An alternative explicit circuit diagram for the quantum search algorithm by implementing a non-unitary gate

TL;DR

This paper proposes alternative explicit circuit implementations for the quantum search algorithm using non-unitary gates, including unitary approximations, to offer potentially useful options for specific quantum platforms.

Contribution

It introduces multiple explicit unitary circuit implementations of the quantum search algorithm based on non-unitary gates and discusses their complexity and potential applications.

Findings

High-probability reading of marked elements with multiple repetitions

Alternative circuit implementations using square roots of non-unitary matrices

Potential integration into different algorithmic schemes

Abstract

Since the final quantum state in the Grover search algorithm is the normalized marked quantum state from the Gram-Schmidt process, Abrams and Lloyd [1] has shown that we can generate this vector by using a non-unitary gate. Following their ideas, in this paper, we present multiple explicit unitary implementations by using the square root of the non-unitary matrix and by a unitary matrix that mimics the Gram-Schmidt process. We also discuss the implementation through a linear combination of unitary matrices or similar methods and how these approximations may change the complexity. The reading of the marked element from the given circuits with high probability still requires multiple repetitions similar to the original algorithm. However, it gives an alternative implementations which may be useful in certain platforms. In addition, in the appendix of the paper, we show that the circuits can be used to group set elements which can be integrated into different algorithmic schemes.