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Comparative analysis for Effect of Noise in Grover’s Quantum Search Algorithm

Dushyant Kumar, Balwinder Raj, Gurmohan Singh

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract In quantum computing, Grover's technique is extremely vulnerable to several kinds of quantum noise in practice, which can seriously impair its functionality. This study examines how the algorithm's success probabilities is affected by several types of noise, such as rotational error, thermal noise, Two-State Purity (TSP) error, decay and decoherence, and depolarization variables. The findings demonstrate that the algorithm can accomplish flawless amplification of the goal state with negligible mistakes in non-target states when it is executed without noise. The most notable decrease among the noise kinds is caused by decay and decoherence errors and depolarization factors, which lower the likelihood of correctly detecting the target state from 1.0 to roughly 0.885 and 0.968, respectively. Thermal noise, TSP error, and rotational error, on the other hand, have less severe impacts and result in lesser deviations in both target and non-target states. These results demonstrate how crucial quantum error correction and noise reduction techniques are to guaranteeing Grover's algorithm's scalability and dependability in noisy quantum environments. To fully utilize the algorithm's promise in practical applications on both present-day and next quantum technology, these issues must be mitigated.

DOI
10.21203/rs.3.rs-8184953/v1

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Abstract In quantum computing, Grover's technique is extremely vulnerable to several kinds of quantum noise in practice, which can seriously impair its functionality. This study examines how the algorithm's success probabilities is affected by several types of noise, such as rotational error, thermal noise, Two-State Purity (TSP) error, decay and decoherence, and depolarization variables. The findings demonstrate that the algorithm can accomplish flawless amplification of the goal state with negligible mistakes in non-target states when it is executed without noise. The most notable decrease among the noise kinds is caused by decay and decoherence errors and depolarization factors, which lower the likelihood of correctly detecting the target state from 1.0 to roughly 0.885 and 0.968, respectively. Thermal noise, TSP error, and rotational error, on the other hand, have less severe impacts and result in lesser deviations in both target and non-target states. These results demonstrate how crucial quantum error correction and noise reduction techniques are to guaranteeing Grover's algorithm's scalability and dependability in noisy quantum environments. To fully utilize the algorithm's promise in practical applications on both present-day and next quantum technology, these issues must be mitigated.
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.
Grover’s quantum search algorithm delivers quadratic speedup over classical counterparts for finding an item in unstructured database. But the accuracy of the algorithm degrades as the number of qubits are increased. Noises of various types affect the accuracy of the Grover’s algorithm. The imperfect measurement on the qubit results in measurement errors appears to be one of the factors which limit the scalability of near-term quantum devices/algorithms. This paper is focused on the implementation of measurement error mitigation technique on Grover’s algorithm up to 4-qubits. The measurement error mitigation model for Grover’s algorithm is developed and implemented on the real-time quantum computer. The accuracy of Grover’s algorithm up to 4-qubits with and without measurement errors is evaluated and compared. The results indicate that measurement error mitigation technique mitigates the measurement errors of the Grover’s algorithm and improves its accuracy.
The quantum singular value transformation (QSVT) algorithm is a general framework to implement most of the known algorithms and provides a way forward for designing new algorithms. In the present work, the impact of noise on the QSVT algorithm is examined for bit flip, phase flip, bit-phase flip, and depolarizing noise models for a small number of qubits. The small number of noisy qubits approximates the currently available noisy quantum computers. For simulation results, the QSVT implementation of the Grover search and quantum phase estimation (QPE) algorithms is considered. These algorithms are among the basic quantum algorithms and form the building blocks of various applications of quantum algorithms. The results showed that the QSVT implementation of the Grover search and QPE algorithms has a consistently worse dependence upon noise than the original implementation for all four noise models. The probability of success of the Grover algorithm and phase measured by the QPE algorithm were found to exponentially depend upon the error probability in the noisy channels but only linearly dependent on the number of qubits.
Isotropic errors have been shown to be immune to conventional error correction techniques. While general theoretical frameworks have been proposed to model such errors, there have been no studies so far analysing their concrete impact on practical use-cases. Here we explore the effect of isotropic errors on the complexity of Grover's search algorithm through numerical simulations, with an analysis of the impact on the algorithm's performance and success probability. The results provide insights into the robustness of Grover's algorithm against isotropic errors, highlighting potential challenges for implementations on noisy quantum hardware. All results presented here are obtained through numerical simulations using the open-source python library \texttt{isotropic} developed as part of this work. The source code, numerical simulations and documentation for the library are available online at https://www.github.com/lazyoracle/isotropic-error-analysis .
It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of nnn qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.