The geometry of quantum computing. (English) Zbl 07910062
Summary: In this expository paper, we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and \(ZX\)-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.
References:
| [1] | Feynman, R. P., Simulating physics with computers, Int. J. Theor. Phys.21 (1982) 467-488. |
| [2] | Feynman, R. P., Quantum mechanical computers, Found. Phys.16(6) (1986) 507-531. |
| [3] | Manin, Y., Computable and Uncomputable (Sovetskoye Radio, Moscow, 1980). · Zbl 0471.03003 |
| [4] | Manin, Y., Topics in Noncommutative Geometry, , (Princeton University Press, Princeton, NJ, 1991). · Zbl 0724.17007 |
| [5] | Deutsch, D., Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. R. Soc. Lond. A Math. Phys. Sci.400(1818) (1985) 97-117. · Zbl 0900.81019 |
| [6] | Shor, P. W., Algorithms for quantum computation, discrete logarithms and factorizing, in Proc. 35th Annual Symp. Foundations of Computer Science (IEEE, 1994). |
| [7] | Shor, P. W., Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A52(4) (1995) R2493. |
| [8] | Preskill, J., Reliable quantum computers, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.454(1969) (1998) 385-410. · Zbl 0915.68051 |
| [9] | Preskill, J., Fault-tolerant quantum computation, in Introduction to Quantum Computation and Information (World Scientific, 1998), pp. 213-269. |
| [10] | Preskill, J., Lecture Notes for Physics 229: Quantum Information and Computation (Barron, 2015). |
| [11] | Ashtekar, A. and Schilling, T. A., Geometrical formulation of quantum mechanics, in On Einstein’s Path, ed. Harvey, A. (Springer, New York, 1999). |
| [12] | Kitaev, A. Y., Fault-tolerant quantum computation by anyons, Ann. Phys.303(1) (2003) 2-30. · Zbl 1012.81006 |
| [13] | Kitaev, A. Y., Periodic table for topological insulators and superconductors, AIP Conf. Proc.1134 (2009) 22. · Zbl 1180.82221 |
| [14] | Dennis, E., Kitaev, A., Landahl, A. and Preskill, J., Topological quantum memory, J. Math. Phys.43(9) (2002) 4452-4505. · Zbl 1060.94045 |
| [15] | Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. and Weinfurther, H., Elementary gates for quantum computation, Phys. Rev. A52(5) (1995) 3457-3467. |
| [16] | Fioresi, R. and Zanchetta, F., Deep learning and geometric deep learning: An introduction for mathematicians and physicists, Int. J. Geom. Methods Mod. Phys.20(12) (2023) 2330006. · Zbl 1531.68069 |
| [17] | Amari, S.-I., Natural gradient works efficiently in learning, Neural Comput.10(2) (1998) 251-276. |
| [18] | Stokes, J., Izaac, J., Killoran, N. and Carleo, G., Quantum natural gradient, Quantum4 (2020) 269. |
| [19] | Helstrom, C. W., Quantum Detection and Estimation Theory (Academic Press, 1976). · Zbl 1332.81011 |
| [20] | Liu, J., Jing, X., Zhong, W. and Wang, X., Quantum Fisher information for density matrices with arbitrary ranks, Commun. Theor. Phys.61(1) (2014) 45-50. · Zbl 1284.81064 |
| [21] | Huybrechts, D., Complex Geometry: An Introduction (Springer GTM, 2005). · Zbl 1055.14001 |
| [22] | X.-Y. Hou, Z. Zhou, X. Wang, H. Guo and C.-C. Chien, Local geometry and quantum geometric tensor of mixed states, Preprint arXiv:2305.07597. |
| [23] | Facchi, P., Kulkarni, R., Manko, V. I., Marmo, G., Sudarshan, E. C. G. and Ventriglia, F., Classical and quantum Fisher information in the geometrical formulation of quantum mechanics, Phys. Lett. A374 (2010) 4801-4803. · Zbl 1238.81013 |
| [24] | Coecke, B. and Duncan, R., Interacting quantum observables, in ICALP 2008: Automata, Languages and Programming, , Vol. 5126 (Springer, Berlin, Heidelberg. 2008), pp. 298-310. · Zbl 1155.81316 |
| [25] | Coecke, B. and Duncan, R., Interacting quantum observables: categorical algebra and diagrammatics, New J. Phys.13 (2011) 043016. · Zbl 1448.81025 |
| [26] | Majid, S., Quantum and braided ZX calculus, J. Phys. A: Math. Theor.55 (2022) 254007. · Zbl 1507.81125 |
| [27] | J. van de Wetering, ZX-calculus for the working quantum computer scientist, (2020), https://api.semanticscholar.org/CorpusID:229680014. |
| [28] | de Beaudrap, N., Kissinger, A. and van de Wetering, J., Circuit extraction for ZX-diagrams can be # P-hard, in 49th Int. Colloquium on Automata, Languages, and Programming (ICALP 2022), , Vol. 229 (Schloss Dagstuhl - Leibniz-Zentrum fr Informatik, 2022), pp. 119:1-119:19. |
| [29] | Montgomery, S., Hopf Algebras and Their Actions on Rings, , Vol. 82 (American Math Society, Providence, RI, 1993). · Zbl 0793.16029 |
| [30] | Kassel, C., Quantum Groups (Springer-Verlag, GTM, 1995). · Zbl 0808.17003 |
| [31] | Majid, S., Foundations of Quantum Groups Theory (Cambridge University Press, 1995). · Zbl 0857.17009 |
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