8,517 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
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
From Quantum Query Complexity to State Complexity
State complexity of quantum finite automata is one of the interesting topics
in studying the power of quantum finite automata. It is therefore of importance
to develop general methods how to show state succinctness results for quantum
finite automata. One such method is presented and demonstrated in this paper.
In particular, we show that state succinctness results can be derived out of
query complexity results.Comment: Some typos in references were fixed. To appear in Gruska Festschrift
(2014). Comments are welcome. arXiv admin note: substantial text overlap with
arXiv:1402.7254, arXiv:1309.773
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
- …