Mathematical Approaches to Quantum Computing Algorithms
Keywords:
Mathematical Foundations, Quantum Gates, Quantum Circuit Design, Quantum Entanglement
Abstract
Quantum computing has emerged as a revolutionary field that leverages quantum mechanics to perform computations exponentially faster than classical computers for certain problems. This paper explores the mathematical foundations and approaches underpinning quantum computing algorithms. The study delves into linear algebra, probability theory, group theory, and tensor calculus, which are integral to quantum algorithm design. Key algorithms such as Shor's algorithm for integer factorization and Grover's search algorithm are examined in detail, highlighting their mathematical structure and computational efficiency. The paper also discusses recent advances in quantum error correction and optimization algorithms for quantum systems.
References
1. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
2. Shor, P. W. (1994). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing.
3. Grover, L. K. (1996). A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing.
4. Preskill, J. (2018). Quantum Computing in the NISQ Era and Beyond. Quantum, 2, 79.
5. Kitaev, A. Y. (1997). Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6), 1191-1249.
6. Arute, F., et al. (2019). Quantum supremacy using a programmable superconducting processor. Nature, 574(7779), 505-510.
7. Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press.
8. Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028.
9. Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), 212-219.
10. Harrow, A. W., & Montanaro, A. (2017). Quantum computational supremacy. Nature, 549(7671), 203-209.
11. Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30.
12. Messiah, A. (1961). Quantum Mechanics. Dover Publications.
13. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
14. Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. Quantum, 2, 79.
15. Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS), 124-134.
2. Shor, P. W. (1994). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing.
3. Grover, L. K. (1996). A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing.
4. Preskill, J. (2018). Quantum Computing in the NISQ Era and Beyond. Quantum, 2, 79.
5. Kitaev, A. Y. (1997). Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6), 1191-1249.
6. Arute, F., et al. (2019). Quantum supremacy using a programmable superconducting processor. Nature, 574(7779), 505-510.
7. Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press.
8. Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028.
9. Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC '96), 212-219.
10. Harrow, A. W., & Montanaro, A. (2017). Quantum computational supremacy. Nature, 549(7671), 203-209.
11. Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30.
12. Messiah, A. (1961). Quantum Mechanics. Dover Publications.
13. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
14. Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. Quantum, 2, 79.
15. Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS), 124-134.
Published
2024-10-15
How to Cite
Bijumon Ramalayathil, & Aneesh Kumar K. (2024). Mathematical Approaches to Quantum Computing Algorithms. Revista Electronica De Veterinaria, 25(2), 1420-1422. https://doi.org/10.69980/redvet.v25i2.1824
Section
Articles
