4 research outputs found
Rational approximations and quantum algorithms with postselection
We study the close connection between rational functions that approximate a
given Boolean function, and quantum algorithms that compute the same function
using postselection. We show that the minimal degree of the former equals (up
to a factor of 2) the minimal query complexity of the latter. We give optimal
(up to constant factors) quantum algorithms with postselection for the Majority
function, slightly improving upon an earlier algorithm of Aaronson. Finally we
show how Newman's classic theorem about low-degree rational approximation of
the absolute-value function follows from these algorithms.Comment: v2: 12 pages LaTeX, to appear in Quantum Information and Computation.
Compared to version 1, the writing has been improved but the results are
unchange
Algorithmic Polynomials
The approximate degree of a Boolean function is
the minimum degree of a real polynomial that approximates pointwise within
. Upper bounds on approximate degree have a variety of applications in
learning theory, differential privacy, and algorithm design in general. Nearly
all known upper bounds on approximate degree arise in an existential manner
from bounds on quantum query complexity. We develop a first-principles,
classical approach to the polynomial approximation of Boolean functions. We use
it to give the first constructive upper bounds on the approximate degree of
several fundamental problems:
- for the -element
distinctness problem;
- for the -subset sum problem;
- for any -DNF or -CNF formula;
- for the surjectivity problem.
In all cases, we obtain explicit, closed-form approximating polynomials that
are unrelated to the quantum arguments from previous work. Our first three
results match the bounds from quantum query complexity. Our fourth result
improves polynomially on the quantum query complexity of the
problem and refutes the conjecture by several experts that surjectivity has
approximate degree . In particular, we exhibit the first natural
problem with a polynomial gap between approximate degree and quantum query
complexity