Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator
We study algorithms for solving three problems on strings. These are sorting of <i>n</i> strings of length <i>k</i>, “the Most Frequent String Search Problem”, and “searching intersection of two sequences of strings”. We construct quantum algorithms that are faster than class...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/3/377 |
| _version_ | 1797486365001121792 |
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| author | Kamil Khadiev Artem Ilikaev Jevgenijs Vihrovs |
| author_facet | Kamil Khadiev Artem Ilikaev Jevgenijs Vihrovs |
| author_sort | Kamil Khadiev |
| collection | DOAJ |
| description | We study algorithms for solving three problems on strings. These are sorting of <i>n</i> strings of length <i>k</i>, “the Most Frequent String Search Problem”, and “searching intersection of two sequences of strings”. We construct quantum algorithms that are faster than classical (randomized or deterministic) counterparts for each of these problems. The quantum algorithms are based on the quantum procedure for comparing two strings of length <i>k</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></semantics></math></inline-formula> queries. The first problem is sorting <i>n</i> strings of length <i>k</i>. We show that classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> for constant size alphabet, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mi>n</mi><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity. The second one is searching the most frequent string among <i>n</i> strings of length <i>k</i>. We show that the classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mi>n</mi><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity. The third problem is searching for an intersection of two sequences of strings. All strings have the same length <i>k</i>. The size of the first set is <i>n</i>, and the size of the second set is <i>m</i>. We show that the classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity. |
| first_indexed | 2024-03-09T23:33:09Z |
| format | Article |
| id | doaj.art-6873653bc38649a480a64537447a5cf2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:33:09Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6873653bc38649a480a64537447a5cf22023-11-23T17:06:31ZengMDPI AGMathematics2227-73902022-01-0110337710.3390/math10030377Quantum Algorithms for Some Strings Problems Based on Quantum String ComparatorKamil Khadiev0Artem Ilikaev1Jevgenijs Vihrovs2Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya 18, 420008 Kazan, RussiaInstitute of Computational Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya 18, 420008 Kazan, RussiaCenter for Quantum Computer Science, Faculty of Computing, University of Latvia, Raina 19, LV-1586 Riga, LatviaWe study algorithms for solving three problems on strings. These are sorting of <i>n</i> strings of length <i>k</i>, “the Most Frequent String Search Problem”, and “searching intersection of two sequences of strings”. We construct quantum algorithms that are faster than classical (randomized or deterministic) counterparts for each of these problems. The quantum algorithms are based on the quantum procedure for comparing two strings of length <i>k</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></semantics></math></inline-formula> queries. The first problem is sorting <i>n</i> strings of length <i>k</i>. We show that classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> for constant size alphabet, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mi>n</mi><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity. The second one is searching the most frequent string among <i>n</i> strings of length <i>k</i>. We show that the classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mi>n</mi><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mi>n</mi><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity. The third problem is searching for an intersection of two sequences of strings. All strings have the same length <i>k</i>. The size of the first set is <i>n</i>, and the size of the second set is <i>m</i>. We show that the classical complexity of the problem is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Θ</mo><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>, but our quantum algorithm has <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>O</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow><msqrt><mi>k</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula> complexity.https://www.mdpi.com/2227-7390/10/3/377quantum computationquantum algorithmsstring processingsorting |
| spellingShingle | Kamil Khadiev Artem Ilikaev Jevgenijs Vihrovs Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator Mathematics quantum computation quantum algorithms string processing sorting |
| title | Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator |
| title_full | Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator |
| title_fullStr | Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator |
| title_full_unstemmed | Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator |
| title_short | Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator |
| title_sort | quantum algorithms for some strings problems based on quantum string comparator |
| topic | quantum computation quantum algorithms string processing sorting |
| url | https://www.mdpi.com/2227-7390/10/3/377 |
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