Representing a Novel Expanded Version of Shor’s Algorithm and a Real-Time Experiment using IBM Q-Experience Platform
Subject Areas : Data MiningSepehr Goodarzi 1 , Afshin Rezakhani 2 , Mahdi Maleki 3
1 - Department of Computer Engineering, Ayatollah Boroujerdi University, Boroujerd, Iran
2 - Department of Computer Engineering, Ayatollah Boroujerdi University, Boroujerd, Iran
3 - Department of Computer Engineering, Ayatollah Boroujerdi University, Boroujerd, Iran
Keywords: Quantum Computer, Quantum Dynamics, Unit Circle, N-dimensional Space, IBM-Q Experience,
Abstract :
The data are stored on the memory of the classical computer in small units of classical bits, which could be either 0 or 1. However, on a Quantum Computer, The Quantum States of each Quantum Bit (Qbit), would be every possible number between 0 and 1, including themselves. By placing the photons on a special state, which is a spot located at the middle of the two-dimensional space vectors (█(1@0)) and (█(1@1)) on the Unit Circle, which is called Superposition and we can take advantage of properties of this state when we place lots of vectors of N-dimensional spaces in superposition and we can do a parallelization and factorization for getting significant speedup. In fact, in Quantum Computing we are taking advantage of Quantum Dynamic Principles to process the data, which Classical Computers lack on, by considering the limitations of logical concepts behind them. Through this paper, we expand a quantum algorithm for the number of n Qbits in a new way and by implementing circuits using IBM-Q Experience, we are going to have some practical results, which are more obvious to be demonstrable. By expanding the Quantum Algorithms and using Linear Algebra, we can manage to achieve the goals at a higher level, the ones that Classical Computers are unable to perform, as machine learning problems with complicated models and by expanding the subject we can mention majors in different sciences like Chemistry (predicting the Structure of proteins with higher percentage accuracy in less period), Astronomy and so on.
[1] P. Steffen, R. Giegerich, and M. Giraud, "GPU parallelization of algebraic dynamic programming," in International Conference on Parallel Processing and Applied Mathematics, 2009: Springer, pp. 290-299.
[2] S. Asano, T. Maruyama, and Y. Yamaguchi, "Performance comparison of FPGA, GPU and CPU in image processing," in 2009 international conference on field programmable logic and applications, 2009: IEEE, pp. 126-131.
[3] N. Lang, H. Mena, and J. Saak, "On the benefits of the LDLT factorization for large-scale differential matrix equation solvers," Linear Algebra and its Applications, vol. 480, pp. 44-71, 2015.
[4] P. Windley, "Transposing matrices in a digital computer," The Computer Journal, vol. 2, no. 1, pp. 47-48, 1959.
[5] D. Zhang, Z.-H. Zhou, and S. Chen, "Non-negative matrix factorization on kernels," in Pacific Rim International Conference on Artificial Intelligence, 2006: Springer, pp. 404-412.
[6] D. Prijatmoko et al., "Early detection of protein depletion in alcoholic cirrhosis: role of body composition analysis," Gastroenterology, vol. 105, no. 6, pp. 1839-1845, 1993.
[7] L. S. Ostrouchov, M. Heath, and C. Romine, "Modeling speedup in parallel sparse matrix factorization," Oak Ridge National Lab., TN (USA), 1990.
[8] K. Likharev, "Classical and quantum limitations on energy consumption in computation," International Journal of Theoretical Physics, vol. 21, no. 3, pp. 311-326, 1982.
[9] W. Wardah, M. G. Khan, A. Sharma, and M. A. Rashid, "Protein secondary structure prediction using neural networks and deep learning: A review," Computational biology and chemistry, vol. 81, pp. 1-8, 2019.
[10] T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio, "Resource theory of superposition," Physical review letters, vol. 119, no. 23, p. 230401, 2017.
[11] G. Gour and C. M. Scandolo, "Dynamical Entanglement," Physical Review Letters, vol. 125, no. 18, p. 180505, 2020.
[12] G. Kumar, R. Saha, M. K. Rai, R. Thomas, and T.-H. Kim, "Proof-of-work consensus approach in blockchain technology for cloud and fog computing using maximization-factorization statistics," IEEE Internet of Things Journal, vol. 6, no. 4, pp. 6835-6842, 2019.
[13] G. Kwasniewski, T. Ben-Nun, A. N. Ziogas, T. Schneider, M. Besta, and T. Hoefler, "On the parallel I/O optimality of linear algebra kernels: near-optimal LU factorization," in Proceedings of the 26th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, 2021, pp. 463-464.
[14] D. O’Malley, V. V. Vesselinov, B. S. Alexandrov, and L. B. Alexandrov, "Nonnegative/binary matrix factorization with a d-wave quantum annealer," PloS one, vol. 13, no. 12, p. e0206653, 2018.
[15] S. Catalán, J. R. Herrero, E. S. Quintana-Ortí, R. Rodríguez-Sánchez, and R. Van De Geijn, "A case for malleable thread-level linear algebra libraries: The LU factorization with partial pivoting," IEEE access, vol. 7, pp. 17617-17633, 2019.
[16] R. Atcheson, "A Generalization of QR Factorization To Non-Euclidean Norms," arXiv preprint arXiv:2101.09830, 2021.
[17] Q. Cao et al., "Extreme-scale task-based cholesky factorization toward climate and weather prediction applications," in Proceedings of the Platform for Advanced Scientific Computing Conference, 2020, pp. 1-11.
[18] I. Yamazaki, A. Ida, R. Yokota, and J. Dongarra, "Distributed-memory lattice h-matrix factorization," The International Journal of High Performance Computing Applications, vol. 33, no. 5, pp. 1046-1063, 2019. [19] N. Heavner, P.-G. Martinsson, and G. Quintana-Ortí, "Computing rank-revealing factorizations of matrices stored out-of-core," arXiv preprint arXiv:2002.06960, 2020.
[20] T. Vander Aa, I. Chakroun, and T. Haber, "Distributed Bayesian probabilistic matrix factorization," Procedia Computer Science, vol. 108, pp. 1030-1039, 2017.
[21] A. Fu, Z. Chen, Y. Mu, W. Susilo, Y. Sun, and J. Wu, "Cloud-based outsourcing for enabling privacy-preserving large-scale non-negative matrix factorization," IEEE Transactions on Services Computing, 2019.
[22] M. Gates, J. Kurzak, P. Luszczek, Y. Pei, and J. Dongarra, "Autotuning batch Cholesky factorization in CUDA with interleaved layout of matrices," in 2017 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), 2017: IEEE, pp. 1408-1417.
[23] A. Abdelfattah et al., "A survey of numerical linear algebra methods utilizing mixed-precision arithmetic," The International Journal of High Performance Computing Applications, p. 10943420211003313, 2021.
[24] M. Tang, M. Gadou, S. Rennich, T. A. Davis, and S. Ranka, "Optimized sparse Cholesky factorization on hybrid multicore architectures," Journal of computational science, vol. 26, pp. 246-253, 2018.
[25] Q. Cao et al., "Performance analysis of tile low-rank cholesky factorization using parsec instrumentation tools," in 2019 IEEE/ACM International Workshop on Programming and Performance Visualization Tools (ProTools), 2019: IEEE, pp. 25-32.
[26] M. Tang, M. Gadou, and S. Ranka, "A Multithreaded Algorithm for Sparse Cholesky Factorization on Hybrid Multicore Architectures," Procedia Computer Science, vol. 108, pp. 616-625, 2017.
[27] M. Green, K. Glover, D. Limebeer, and J. Doyle, "AJ-Spectral Factorization Approach to H_∞," SIAM Journal on Control and Optimization, vol. 28, no. 6, pp. 1350-1371, 1990.
[28] P. Smolensky, "Tensor product variable binding and the representation of symbolic structures in connectionist systems," Artificial intelligence, vol. 46, no. 1-2, pp. 159-216, 1990.
[29] D. P. DiVincenzo, "Quantum gates and circuits," Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 454, no. 1969, pp. 261-276, 1998.
[30] D. M. Zajac et al., "Resonantly driven CNOT gate for electron spins," Science, vol. 359, no. 6374, pp. 439-442, 2018.
[31] D. Riste et al., "Detecting bit-flip errors in a logical qubit using stabilizer measurements," Nature communications, vol. 6, no. 1, pp. 1-6, 2015.
[32] F. Benatti and R. Floreanini, Irreversible quantum dynamics. Springer Science & Business Media, 2003.
[33] S. Gulde et al., "Implementation of the Deutsch–Jozsa algorithm on an ion-trap quantum computer," Nature, vol. 421, no. 6918, pp. 48-50, 2003.
[34] M. Wolf and D. Terrell, "The high-tech industry, what is it and why it matters to our economic future," 2016.
[35] A. P. Carnevale, N. Smith, and M. Melton, "STEM: Science Technology Engineering Mathematics," Georgetown University Center on Education and the Workforce, 2011.
[36] A. Cross, "The IBM Q experience and QISKit open-source quantum computing software," in APS March Meeting Abstracts, 2018, vol. 2018, p. L58. 003.
