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arXiv:1712.04717

The influence of quantum noise on the Grover algorithm and quantum Fourier transform: quantum operations theory approach

Yu. I. Bogdanov, Andrey Yu. Chernyavskiy, B. I. Bantysh, Dmitriy V. Fastovets, V. F. Likichev

Published Dec 13, 2017• 0 citations
arXivHTMLPDF

Abstract

The method of noisy multiqubit quantum circuits modeling is proposed. The analytical formulas for the dependence of quantum algorithms accuracy on qubits count and noise level are obtained for Grover algorithm and quantum Fourier transform. It is shown that the proposed approach is very much in line with results obtained by Monte Carlo statistical modeling method. The developed theory makes it possible to predict the influence of quantum noise on the accuracy of the forward-looking multiqubit quantum systems under development.

DOI
10.48550/arxiv.1712.04717

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The method of noisy multiqubit quantum circuits modeling is proposed. The analytical formulas for the dependence of quantum algorithms accuracy on qubits count and noise level are obtained for Grover algorithm and quantum Fourier transform. It is shown that the proposed approach is very much in line with results obtained by Monte Carlo statistical modeling method. The developed theory makes it possible to predict the influence of quantum noise on the accuracy of the forward-looking multiqubit quantum systems under development.
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Abstract The effectiveness of the implementation of the quantum Fourier transform (QFT) and the Grover algorithm on qudit systems is investigated. A comparative analysis of computing resources for quantum states of N ≥ 1024 dimension is carried out. It is shown that using qudits can significantly reduce the number of quantum operations in Grover’s algorithm compared to its qubit counterparts, while for QFT the main advantage is to reduce the number of physical carriers. The effect of noise using the Monte Carlo method on the accuracy of algorithms for qubit systems of various dimensions is analyzed.
We propose using variational quantum algorithms (VQAs) to simulate established quantum algorithms under realistic noise conditions, aiming to surpass the fidelity of theoretical circuits in noisy environments. Focusing on the Quantum Fourier Transform (QFT), we perform numerical simulations for two qubits under both coherent and incoherent noise. To enhance generalization, we further introduce the use of Mutually Unbiased Bases (MUBs) during the optimization. Our results show that the variational circuit can reproduce the QFT with higher fidelity in scenarios dominated by coherent noise. This demonstrates the potential of the approach as an effective error-mitigation strategy for small- to medium-scale quantum systems, particularly in settings where coherent noise strongly impacts performance. Beyond mitigating noise and improving fidelity, the method can be adapted to the noise profile of a specific device, providing a versatile and practical route to enhance the reliability of quantum algorithms in near-term quantum hardware.
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The simulation of quantum systems on a quantum computer using the Zalka–Wiesner algorithm including quantum noise has been considered. The efficiency of developed methods and algorithms has been demonstrated by example of the solution of the time-dependent Schrödinger equation for a particle in the Pöschl–Teller potential. The developed analytical theory of the effect of quantum noise on the accuracy of simulation has been compared with Monte Carlo simulations. The accuracy of the solution of the Schrödinger equation for a many-electron system has been forecasted as a function of the number of electrons and for different levels of noise.
In order to understand the bounds of utilization of Grover's search algorithm for large unstructured data in the presence of quantum computer noise, we undertake a series of simulations by inflicting various types of noise, modeled by the IBM qiskit. We apply three forms of Grover's algorithm: (i) the standard one, with 4--10 qubits; (ii) a recently published modified Grover's algorithm, set to reduce the circuit depth; and (iii) the algorithms in (i) and (ii) with multicontrol Toffoli's gates modified by the addition of an ancilla qubit. Based on these simulations, we find the upper bound of noise for these cases, establish its dependence on the quantum depth of the circuit, and provide comparison among them. By extrapolation of the fitted thresholds, we predict what would be the typical gate error bounds when applying Grover's algorithms for the search of data in a data set as large as 32 000.