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Performance Analysis of the Hardware-Efficient Quantum Search Algorithm

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Abstract

This article explores the Hardware-Efficient Quantum Search Algorithm and compares it with other well-known counterparts. Escalating the count of qubits may elevate susceptibility to errors, particularly in iterative algorithms such as Grover’s. Conversely, Noisy-Intermediate-Scale-Quantum (NISQ) computers encounter limitation in the number of gates necessary for excecution of any quantum queries. Thus, we utilize hardware-efficient quantum search algorithm for further investigation due to its optimized circuit depth. Moreover, the Qiskit library and Matlab are used for validation of the analysis. Furthermore, the noise effects, encompassing phase-damping (PD) and amplitude-damping (AD) noises, are explored to present a comparative analsysis of various search algorithms.

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Data Availability

The codes that support the plots within the paper are publicly available on a Github repository at: https://github.com/paymanprime/QSearch_HE.git

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Authors and Affiliations

  1. School of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

    Armin Ahmadkhaniha, Yousef Mafi, Payman Kazemikhah, Hossein Aghababa & Mohammadreza Kolahdouz

  2. Faculty of Engineering, College of Farabi, University of Tehran, Tehran, Iran

    Hossein Aghababa

  3. Swanson School of Engineering, Electrical Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Masoud Barati

Authors
  1. Armin Ahmadkhaniha
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  2. Yousef Mafi
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  3. Payman Kazemikhah
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  4. Hossein Aghababa
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  6. Mohammadreza Kolahdouz
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Contributions

Armin Ahmadkhaniha: Conceptualization, Implementation, Methodology, Investigation, Writing initial Draft. Yousef Mafi: Methodology, Investigation, Noise Analysis. Payman Kazemikhah: Implementation, Validation, Literature survey, Writing - editing, Data Curation. Hossein Aghababa: Conceptualization, Project Administration, Resources, Supervision, Writing – Review & Editing. Masoud Barati: Investigation, Validation, Literature survey, Supervision. Mohammadreza Kolahdouz: Investigation, Literature survey, Supervision.

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Correspondence to Hossein Aghababa.

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Appendices

Appendix A

Eigenvectors of Gd and Gq − d are given in Eqs. 9 and 12, respectively. Hardware efficient basis sets are evaluated by considering these equations as given in Eqs. 24 and 25.

$${\displaystyle \begin{array}{l}\begin{array}{l}\left|v\right\rangle =\frac{1}{\sqrt{2}}\left(\left|{\psi}_d^{+}\right\rangle +\left|{\psi}_d^{-}\right\rangle \right)\\ {}\left|n{v}_d\right\rangle =\frac{-i}{\sqrt{2}}\left(\left|{\psi}_d^{+}\right\rangle -\left|{\psi}_d^{-}\right\rangle \right)\end{array}\\ {}\left|n{v}_{q-d}\right\rangle =\sin \left(\alpha \right)\left|{\psi}_d^0\right\rangle +\cos \left(\alpha \right)\left|{\psi}_d^1\right\rangle \\ {}\left|o\right\rangle =\cos \left(\alpha \right)\left|{\psi}_d^0\right\rangle -\sin \left(\alpha \right)\left|{\psi}_d^1\right\rangle \end{array}}$$
(24)
$${\displaystyle \begin{array}{l}\begin{array}{l}\left|v\right\rangle =\frac{1}{\sqrt{2}}\left(\left|{\psi}_{q-d}^{+}\right\rangle +\left|{\psi}_{q-d}^{-}\right\rangle \right)\\ {}\left|n{v}_d\right\rangle =\sin \left(\beta \right)\left|{\psi}_{q-d}^0\right\rangle +\cos \left(\beta \right)\left|{\psi}_{q-d}^1\right\rangle \end{array}\\ {}\left|n{v}_{q-d}\right\rangle =\frac{i}{\sqrt{2}}\left(\left|{\psi}_{q-d}^{+}\right\rangle -\left|{\psi}_{q-d}^{-}\right\rangle \right)\\ {}\left|o\right\rangle =\cos \left(\beta \right)\left|{\psi}_{q-d}^0\right\rangle -\sin \left(\beta \right)\left|{\psi}_{q-d}^1\right\rangle \end{array}}$$
(25)

Appendix B

GRK Partial search algorithm is another search algorithm employed in this study to compare its results with the Grover and the Hardware efficient algorithms. Figure 9 shows the quantum circuit considered for investigations in this study. One can use this algorithm to find the output with fewer queries (oracle complexity of the Partial search algorithm is \(O\left(\sqrt{N}-\sqrt{d}\right)\)).

Fig. 9
figure 9

First two iterations of the 4-qubit Partial search algorithm

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Ahmadkhaniha, A., Mafi, Y., Kazemikhah, P. et al. Performance Analysis of the Hardware-Efficient Quantum Search Algorithm. Int J Theor Phys 62, 168 (2023). https://doi.org/10.1007/s10773-023-05424-7

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