Description
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, \(\{\gamma_i,\gamma_j\} = \delta_{ij}\).
Encoding
Parent
- Auxiliary qubit mapping (AQM) code — The AQM fermion-into-qubit code reduces to the JW transformation code when the outer code is trivial.
Cousins
- Bravyi-Kitaev transformation (BKT) code — The weight of a Majorana operator in the BKT (JW transformation) code scales logarithmically (linearly) with \(n\), with the former demonstrating an exponential imporvement [6].
- 2D bosonization code — The 2D bosonization code can be converted into chains of JW-transformed qubits by a linear-depth circuit [7; Fig. 24].
References
- [1]
- P. Jordan and E. P. Wigner, “Über das Paulische Äquivalenzverbot”, The Collected Works of Eugene Paul Wigner 109 (1993) DOI
- [2]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [3]
- R. Somma et al., “Simulating physical phenomena by quantum networks”, Physical Review A 65, (2002) arXiv:quant-ph/0108146 DOI
- [4]
- 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
- [5]
- M. Chiew and S. Strelchuk, “Discovering optimal fermion-qubit mappings through algorithmic enumeration”, Quantum 7, 1145 (2023) arXiv:2110.12792 DOI
- [6]
- 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
- [7]
- Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
Page edit log
- Victor V. Albert (2024-03-20) — most recent
Cite as:
“Jordan-Wigner transformation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/jw