nt - p h / 06 03 13 6 v 1 15 M ar 2 00 6 Sure success partial search

In 1985 Deutsch designed a quantum algorithm which evaluates whether the two outputs of a Boolean function are the same or not using only one function evaluation [1]. Deutsch and Jozsa generalized this algorithm for a more general case such as whether a given Boolean function is constant or balanced. This algorithm demonstrated an exponential speed-up on a quantum machine compared to the best performance on classical machines [2]. The most important contribution in this field was achieved when Shor discovered a polynomial-time quantum algorithm for factoring and computing discrete logarithms — yielding an exponentially faster algorithm than the best known classical ones [3]. After this breakthrough many researchers started to find various applications, especially in cryptanalysis. On the other hand, Grover discovered the quantum (virtual) database search algorithm which yields a quadratic speed-up compared to classical database searches [4]. Since the database search algorithm is one of the most widely used algorithms in computer applications, the scientific impact is huge and many researchers have been interested in various applications of the quantum database search algorithm. The work presented here is concerned with a variation of the Grover search algorithm. Recently, several researchers have investigated a partial search where instead of seeking the exact location of a unique target solution, they are interested in finding which ‘target block’ the solution sits in [5, 6]. Indeed, because only a partial search is being performed an improvement in speed over the full search is expected. Indeed, recently proposed algorithms achieve meaningful performance improvements over full search [7]. Meanwhile, until now these works have considered only optimizing the performance at the expense of finding the target block with unit probability. One might hope that partial search could become an important component or subroutine of larger quantum algorithms if a sure success (unit probability) formulation could be found. The idea would be to perform the search on successively smaller block sizes with each partial search successively revealing more information about the location of the target. Indeed, in Ref. 6 the idea for a sure success partial search has been mentioned. In order to achieve this goal we utilized the scheme for partial search as described in Ref. 8 which reduces the problem to one essentially involving rotations in a three-dimensional Hilbert space. In this way we find that a simple modification, involving introducing additional phases in the final step, allows us to construct a sure success partial search algorithm. This paper is organized as follows. Firstly, we review an optimal version of the partial search algorithm, known as GRK algorithm [7]. Secondly, we propose a modification to the phases for the final step to guarantee sure success of the partial search algorithm. We derive a phase condition that must be satisfied for this modification to yield sure success and finally we show numerically that this condition may easily be solved. We conclude with a consideration of other problems that might be extensions suitable for further study.