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arXiv:quant-ph/0112035 · Physical Review A

General SU(2) formulation for quantum searching with certainty

Jin-Yuan Hsieh, Che‐Ming Li

Published May 7, 2002• 16 citations
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Abstract

A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel are also studied.

DOI
10.1103/physreva.65.052322

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arXiv:quant-ph/0010077•Physical Review A•Dec 13, 2000•72 citations
A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To search a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a complex phase in its amplitude. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel are also studied.
Each iteration in Grover’s original quantum search algorithm contains 4 steps: two Hadamard-Walsh transformations and two amplitudes inversions. When the inversion of the marked state is replaced by arbitrary phase rotation θ and the inversion for the prepared state |γ 〉 is replaced by rotation through φ, we found that these phase rotations must satisfy a matching condition θ = φ. Approximate formula for the amplitude of the marked state after an arbitrary number of iterations are also derived. When phase matching is obtained, we found that the generalized quantum search is still a rotation in 2-dimensional space, but with a small angle β ′ ≈ (2sin θ 2)β, where θ is the angle of phase rotation of the marked state and 2β is the angle of U(2) rotation when the phase rotations are the inversions in the original version. We give also a simple explanation of the phase matching requirement.
A general consideration to the relation between the general SU(2) formulation for quantum searching with certainty and its corresponding Hamiltonian is undertaken in this work. We have derived the general Hamiltonian from the SU(2) formulation and reduced it to all known search Hamiltonians. A quantum mechanical algorithm is a complete and straightforward set of directions, based on the quntum physical process which possess the unique features of superpositions and entanglement, to accomplish a computation in a finite number of steps. Each step in the algorithm is designed by a sequence of unitary operators that transform the initial state to a desired one. We usually consider that these transformations are composed of several universal quantum logic gates. Therefore, this way to think about the construction of a quantum mechanical algorithm is called the quantum circuit model for quantum computing. Famous examples include the quantum factorizing algorithm discorvered by Shor[1] and the quantum searching algorithm well developed by Grover[2]. The latter is what we intend to discuss in this work.
We present a general phase matching condition for the quantum search algorithm with arbitrary unitary transformation and arbitrary phase rotations. We show by an explicit expression that the phase matching condition depends both on the unitary transformation U and the initial state. Assuming that the initial amplitude distribution is an arbitrary superposition sinθ_0 |1> + cosθ_0 e^{iδ} |2> with |1> = {1 / sinβ} \sum_k |τ_k> and |2> = {1 / cosβ} \sum_{i \ne τ}|i> , where |τ_k> is a marked state and \sinβ= \sqrt{\sum_k|U_{τ_k 0}|^2} is determined by the matrix elements of unitary transformation U between |τ_k> and the |0> state, then the general phase matching condition is tan{θ/ 2} [cos 2β+ tanθ_0 cosδsin 2β]= tan{ϕ/ 2} [1-tanθ_0 sinδsin 2βtan{θ/ 2}], where θand ϕare the phase rotation angles for |0> and |τ_k>, respectively. This generalizes previous conclusions in which the dependence of phase matching condition on UUU and the initial state has been disguised. We show that several phase conditions previously discussed in the literature are special cases of this general one, which clarifies the question of which condition should be regarded as exact.
An SO(3) picture of the generalized Grover's quantum searching algorithm,with arbitrary unitary transformation and with arbitrary phase rotations, is constructed. In this picture, any quantum search operation is a rotation in a 3 dimensional space. Exact formulas for the rotation angle and rotational axis are given. The probability of finding the marked state is just (z+1)/2(z+1)/2(z+1)/2, where z is the z-component of the state vector. Exact formulas for this probability is easily obtained. The phase matching requirement and the failure of algorithm when phase mismatches are clearly explained.
Abstract We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ , at the cost of decreased efficiency in the smaller λ region.
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution, is expressed using first-order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, Pmax.{P}_{\mathrm{max}}.Pmax​. Both T and Pmax{P}_{\mathrm{max}}Pmax​ are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.