  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.
