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Fortschritte der Physik

Noisy Effect on Nonlinear Quantum Search Algorithm

J.-P. Tchapet Njafa, S. G. Nana Engo

Published Apr 7, 2021• 0 citations
Publisher

Abstract

Abstract In this paper, we investigate the effects of three single‐qubit quantum noise channels on the quantum nonlinear search algorithm proposed by Abrams and Lloyd. We focus on the gates of the nonlinear part of the algorithm and consider the case where the quantum noise channel only affects the qubit on which the gate is proceeding. Assuming that quantum noise effects occur after using the gates of the nonlinear part of the algorithm, we find that the quantum nonlinear search algorithm appears to be very resilient to the quantum noise when the desired state does not exist in the first register and is subject to the phase damping channel or the depolarising channel.

DOI
10.1002/prop.202000049

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Abstract In this paper, we investigate the effects of three single‐qubit quantum noise channels on the quantum nonlinear search algorithm proposed by Abrams and Lloyd. We focus on the gates of the nonlinear part of the algorithm and consider the case where the quantum noise channel only affects the qubit on which the gate is proceeding. Assuming that quantum noise effects occur after using the gates of the nonlinear part of the algorithm, we find that the quantum nonlinear search algorithm appears to be very resilient to the quantum noise when the desired state does not exist in the first register and is subject to the phase damping channel or the depolarising channel.
Entanglement lies at the heart of quantum mechanics and has no classical analogue. It is central to the speed up achieved by quantum algorithms over their classical counterparts. The Grover's search algorithm is one such algorithm which enables us to achieve a quadratic speed up over any known classical algorithm that searches for an element in an unstructured database. Here, we analyse and quantify the effects of entanglement in the generalized version of this algorithm for two qubits. By 'generalized', it is meant that the use of any arbitrary single qubit unitary gate is permitted to create superposed states. Our analysis has been firstly on a noise free environment and secondly in the presence of noise. In the absence of noise, we establish a relation between the concurrence and the amplitude of the final state thereby showing the explicit effects of entanglement on the same. Moreover, the effects of noisy channels, namely amplitude and phase damping channels are studied. We investigate the amount of quantum correlation in the states obtained after the phase inversion stage of the algorithm followed by interaction of those states with the noisy environment. The quantum correlations are quantified by geometric discord. It has been revealed that the states generated after the effect of amplitude damping on the phase inverted states of the quantum search algorithm possess non-zero quantum correlation even when entanglement is absent. However, this is absent in the phase damping scenario.
The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate W, which is an essential component in each iteration of the quantum search process. To this end W is replaced by a noisy Hadamard gate U. The parameters of U at each iteration are taken from an arbitrary probability distribution (e.g., a Gaussian distribution) and are characterized by their statistical moments around the parameters of W. For simplicity, we assume that the noise is unbiased and isotropic, namely, all noise variables in the parametrization we use have zero average and the same standard deviation $\ensuremath{\epsilon}.$ The noise terms at different calls to U are assumed to be uncorrelated. For a search space of size N=2n{N=2}^{n}N=2n (where n is the number of qubits used to span this space) it is found that as long as $\ensuremath{\epsilon}<{O(n}^{\ensuremath{-}1/2}{N}^{\ensuremath{-}1/4})$ the algorithm maintains significant efficiency, while above this noise level its operation is hampered completely. It is also found that below this noise threshold, when the search fails, it is likely to provide a state that differs from the marked state by only a few bits. This feature can be used to search for the marked state by a classical postprocessing, even if the quantum search has failed, thus improving the success rate of the search process.
Noise is an inherent part of quantum computation. Although there exists a fault-tolerant approach to quantum computing, this requires many computational resources. Hence, it is important to analyze how noise affects well-known algorithms. In this paper the effect of noise in Grover?s search algorithm is studied. The noise is modeled as both total depolarizing channel (TDCh), and local depolarizing channel (LDCh) in every qubit. An analysis of the order has been made analytically for the TDCh, and an approximation has been done for the LDCh.
In this work we study several models of decoherence and how different quantum maps and algorithms react when perturbed by them. Following closely Ref. [1], generalizations of the three paradigmatic one single qubit quantum channels (these are the depolarizing channel, the phase damping channel and the amplitude damping channel) for the case of an arbitrarily-sized finite-dimensional Hilbert space are presented, as well as other types of noise in phase space. More specifically, Grover's search algorithm's response to decoherence is analyzed; together with those of a family of quantum versions of chaotic and regular classical maps (the baker's map and the cat maps). A relationship between how sensitive to decoherence a quantum map is and the degree of complexity in the dynamics of its associated classical counterpart is observed; resulting in a clear tendency to react the more decoherently the more complex the associated classical dynamics is.
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.
Abstract Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.