A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.
In this paper, we study decoherence in Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers ${\ensuremath{\sigma}}_{z}$ error with probability
p(0< p< 1) independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of
2Mnp and derive its matrix element order by order under the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ limit. (In this large-n limit, we assume p is small enough, so that
2Mnp(> 0) can take any real positive value or
0.) This approach gives us an interpretation about creation of new modes caused by ${\ensuremath{\sigma}}_{z}$ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of
2Mnp (perturbative parameter), where the algorithm finds the correct item with a threshold of probability
Pth or more. It satisfies $2Mnp<(8/5)(1\ensuremath{-}{P}_{\mathrm{th}})$ around $2Mnp\ensuremath{\simeq}0$ and ${P}_{\mathrm{th}}\ensuremath{\simeq}1,$ and this linear relation is applied to a wide range of
Pth approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.
In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability
p independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an
n-qubit density operator to which Grover's iterative operation is applied
M times, we expand it in powers of
2Mnp and derive its matrix element order by order under the large-
n limit. [In this large-
n limit, we assume
p is small enough, so that
2Mnp can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter
x at which the algorithm finds a correct item with a given threshold of probability
Pth or more. (We refer to this maximum value of
x as
xc, a critical point of
x.) We obtain a curve of
xc as a function of
Pth by repeating this numerical calculation for many points of
Pth and find the following facts: a tangent of the obtained curve at
Pth=1 is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}>\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near
Pth=0.
Abstract We analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set ofmsuch ‘chosen’ elements (out of
n>m) n>m) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change
φ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers
φ φ and
α α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on
α α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change
φ, φ, and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm (
m=1,ϕ=π m=1,ϕ=π ), which are of the form, the found previously, unitary operators.
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
n qubits is obtained by a purification procedure into a pure state of
2n 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.
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.