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Phase shift and multi-controlled Z-type gates

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A Correction to this article was published on 11 December 2023

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Abstract

One of the most famous algorithms in quantum computations is the Grover search algorithm. Under certain assumptions, this algorithm provides quantum speedup for the search problem. We always assume that we can build it efficiently, but assume for a moment that you are limited in the gates you are allowed to use for the implementation of the Grover diffusion operator. For example, if you need to use at most two-channel gates, what would be the complexity of the decomposition circuit for the diffusion operator itself? Here, we show that it is sufficient to use just only the Z-base operators (such \(\root 2^n \of {Z}\) and controlled CZ) and Hadamard operators, but also we show that in this case, the number of used gates can grow exponentially. At least, the number of the multi-controlled Z gates needed to build diffusion operator decomposition circuit only of the following gates: Z, controlled Z, multi-controlled Z gates, or is summed into \(2^n-1\) for the n-dimensional phase shift and cannot be decreased if we will not allow other gates.

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Acknowledgements

The contribution of the first author (Andrei Novikov) in this work was supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. The contribution of the second author (Ramil Zainullin) in this work was supported by the development program of the Volga Region Mathematical Center under agreement No. 075-02-2020-1478 with the Ministry of Science and Higher Education of the Russian Federation.

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Authors and Affiliations

  1. Innovation Department, Sobolev Institute of Mathematics, Siberian branch of Russian Academy of Sciences, Novosibirsk, Russia

    Andrei Novikov

  2. Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Tatarstan, Russia

    Ramil Zainulin

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  1. Andrei Novikov
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  2. Ramil Zainulin
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Correspondence to Andrei Novikov.

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The original online version of this article was revised: the given name and family name are swapped for both the authors and the typo in surname of co-author has been corrected.

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Novikov, A., Zainulin, R. Phase shift and multi-controlled Z-type gates. Quantum Inf Process 22, 269 (2023). https://doi.org/10.1007/s11128-023-04005-1

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