• DOI:10.1145/1073970.1073997
• Corpus ID: 6360549

Is partial quantum search of a database any easier?

@inproceedings{Grover2004IsPQ,
  title={Is partial quantum search of a database any easier?},
  author={Lov K. Grover and Jaikumar Radhakrishnan},
  booktitle={ACM Symposium on Parallelism in Algorithms and Architectures},
  year={2004}
}

• Lov K. Grover, J. Radhakrishnan
• Published in
  ACM Symposium on Parallelism…
  15 July 2004
• Computer Science

We consider the partial database search problem where given a quantum database <i>f</i> : {0,1}ⁿ→{0,1} such that <i>f</i>(<i>x</i>) =1 for a unique <i>x</i> ∈ {0,1}ⁿ, we are required to determine <i>only</i> the first <i>k</i> bits of the address <i>x</i>. We present an algorithm and derive a lower bound for this problem. Let <i>q</i>(<i>k,n</i>) be the minimum number of queries needed to find the first <i>k</i> bits of the required address <i>x</i> with certainty (or with very high probability, say 1--<i>O</i>(<i>N</i>^--¼)). We show that there exist constants <i>c<inf>k</inf></i> (corresponding to the algorithm) and <i>d<inf>k</inf></i> (corresponding to the lower bound) such that πover4 (1--<i>d<inf>k</inf></i>over√<i>K</i>) √<i>N</i> ≤ <i>q</i>(<i>k,n</i>) ≤ πover4 (1--<i>c<inf>k</inf></i>over√<i>K</i>) √<i>N</i>, where <i>K</i>=2^<i>k</i> and <i>N</i>=2^<i>n</i>. Our algorithm returns the correct answer with probability 1--<i>O</i>(<i>N</i>^--½), and can be easily modified to give the correct answer with certainty. The lower bound for algorithms that return the correct answer with certainty is proved by reducing the usual database search problem to this partial search problem, and invoking Zalka's lower bound showing that Grovers algorithm is optimal for the usual database search problem. We then derive a lower bound that is applicable for database search algorithms that err with small probability, and use it to show that our lower bound also applies to partial search algorithms that return the correct answer with probability at least 1--<i>O</i>(<i>N</i>^--¼).