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arXiv:quant-ph/0307141 · Physical Review A

Analysis of Grover’s quantum search algorithm as a dynamical system

Ofer Biham, Daniel Shapira, Yishai Shimoni

Published Aug 29, 2003• 15 citations
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Abstract

Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\ensuremath{\varphi}〉$ of n qubits. It is shown that the optimal time to perform the measurement is independent of $|\ensuremath{\varphi}〉,$ namely, it is identical to the optimal time in the original algorithm in which $|\ensuremath{\varphi}〉=|0〉,$ with the same number of marked states, r. The probability of success ${P}_{\mathrm{s}}$ is obtained in terms of the amplitudes of the state $|\ensuremath{\varphi}〉$ and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.

DOI
10.1103/physreva.68.022326

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Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state ∣\ensuremathφ〉|\ensuremath{\varphi}〉∣\ensuremathφ〉 of n qubits. It is shown that the optimal time to perform the measurement is independent of ∣\ensuremathφ〉,|\ensuremath{\varphi}〉,∣\ensuremathφ〉, namely, it is identical to the optimal time in the original algorithm in which ∣\ensuremathφ〉=∣0〉,|\ensuremath{\varphi}〉=|0〉,∣\ensuremathφ〉=∣0〉, with the same number of marked states, r. The probability of success Ps{P}_{\mathrm{s}}Ps​ is obtained in terms of the amplitudes of the state ∣\ensuremathφ〉|\ensuremath{\varphi}〉∣\ensuremathφ〉 and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.
The Groverian entanglement measure, introduced earlier for pure quantum states of multiple qubits [O. Biham, M.A. Nielsen, and T. Osborne, Phys. Rev. A 65, 062312 (2002)], is generalized to the case of mixed states. The Groverian measure of a mixed state of nnn qubits is obtained by a purification procedure into a pure state of 2n2n2n qubits, followed by an optimization process, before the resulting state is fed into Grover's search algorithm. It is expressed in terms of the maximal success probability of the algorithm and in this sense provides an operational measure of entanglement.
In this paper, Grover's quantum search algorithm is analyzed using a classical computer by calculating the amplitudes and the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, and 27 qubit states.The calculations show that the marked state can be found in 𝒪(√𝑁) iterations, where 𝑁 = 2 ( is the number of items.The possibility of improving Grover's search algorithm to find a single item in N search elements is discussed by calculating the amplitudes and hence the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 qubit states.The calculations showed that the marked state could be found with sufficiently high probability in 𝒪(ln (𝑁)) iterations.This is quite a remarkable speed-up that can be achieved to find a single marked element in an unsorted N search element.
Evolution of entanglement with the proceeding of quantum algorithms affects the outcome of the algorithm. Particularly, the performance of Grover's search algorithm gets worsened if the initial state of the algorithm is an entangled one. The success probability of search can be seen as an operational measure of entanglement. This paper demonstrates an entanglement measure based on the performance of Grover's search algorithm for three and five qubit systems. We also show that although the overall pattern shows growth of entanglement, its rise to a maximum and then consequent decay, the presence of local fluctuation within each iterative step is likely.
An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state $\ensuremath{\mid}\ensuremath{\psi}⟩$ of nnn qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. It unifies and generalizes earlier results on the time evolution of the amplitudes during the search, the optimal number of iterations, and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed state, providing an exact formula for the success probability.
Grover's algorithm for quantum searching is generalized to deal with arbitrary initial complex amplitude distributions. First-order linear difference equations are found for the time evolution of the amplitudes of the marked and unmarked states. These equations are solved exactly. New expressions are derived for the optimal time of measurement and the maximal probability of success. They 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. Our results imply that Grover's algorithm is robust against modest noise in the amplitude initialization procedure.
Grover proposed how a quantum computer can find a single marked object in a database of size N by using only O(N) queries of the oracle that identifies the object with the probability of finding the marked object being not exactly 1.The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state.The probability of finding the marked object is calculated and it is found that it depends strongly on the specific initial state.Our results contribute to analyze the influence of quantum noise on Grover algorithm based on pure state.Finally,we give few examples to show that for some extremely mixed initial states.