Jordan-Wigner Transformation and Qubits with Nontrivial Exchange Rule
( Pp. 23-28)

Well-known (spinless) fermionic qubits may need more subtle consideration in comparison with usual (spinful) fermions. Taking into account a model with local fermionic modes, formally only the ’occupied’ states |1〉 could be relevant for antisymmetry with respect to particles interchange, but ‘vacuum’ state |0〉 is not. Introduction of exchange rule for such fermionic qubits indexed by some ‘positions’ may look questionable due to general super-selection principle. However, a consistent algebraic construction of such ‘super-indexed’ qubits is presented in this work. Considered method has some relation with construction of super-spaces, but it has some differences with standard definition of supersymmety sometimes used for generalizations of qubit model.

Vlasov A.Y., (2021), JORDAN-WIGNER TRANSFORMATION AND QUBITS WITH NONTRIVIAL EXCHANGE RULE. Computational Nanotechnology, 3 => 23-28.

Feynman R.P. Simulating physics with computers. Int. J. Theor. Phys. 1982. No. 21. Pp. 467-488.
Feynman R.P. Quantum-mechanical computers. Found. Phys. 1986. No. 16. Pp. 507-531.
Jordan P., Wigner E. Uber das Paulische Aquivalenzverbot. Zeitschrift fur Physik. 1928. No. 47. Pp. 631-651.
Bravyi S., Kitaev A. Fermionic quantum computation. Ann. Phys. 2002. No. 298. Pp. 210-226. arXiv:quant-ph/0003137
Gilbert J.E., Murray M.A.M. Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge: Cambridge University Press, 1991.
DiVincenzo D.P. Two-bit gates are universal for quantum computation. Phys. Rev. A. 1995. No. 51. Pp. 1015-1022. arXiv:cond-mat/9407022
Deutsch D., Barenco A., Ekert A. Universality in quantum computation. Proc. R. Soc. Lond. A. 1995. No. 449. Pp. 669-677. arXiv:quant-ph/9505018
Vlasov A.Yu. Clifford algebras and universal sets of quantum gates. Phys. Rev. A. 2001. No. 63. P. 054302. arXiv:quant-ph/0010071
Lounesto P. Clifford Algebras and Spinors. Cambridge: Cambridge University Press, 2001.
Postnikov M.M. Lie Groups and Lie Algebras. Moscow: Mir Publishers, 1986.
Varadarajan V. Supersymmetry for mathematicians: An introduction. Providence: AMS, 2004.
Deligne P., Morgan J.W. Notes on supersymmetry (following Joseph Bernstein). Quantum Fields and Strings: A Course for Mathematicians, 1. P. Deligne et al. (eds.). Providence: AMS, 1999. Pp. 41-97.
Vlasov A.Yu. Quantum circuits and Spin (3n) groups. Quant. Inf. Comp. 2015. No. 15. Pp. 235-259. arXiv:1311.1666 quant-ph
Frydryszak A.M. Qubit, superqubit and squbit. J. Phys.: Conf. Ser. 2013. No. 411. P. 012015. arXiv:1210.0922 math-ph

Jordan-Wigner transformation, qubits with nontrivial exchange rule, Quantum Informatics.