We describe several problems that are impossible to solve classically
but can be solved by a quantum algorithm using a single query; we show that
such infinity-vs-one separations between classical and quantum query
complexities can be constructed from much weaker separations.
We also give conditions to determine when oracle problems---either in the
standard model, or in any of the generalizations we consider---cannot be solved
with success probability better than random guessing would achieve. In the
oracle model with internal randomness where the goal is to gain any nonzero
advantage over guessing, we prove (roughly speaking) that quantum queries
are equivalent in power to classical queries, thus extending results of
Meyer and Pommersheim.Comment: 18 pages. v2. shortened, presentation improved, same result