{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T01:25:46Z","timestamp":1776129946695,"version":"3.50.1"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,5,1]],"date-time":"2021-05-01T00:00:00Z","timestamp":1619827200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"content-version":"vor","delay-in-days":25,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We analyze the mathematical structure of the classical Grover\u2019s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one \u2018chosen\u2019 element (sometimes called a \u2018solution\u2019) of the dataset, but a set of<jats:italic>m<\/jats:italic>such \u2018chosen\u2019 elements (out of<jats:inline-formula><jats:alternatives><jats:tex-math>$$n&gt;m)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that \u2018marks,\u2019 by a suitable phase change<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varphi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03c6<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, all these \u2018chosen\u2019 elements. In the first part of the paper, we construct a unique unitary operator that selects all \u2018chosen\u2019 elements in one step. The constructed operator is uniquely defined by the numbers<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varphi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03c6<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, \u2018convenient\u2019 phase change<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varphi ,$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u03c6<\/mml:mi><mml:mo>,<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and by sequentially applying the so-constructed operator, we find the number of steps to find these \u2018chosen\u2019 elements with great probability. We apply this knowledge to study the generalizations of Grover\u2019s algorithm (<jats:inline-formula><jats:alternatives><jats:tex-math>$$m=1,\\phi =\\pi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>\u03d5<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>\u03c0<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>), which are of the form, the found previously, unitary operators.<\/jats:p>","DOI":"10.1007\/s11128-021-03125-w","type":"journal-article","created":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T06:02:33Z","timestamp":1622008953000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Understanding mathematics of Grover\u2019s algorithm"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3013-5163","authenticated-orcid":false,"given":"Pawe\u0142 J.","family":"Szab\u0142owski","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,26]]},"reference":[{"issue":"2","key":"3125_CR1","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1103\/PhysRevLett.79.325","volume":"79","author":"Lov K Grover","year":"1997","unstructured":"Grover, Lov K.: Quantum Mechanics Helps in Searching for a Needle in a Haystack. 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