Jordan-Wigner transformation code[13] 


A mapping between qubit Pauli strings and Majorana operators that can be thought of as a trivial \([[n,n,1]]\) code. The mapping is best described as converting a chain of \(n\) qubits into a chain of \(2n\) Majorana modes (i.e., \(n\) fermionic modes). It maps Majorana operators into Pauli strings of weight \(O(n)\).

The Majorana modes \(\{\gamma_j\}\) are defined from Pauli strings as follows, \begin{align} \begin{split} \gamma_{0}&=\phantom{X}Z\\ i\gamma_{1}&= X Z\\ \gamma_{2}&=X\phantom{Z}\otimes\phantom{X}Z\\\gamma_{3}&=X\phantom{Z}\otimes XZ\\ i\gamma_{4}&=X\phantom{Z}\otimes X\phantom{Z}\otimes\phantom{X}Z\\ &\vdots \end{split} \tag*{(1)}\end{align} The \(X\)-type Pauli strings ensure that the resulting Majorana operators satisfy the appropriate anti-commutation relations, namely, \(\{\gamma_i,\gamma_j\} = \delta_{ij}\).


Circuit of depth linear in the number of qubits \(n\). The depth can be reduced for particle-preserving systems [4] and in other contexts [5].




P. Jordan and E. P. Wigner, “Über das Paulische Äquivalenzverbot”, The Collected Works of Eugene Paul Wigner 109 (1993) DOI
A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
R. Somma et al., “Simulating physical phenomena by quantum networks”, Physical Review A 65, (2002) arXiv:quant-ph/0108146 DOI
B. Harrison et al., “Reducing the qubit requirement of Jordan-Wigner encodings of \(N\)-mode, \(K\)-fermion systems from \(N\) to \(\lceil \log_2 {N \choose K} \rceil\)”, (2023) arXiv:2211.04501
M. Chiew and S. Strelchuk, “Discovering optimal fermion-qubit mappings through algorithmic enumeration”, Quantum 7, 1145 (2023) arXiv:2110.12792 DOI
Z. Jiang et al., “Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning”, Quantum 4, 276 (2020) arXiv:1910.10746 DOI
Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
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“Jordan-Wigner transformation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_jw, title={Jordan-Wigner transformation code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Jordan-Wigner transformation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.