Zero-pi qubit code

Zero-pi qubit code[1–3]

Description

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)}\textnormal{d}\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}\textnormal{d}\phi\left|\phi,\phi+\pi\right\rangle~. \end{split} \tag*{(2)}\end{align}

Protection

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

Gates

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].

Realizations

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

Notes

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

Parents

Cousin

Square-lattice GKP code — Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states [1,6].

References

[1]: P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits”, Physical Review A 87, (2013) arXiv:1302.4122 DOI
[2]: J. M. Dempster, B. Fu, D. G. Ferguson, D. I. Schuster, and J. Koch, “Understanding degenerate ground states of a protected quantum circuit in the presence of disorder”, Physical Review B 90, (2014) arXiv:1402.7310 DOI
[3]: C. Vuillot, A. Ciani, and B. M. Terhal, “Homological Quantum Rotor Codes: Logical Qubits from Torsion”, Communications in Mathematical Physics 405, (2024) arXiv:2303.13723 DOI
[4]: S. M. Girvin, “Circuit QED: superconducting qubits coupled to microwave photons”, Quantum Machines: Measurement and Control of Engineered Quantum Systems 113 (2014) DOI
[5]: P. Groszkowski, A. D. Paolo, A. L. Grimsmo, A. Blais, D. I. Schuster, A. A. Houck, and J. Koch, “Coherence properties of the 0-πqubit”, New Journal of Physics 20, 043053 (2018) arXiv:1708.02886 DOI
[6]: A. Kitaev, “Protected qubit based on a superconducting current mirror”, (2006) arXiv:cond-mat/0609441
[7]: A. Gyenis, P. S. Mundada, A. Di Paolo, T. M. Hazard, X. You, D. I. Schuster, J. Koch, A. Blais, and A. A. Houck, “Experimental Realization of a Protected Superconducting Circuit Derived from the 0 – π Qubit”, PRX Quantum 2, (2021) arXiv:1910.07542 DOI
[8]: 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
[9]: 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

Page edit log

Victor V. Albert (2023-04-12) — most recent

Cite as:

“Zero-pi qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/zero_pi

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/rotors/stabilizer/css/zero_pi.yml.