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arXiv:quant-ph/0101126

Universal construction for the unsorted quantum search algorithms

Xijia Miao

Published Jan 26, 2001• 4 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.quant-ph/0101126

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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.
The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle the conventional NMR multiple-quantum spectroscopy has been developed to solve experimentally the search problem. The solution information of the search problem is first loaded on the unitary evolution propagator which is constructed with the oracle unitary operation and oracle-independent unitary operations and is used to excite the multiple-quantum coherence in a spin ensemble. Then the multiple- quantum spectroscopy is used to extract experimentally the solution information. It has been discussed how to enhance the output NMR signal of the quantum search NMR multiple-quantum pulse sequence and some approaches to enhancing the NMR signal are also proposed. The present work could be helpful for conventional high field NMR machines to solve efficiently the quantum search problem.
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qubit gates may be constructed physically with one- and two-body interactions in a two-state quantum system and hence could be conveniently realized experimentally. The present work will be helpful for implementing generally any N-qubit quantum computation in those feasible two-state quantum systems and determining conveniently the time evolution of these systems in course of quantum computation.
Currently,the NMR technology is one of the effective physical systems to realize quantum computation.As the multiple-quantum operator algebra theory mentions,any unitary transformation can be decomposed into a sequence of a limited number of one-qubit quantum gates and two-qubit diagonal gates.Based on these backgrounds,we propose a method to form NMR pulse sequences to realize the multi-qubit Grover quantum search algorithm.Experiments finish the algorithm with three-qubit,on a Quantum Computer Emulator.
A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.
Any quantum computational network can be constructed with a sequence of the two-qubit diagonal quantum gates and one-qubit gates in two-state quantum systems. The universal construction of these quantum gates in the quantum systems and of the quantum computational networks with these gates may be achieved with the help of the operator algebra structure of Hamiltonians of the systems and the properties of the multiple-quantum operator algebra subspaces of the Liouville operator space and the specific properties of the quantum algorithm corresponding to the quantum network. As an example, the two-qubit diagonal gates are exactly prepared in detail in superconducting Josephson junctions.
A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.