3,016 research outputs found
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
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
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
On the sum-of-squares degree of symmetric quadratic functions
We study how well functions over the boolean hypercube of the form
can be approximated by sums of squares of low-degree
polynomials, obtaining good bounds for the case of approximation in
-norm as well as in -norm. We describe three
complexity-theoretic applications: (1) a proof that the recent breakthrough
lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite
extension complexity of the correlation and TSP polytopes cannot be improved
further by showing better sum-of-squares degree lower bounds on
-approximation of ; (2) a proof that Grigoriev's lower bound on
the degree of Positivstellensatz refutations for the knapsack problem is
optimal, answering an open question from his work; (3) bounds on the query
complexity of quantum algorithms whose expected output approximates such
functions.Comment: 33 pages. Second version fixes some typos and adds reference
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
Communication Complexity Lower Bounds by Polynomials
The quantum version of communication complexity allows the two communicating
parties to exchange qubits and/or to make use of prior entanglement (shared
EPR-pairs). Some lower bound techniques are available for qubit communication
complexity, but except for the inner product function, no bounds are known for
the model with unlimited prior entanglement. We show that the log-rank lower
bound extends to the strongest model (qubit communication + unlimited prior
entanglement). By relating the rank of the communication matrix to properties
of polynomials, we are able to derive some strong bounds for exact protocols.
In particular, we prove both the "log-rank conjecture" and the polynomial
equivalence of quantum and classical communication complexity for various
classes of functions. We also derive some weaker bounds for bounded-error
quantum protocols.Comment: 16 pages LaTeX, no figures. 2nd version: rewritten and some results
adde
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Polynomial approximations to boolean functions have led to many positive
results in computer science. In particular, polynomial approximations to the
sign function underly algorithms for agnostically learning halfspaces, as well
as pseudorandom generators for halfspaces. In this work, we investigate the
limits of these techniques by proving inapproximability results for the sign
function.
Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput.
2008) shows that halfspaces can be learned with respect to log-concave
distributions on in the challenging agnostic learning model. The
power of this algorithm relies on the fact that under log-concave
distributions, halfspaces can be approximated arbitrarily well by low-degree
polynomials. We ask whether this technique can be extended beyond log-concave
distributions, and establish a negative result. We show that polynomials of any
degree cannot approximate the sign function to within arbitrarily low error for
a large class of non-log-concave distributions on the real line, including
those with densities proportional to .
Secondly, we investigate the derandomization of Chernoff-type concentration
inequalities. Chernoff-type tail bounds on sums of independent random variables
have pervasive applications in theoretical computer science. Schmidt et al.
(SIAM J. Discrete Math. 1995) showed that these inequalities can be established
for sums of random variables with only -wise independence,
for a tail probability of . We show that their results are tight up to
constant factors.
These results rely on techniques from weighted approximation theory, which
studies how well functions on the real line can be approximated by polynomials
under various distributions. We believe that these techniques will have further
applications in other areas of computer science.Comment: 22 page
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