---
title: "Aanderaa–Karp–Rosenberg conjecture"
chunk: 4/4
source: "https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:01:55.807273+00:00"
instance: "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)