A simpler quantum counting algorithm based on amplitude amplification is presented. This algorithm is bounded by O(sqrt(N/M)) calls to the controlled-Grover operator where M is the number of marked states and N is the total number of states in the search space. This algorithm terminates within log(sqrt(N/M)) consecutive measurement steps when the probability p1 of measuring the state |1> is at least 0.5, and the result from the final step is used in estimating M by a classical post processing. The purpose of controlled-Grover iteration is to increase the probability p1. This algorithm requires less quantum resources in terms of the width and depth of the quantum circuit, produces a more accurate estimate of M, and runs significantly faster than the phase estimation-based quantum counting algorithm when the ratio M/N is small. We compare the two quantum counting algorithms by simulating various cases with a different M/N ratio, such as M/N > 0.125 or M/N < 0.001.
Abstract When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.
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.
Implementations of the quantum phase estimation and quantum counting algorithms with qudits are proposed. The construction of the basic building blocks of the algorithms is described in detail and the fidelity of the quantum counting algorithm with qubits and qudits is simulated numerically. Along with the exponential increase of the size of the target register, the simulations with qudits demonstrate significantly higher probability of finding the number of solutions than qubits and more consistent performance for a different number of solutions.
The quantum phase estimation (QPE) is one of the fundamental algorithms based on the quantum Fourier transform. It has applications in order-finding, factoring, and finding the eigenvalues of unitary operators. The major challenge in running QPE and other quantum algorithms is the noise in quantum computers. In the present work, we study the impact of incoherent noise on QPE, modeled as trace-preserving and completely positive quantum channels. Different noise models such as depolarizing, phase flip, bit flip, and bit-phase flip are taken to understand the performance of the QPE in the presence of noise. The simulation results indicate that the standard deviation of the eigenvalue of the unitary operator has strong exponential dependence upon the error probability of individual qubits. However, the standard deviation increases only linearly with the number of qubits for fixed error probability when that error probability is small.