Hardware-Efficient Bosonic Quantum Error-Correcting Codes Based on Symmetry Operators

Understand

We establish a symmetry-operator framework for designing quantum error correcting~(QEC) codes based on fundamental properties of the underlying system dynamics.

• Based on this framework, we propose three hardware-efficient bosonic QEC codes that are suitable for $\chi^{(2)}$-interaction based quantum computation: the $\chi^{(2)}$ parity-check code, the $\chi^{(2)}$ embedded error-correcting code, and the $\chi^{(2)}$ binomial code, all of which detect photon-loss or photon-gain errors by means of photon-number parity measurements and then correct them via $\chi^{(2)}$ Hamiltonian evolutions and linear-optics transformations.
• Our symmetry-operator framework provides a systematic procedure for finding QEC codes that are not stabilizer codes.
• The $\chi^{(2)}$ binomial code is of special interest because, with $m\le N$ identified from channel monitoring, it can correct $m$-photon loss errors, $m$-photon gain errors, and $(m-1)$th-order dephasing errors using logical qudits that are encoded in $O(N)$ photons.