Zero-pi qubit code[13] 


A \([[2,(0,2),(2,1)]]_{\mathbb{Z}}\) homological rotor code on the smallest tiling of the projective plane \(\mathbb{R}P^2\). The ideal code can be obtained from a four-rotor Josephson-junction [4] system after a choice of grounding [3].

Logical codewords can be expressed in the basis of angular momentum states as \begin{align} \begin{split} |\overline{0}\rangle&=\sum_{\ell\in\mathbb{Z}}\left|2\ell,-2\ell\right\rangle \\|\overline{1}\rangle&=\sum_{\ell\in\mathbb{Z}}\left|2\ell+1,-2\ell-1\right\rangle~. \end{split} \tag*{(1)}\end{align} An alternative codeword basis in terms of angular position states is \begin{align} \begin{split} |\overline{+}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi+\pi\right\rangle~. \end{split} \tag*{(2)}\end{align}


Protection in the context of superconducting circuits investigated in Refs. [5].


One- and two-qubit phase gates utilizing ancillary osillators in GKP states [1,6].

Fault Tolerance

One- and two-qubit phase gate errors can be suppressed [1].


A related superconducting circuit has been realized by the Houck group [7].


The zero-pi qubit is based on earlier blueprints for protected subspaces using superconducting circuits [8,9].




P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits”, Physical Review A 87, (2013) arXiv:1302.4122 DOI
J. M. Dempster et al., “Understanding degenerate ground states of a protected quantum circuit in the presence of disorder”, Physical Review B 90, (2014) arXiv:1402.7310 DOI
C. Vuillot, A. Ciani, and B. M. Terhal, “Homological Quantum Rotor Codes: Logical Qubits from Torsion”, (2023) arXiv:2303.13723
S. M. Girvin, “Circuit QED: superconducting qubits coupled to microwave photons”, Quantum Machines: Measurement and Control of Engineered Quantum Systems 113 (2014) DOI
P. Groszkowski et al., “Coherence properties of the 0-πqubit”, New Journal of Physics 20, 043053 (2018) arXiv:1708.02886 DOI
A. Kitaev, “Protected qubit based on a superconducting current mirror”, (2006) arXiv:cond-mat/0609441
A. Gyenis et al., “Experimental Realization of a Protected Superconducting Circuit Derived from the 0 – π Qubit”, PRX Quantum 2, (2021) arXiv:1910.07542 DOI
B. Douçot and J. Vidal, “Pairing of Cooper Pairs in a Fully Frustrated Josephson-Junction Chain”, Physical Review Letters 88, (2002) arXiv:cond-mat/0202115 DOI
L. B. Ioffe and M. V. Feigel’man, “Possible realization of an ideal quantum computer in Josephson junction array”, Physical Review B 66, (2002) arXiv:cond-mat/0205186 DOI
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Zoo Code ID: zero_pi

Cite as:
“Zero-pi qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_zero_pi, title={Zero-pi qubit code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Zero-pi qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.