Single-mode bosonic error correcting codes with rotation symmetry

Matti Silveri, Kari Mäkinen

Benjamin Marinoff, Miles Bush, Joshua Combes

Bosonic codes with rotational symmetry are currently one of the best-performing quantum error-correcting codes. Little is known about error propagation and code distance for these rotation codes in contrast with qubit codes and bosonic codes with translation symmetry. We use a general purpose error basis that is naturally suited to codes with rotation symmetry to compute how errors propagate through gates. This error basis allows us to give an explicit error detection, decoding, and correction scheme for any code with rotation symmetry. We also prove that codes with an $N$-fold rotation symmetry have a distance of $({d}_{n},{d}_{\ensuremath{\theta}})=(N,\ensuremath{\pi}/N)$ with respect to number and rotation errors.

Dong-Long Hu, Weizhou Cai, Chang‐Ling Zou +1 more

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.

Luyan Sun, Ling Hu, Yuwei Ma +10 more

Saurabh Totey, Akira Kyle, Liu, Steven +3 more

Bosonic error correcting codes utilize the infinite dimensional Hilbert space of a harmonic oscillator to encode a qubit. Bosonic rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes.We define two different notions of random bosonic rotation codes and numerically explore their performance against loss and dephasing. We find that the best random rotation codes can outperform cat and binomial codes in a certain parameter regime where loss is large and dephasing errors are small.

Shushen Qin, Long D. H. My, Adrian Copetudo +3 more

Bosonic error correction provides a hardware-efficient route to protected qubits for accurate quantum information processing. A critical component for error correction of rotation-symmetric bosonic (RSB) codes, like the well-known cat codes, is the two-mode controlled-rotation (CROT) operation. The CROT gate is useful because of its code-agnostic generality and its error-propagation properties. While the CROT gate can in principle be composed from existing bosonic physical primitives, the desirable properties are typically lost when composed as a sequence of imperfect primitive operations. Past work has focused on direct implementations that rely on features of specific codes. Here, we present a direct route to CROT within the circuit-QED architecture that retains its code-agnostic and error-propagation characteristics. By driving a transmon simultaneously coupled to two microwave cavities, we controllably enhance the effective nonlinearities between the microwave cavities and thus engineer the necessary two-mode interaction underlying the CROT gate. Using only a single drive frequency, we can achieve an on-off ratio sufficient for gate implementation, while minimizing mode distortions. We provide simple analytical formulas that permit the identification of potential working regimes, further refined by exact numerics, and illustrate the efficacy of our CROT approach in an error-correction example involving different RSB codes within the same circuit. Our CROT gate adds to the arsenal of direct two-mode gates available for general bosonic information processing.