10,058 research outputs found
A lower bound on the quantum query complexity of read-once functions
We establish a lower bound of on the bounded-error
quantum query complexity of read-once Boolean functions, providing evidence for
the conjecture that is a lower bound for all Boolean
functions. Our technique extends a result of Ambainis, based on the idea that
successful computation of a function requires ``decoherence'' of initially
coherently superposed inputs in the query register, having different values of
the function. The number of queries is bounded by comparing the required total
amount of decoherence of a judiciously selected set of input-output pairs to an
upper bound on the amount achievable in a single query step. We use an
extension of this result to general weights on input pairs, and general
superpositions of inputs.Comment: 12 pages, LaTe
Negative weights make adversaries stronger
The quantum adversary method is one of the most successful techniques for
proving lower bounds on quantum query complexity. It gives optimal lower bounds
for many problems, has application to classical complexity in formula size
lower bounds, and is versatile with equivalent formulations in terms of weight
schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on
the principle that if an algorithm successfully computes a function then, in
particular, it is able to distinguish between inputs which map to different
values.
We present a stronger version of the adversary method which goes beyond this
principle to make explicit use of the stronger condition that the algorithm
actually computes the function. This new method, which we call ADV+-, has all
the advantages of the old: it is a lower bound on bounded-error quantum query
complexity, its square is a lower bound on formula size, and it behaves well
with respect to function composition. Moreover ADV+- is always at least as
large as the adversary method ADV, and we show an example of a monotone
function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing
that ADV+- does not face limitations of ADV like the certificate complexity
barrier and the property testing barrier.Comment: 29 pages, v2: added automorphism principle, extended to non-boolean
functions, simplified examples, added matching upper bound for AD
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
- …