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 ${\chi }^{\left(2\right)}$-interaction based quantum computation in multimode Fock bases: the ${\chi }^{\left(2\right)}$ parity-check code, the ${\chi }^{\left(2\right)}$ embedded error-correcting code, and the ${\chi }^{\left(2\right)}$ binomial code. All of these QEC codes detect photon-loss or photon-gain errors by means of photon-number parity measurements, and then correct them via ${\chi }^{\left(2\right)}$ Hamiltonian evolutions and linear-optics transformations. Our symmetry-operator framework provides a systematic procedure for finding QEC codes that are not stabilizer codes, and it enables convenient extension of a given encoding to higher-dimensional qudit bases. The ${\chi }^{\left(2\right)}$ binomial code is of special interest because, with $m\le N$ identified from channel monitoring, it can correct $m$-photon-loss errors, or $m$-photon-gain errors, or $(m-1)\mathrm{th}$-order dephasing errors using logical qudits that are encoded in $O\left(N\right)$ photons. In comparison, other bosonic QEC codes require $O\left(N^2\right)$ photons to correct the same degree of bosonic errors. Such improved photon efficiency underscores the additional error-correction power that can be provided by channel monitoring. We develop quantum Hamming bounds for photon-loss errors in the code subspaces associated with the ${\chi }^{\left(2\right)}$ parity-check code and the ${\chi }^{\left(2\right)}$ embedded error-correcting code, and we prove that these codes saturate their respective bounds. Our ${\chi }^{\left(2\right)}$ QEC codes exhibit hardware efficiency in that they address the principal error mechanisms and exploit the available physical interactions of the underlying hardware, thus reducing the physical resources required for implementing their encoding, decoding, and error-correction operations, and their universal encoded-basis gate sets.

• Received 23 September 2017
• Revised 22 December 2017

DOI:https://doi.org/10.1103/PhysRevA.97.032323