2,371 research outputs found
Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative -matrix~ is
the minimum number~ such that there exist positive semidefinite -matrices , such that S(k,\ell) =
\mbox{tr}(A_k^* B_\ell).
The most important, lower bound technique for nonnegative rank is solely
based on the support of the matrix S, i.e., its zero/non-zero pattern. In this
paper, we characterize the power of lower bounds on positive semidefinite rank
based on solely on the support.Comment: 9 page
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
The quantum adversary method and classical formula size lower bounds
We introduce two new complexity measures for Boolean functions, or more
generally for functions of the form f:S->T. We call these measures sumPI and
maxPI. The quantity sumPI has been emerging through a line of research on
quantum query complexity lower bounds via the so-called quantum adversary
method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the
realization that these many different formulations are in fact equivalent.
Given that sumPI turns out to be such a robust invariant of a function, we
begin to investigate this quantity in its own right and see that it also has
applications to classical complexity theory.
As a surprising application we show that sumPI^2(f) is a lower bound on the
formula size, and even, up to a constant multiplicative factor, the
probabilistic formula size of f. We show that several formula size lower bounds
in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93],
including a key lemma of [Has98], are in fact special cases of our method.
The second quantity we introduce, maxPI(f), is always at least as large as
sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a
lower bound on formula size. While sumPI(f) is always a lower bound on the
quantum query complexity of f, this is not the case in general for maxPI(f). A
strong advantage of sumPI(f) is that it has both primal and dual
characterizations, and thus it is relatively easy to give both upper and lower
bounds on the sumPI complexity of functions. To demonstrate this, we look at a
few concrete examples, for three functions: recursive majority of three, a
function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200
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