188 research outputs found
An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation
This paper employs a powerful argument, called an algorithmic argument, to
prove lower bounds of the quantum query complexity of a multiple-block ordered
search problem in which, given a block number i, we are to find a location of a
target keyword in an ordered list of the i-th block. Apart from much studied
polynomial and adversary methods for quantum query complexity lower bounds, our
argument shows that the multiple-block ordered search needs a large number of
nonadaptive oracle queries on a black-box model of quantum computation that is
also supplemented with advice. Our argument is also applied to the notions of
computational complexity theory: quantum truth-table reducibility and quantum
truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the
29th International Symposium on Mathematical Foundations of Computer Science,
Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27,
200
Adversary lower bounds for nonadaptive quantum algorithms
International audienceWe present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis
Quantum Certificate Complexity
Given a Boolean function f, we study two natural generalizations of the
certificate complexity C(f): the randomized certificate complexity RC(f) and
the quantum certificate complexity QC(f). Using Ambainis' adversary method, we
exactly characterize QC(f) as the square root of RC(f). We then use this result
to prove the new relation R0(f) = O(Q2(f)^2 Q0(f) log n) for total f, where R0,
Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error
quantum query complexities respectively. Finally we give asymptotic gaps
between the measures, including a total f for which C(f) is superquadratic in
QC(f), and a symmetric partial f for which QC(f) = O(1) yet Q2(f) = Omega(n/log
n).Comment: 9 page
Optimal query complexity for estimating the trace of a matrix
Given an implicit matrix with oracle access for any
, we study the query complexity of randomized algorithms for
estimating the trace of the matrix. This problem has many applications in
quantum physics, machine learning, and pattern matching. Two metrics are
commonly used for evaluating the estimators: i) variance; ii) a high
probability multiplicative-approximation guarantee. Almost all the known
estimators are of the form for being i.i.d. for some special distribution.
Our main results are summarized as follows. We give an exact characterization
of the minimum variance unbiased estimator in the broad class of linear
nonadaptive estimators (which subsumes all the existing known estimators). We
also consider the query complexity lower bounds for any (possibly nonlinear and
adaptive) estimators: (1) We show that any estimator requires
queries to have a guarantee of variance at most
. (2) We show that any estimator requires
queries to achieve a
-multiplicative approximation guarantee with probability at
least . Both above lower bounds are asymptotically tight.
As a corollary, we also resolve a conjecture in the seminal work of Avron and
Toledo (Journal of the ACM 2011) regarding the sample complexity of the
Gaussian Estimator.Comment: full version of the paper in ICALP 201
Quantum Query Lower Bounds for Key Recovery Attacks on the Even-Mansour Cipher
The Even-Mansour (EM) cipher is one of the famous constructions for a block
cipher. Kuwakado and Morii demonstrated that a quantum adversary can recover
its -bit secret keys only with nonadaptive quantum queries. While the
security of the EM cipher and its variants is well-understood for classical
adversaries, very little is currently known of their quantum security. Towards
a better understanding of the quantum security, or the limits of quantum
adversaries for the EM cipher, we study the quantum query complexity for the
key recovery of the EM cipher and prove every quantum algorithm requires
quantum queries for the key recovery even if it is allowed to make
adaptive queries. Therefore, the quantum attack of Kuwakado and Morii has the
optimal query complexity up to a constant factor, and we cannot asymptotically
improve it even with adaptive quantum queries
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