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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aanderaa–Karp–Rosenberg conjecture | 4/4 | https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture | reference | science, encyclopedia | 2026-05-05T11:01:55.807273+00:00 | kb-cron |
== Quantum query complexity == For bounded-error quantum query complexity, the best known lower bound is
Ω
(
n
2
/
3
(
log
n
)
1
/
6
)
{\displaystyle \Omega {\bigl (}n^{2/3}(\log n)^{1/6}{\bigr )}}
as observed by Andrew Yao. It is obtained by combining the randomized lower bound with the quantum adversary method. The best possible lower bound one could hope to achieve is
Ω
(
n
)
{\displaystyle \Omega (n)}
, unlike the classical case, due to Grover's algorithm which gives an
O
(
n
)
{\displaystyle O(n)}
-query algorithm for testing the monotone property of non-emptiness. Similar to the deterministic and randomized case, there are some properties which are known to have an
Ω
(
n
)
{\displaystyle \Omega (n)}
lower bound, for example non-emptiness (which follows from the optimality of Grover's algorithm) and the property of containing a triangle. There are some graph properties which are known to have an
Ω
(
n
3
/
2
)
{\displaystyle \Omega (n^{3/2})}
lower bound, and even some properties with an
Ω
(
n
2
)
{\displaystyle \Omega (n^{2})}
lower bound. For example, the monotone property of nonplanarity requires
Θ
(
n
3
/
2
)
{\displaystyle \Theta (n^{3/2})}
queries, and the monotone property of containing more than half the possible number of edges (also called the majority function) requires
Θ
(
n
2
)
{\displaystyle \Theta (n^{2})}
queries.
== Notes ==
== References ==
== Further reading == Bollobás, Béla (2004), "Chapter VIII. Complexity and packing", Extremal Graph Theory, New York: Dover Publications, pp. 401–437, ISBN 978-0-486-43596-1. Lovász, László; Young, Neal E. (2002), "Lecture Notes on Evasiveness of Graph Properties", arXiv:cs/0205031v1 Chronaki, Catherine E (1990), A survey of Evasiveness: Lower Bounds on the Decision-Tree Complexity of Boolean Functions, CiteSeerX 10.1.1.37.1041. Michael Saks, Decision Trees: Problems and Results, Old and New (PDF)