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International Journal of Quantum Information

FIXED-POINT SEARCH WITH TWO ARBITRARY EQUAL PHASE SHIFTS FOR ANY NUMBER OF ITERATIONS

Xiangrong Li, Dafa Li, Hongtao Huang, Xinxin Li

Published Oct 1, 2010• 0 citations
Publisher

Abstract

In this paper, we investigate the convergence behavior of the fixed-point search algorithm with two arbitrary equal phase shifts for any number of iterations. We show that no matter what the initial deviation is, the algorithm converges to the target state with certainty for phase shifts between 0 and π∕2, and to the target state with the success probabilities of at least 80% and at least 66% for phase shifts between π∕2 and arccos(-1/4) and between arccos(-1/4) and 2π/3, respectively, while the algorithm does not converge for phase shifts between 2π/3 and π.

DOI
10.1142/s0219749910006745

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In this paper, we investigate the convergence behavior of the fixed-point search algorithm with two arbitrary equal phase shifts for any number of iterations. We show that no matter what the initial deviation is, the algorithm converges to the target state with certainty for phase shifts between 0 and π∕2, and to the target state with the success probabilities of at least 80% and at least 66% for phase shifts between π∕2 and arccos(-1/4) and between arccos(-1/4) and 2π/3, respectively, while the algorithm does not converge for phase shifts between 2π/3 and π.
Grover presented the fixed-point search by replacing the selective inversions by selective phase shifts of π/3π/3π/3. In this paper, we investigate the convergence behavior of the fixed-point search algorithm with general but equal phase shifts for any iterations.
One iterative in Grover's original quantum search algorithm consists of two Hadamard–Walsh transformations, a selective amplitude inversion and a diffusion amplitude inversion. We concentrate on the relation among the probability of success of the algorithm, the phase shifts, the number of target items and the number of iterations via replacing the two amplitude inversions by phase shifts of an arbitrary ϕ = ϕ(0 ⩽ ϕ, ϕ ⩽ 2π). Then, according to the relation we find out the optimal phase shifts when the number of iterations is given. We present a new quantum search algorithm based on the optimal phase shifts of 1.018 after 0.5π/ iterations. The new algorithm can obtain either a single target item or multiple target items in the search space with the probability of success at least 93.43%.
We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.
The quantum search algorithm consists of an iterative sequence of selective inversions and diffusion type operations, as a result of which it is able to find a state with desired properties (target state) in an unsorted database of size N in only sqrt[N] queries. This is achieved by designing the iterative transformations in a way that each iteration results in a small rotation of the state vector in a two-dimensional Hilbert space that includes the target state; if we choose the right number of iterative steps, we will stop just at the target state. This Letter shows that by replacing the selective inversions by selective phase shifts of pi/3, the algorithm preferentially converges to the target state irrespective of the step size or number of iterations. This feature leads to robust search algorithms and also to new schemes for quantum control and error correction.
When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.
The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-\pi/3 search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from \epsilon to \epsilon^{2q+1} using q oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-\pi/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives \epsilon^{2q+1} convergence for all integral q, whereas the Phase-\pi/3 search algorithm requires q to be (3^{n}-1)/2 with n a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.