Universal construction for the unsorted quantum search algorithms

TL;DR

This paper presents a universal quantum search algorithm based on multiple-quantum operator algebra, decomposing the exponential propagator into elementary gates, and proposes an NMR implementation for efficient unsorted search.

Contribution

It introduces a general construction method for quantum search algorithms using operator algebra and symmetry, enabling practical implementation with elementary gates.

Findings

Decomposed the exponential propagator into elementary quantum gates.

Proposed an NMR-based implementation for polynomial-time unsorted search.

Showed the algorithm's complexity depends on multidimensional integration.

Abstract

The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.