Generalized number-phase lattice encoding of a bosonic mode for quantum error correction

Abstract

Bosonic systems offer unique advantages for quantum error correction, as a single bosonic mode provides a large Hilbert space to redundantly encode quantum information. However, previous studies have been limited to exploiting symmetries in the quadrature phase space. Here we introduce a unified framework for encoding a qubit utilizing the symmetries in the phase space of number and phase variables of a bosonic mode. The logical codewords form lattice structures in the number-phase space, resulting in rectangular, oblique, and diamond-shaped lattice codes. Notably, oblique and diamond codes exhibit a number-phase vortex effect, where number-shift errors induce discrete phase rotations as syndromes, enabling efficient correction via phase measurements. These codes show significant performance advantages over conventional quadrature codes against dephasing noise in the potential one-way quantum communication applications. Our generalized number-phase codes open up new possibilities for fault-tolerant quantum computation and extending the quantum communication range with bosonic systems. Current bosonic quantum error correction codes exploit displacement or rotational symmetries in the quadrature phase space. Here, the authors generalise the concept by looking at potential symmetries and encoding in a broader number-phase space, where the known cat and binomial codes would correspond to rectangular lattices and be completed by other lattice codes like oblique and diamond.