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Implementing the Grover algorithm in homomorphic encryption schemes Fernández Ortiz, Pablo Martín-Delgado Alcántara, Miguel Ángel PRE2019-090517; W911NF14-1-0103 We apply quantum homomorphic encryption (QHE) schemes suitable for circuits with a polynomial number of 𝑇+𝑇† gates to Grover's algorithm, performing a simulation in Qiskit of a Grover circuit that contains three qubits. The 𝑇+𝑇†-gate complexity of Grover's algorithm is also analyzed in order to show that any Grover circuit can be evaluated homomorphically in an efficient manner. We discuss how to apply these QHE schemes to allow for the efficient homomorphic evaluation of any Grover circuit composed of 𝑛 qubits using 𝑛−2 extra ancilla qubits. We also show how the homomorphic evaluation of the special case where there is only one marked item can be implemented using an algorithm that makes the decryption process more efficient compared with the standard Grover algorithm. 2026-06-02T18:17:49Z 2026-06-02T18:17:49Z 2024-11-06 journal article Fernández, Pablo, y Miguel A. Martin-Delgado. «Implementing the Grover Algorithm in Homomorphic Encryption Schemes». Physical Review Research, vol. 6, n.o 4, noviembre de 2024, p. 043109. DOI.org (Crossref), https://doi.org/10.1103/PhysRevResearch.6.043109. 10.1103/PhysRevResearch.6.043109 https://hdl.handle.net/20.500.14352/137119 2643-1564 https://dx.doi.org/10.1103/PhysRevResearch.6.043109 https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.6.043109 eng PID2021-122547NB-I00 MaDQuantum-CM Quantum Spain http://creativecommons.org/licenses/by/4.0/ open access Attribution 4.0 International American Physical Society