We introduce a family of multimode, rotationally symmetric bosonic codes inspired by the group-theoretic framework introduced by Denys and Leverrier []. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction results in codes that display rotational symmetry across multiple modes, while enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multimode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial code instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.
We introduce a family of multimode, rotationally symmetric bosonic codes inspired by the group-theoretic framework introduced by Denys and Leverrier []. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction results in codes that display rotational symmetry across multiple modes, while enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multimode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial code instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.
Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger SU(2) symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.
Bosonic codes offer a promising approach towards hardware-efficient fault tolerance, with many leading examples such as the cat-code and the GKP code organized by an underlying symmetry. We build on a representation-theoretic framework of quantum error correction to formalize the construction and analysis of multimode symmetric bosonic codes, focusing on finite groups. Here, a physical, logical representation and an initial seed state together define the codewords via a covariant encoding. A central observation is that QEC matrices can be viewed as morphisms of group representations, so that Schur's lemma forces entire irreducible representations blocks to vanish. We recover several known single- and two-mode bosonic codes and build and analyze the error protection capabilities of three codes based on finite subgroups of SU(2). We analyze the impact of reducibility of the logical representation and optimize the seed state using the near-optimal fidelity.
Tiger codes form a family of multimode bosonic quantum codes that unify several previously known constructions, including cat, paircat, and the two-mode binomial code. In this work, we give a rigorous algebraic treatment of these codes. Starting from a kernel definition of the codespace, we prove that the annihilation-type constraints admit a finite generating set, construct an explicit orthonormal basis, and show that the logical structure of the code is governed by the homology of an underlying chain complex, as expected in the original work on Tiger codes of Xu et al. We then develop a Fourier transform over the codespace to prove that the span of phase-rotated projected coherent states is dense therein, and to yield dual X- and Z-type descriptions of the code. We further extend the framework to non-linear number constraints, encompassing codes such as the four-legged cat or the repetition cat code. Finally, we investigate the implementation of logical operations. We first generalise the construction of logical Pauli operators proposed by Xu et al. to arbitrary logical spaces, and then construct non-Clifford gates using physical polynomial phase rotations of the form eiP(n^). We derive criteria on the real polynomial P which, for positive single-logical-qubit Tiger codes satisfying an additional sign assumption, such as the paircat code, characterise the polynomials P that preserve the codespace by decomposing them into a family of univariate polynomials. Through this decomposition, we relate the degrees of the resulting components to the induced logical action in the Clifford hierarchy. These results establish Tiger codes as a mathematically robust framework for describing a broad class of bosonic encodings.
We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The n-rotor Clifford group, $\text{U}{(1)}^{n(n+1)/2}\ensuremath{\rtimes}{\text{GL}}_{n}(\mathbb{Z})$, is represented by continuous U(1) gates generated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) gates generated by momentum sign-flip and sum gates. Our understanding of this group allows us to establish connections between homological rotor error-correcting codes [Vuillot, Ciani, and Terhal, Commun. Math. Phys. 405, 53 (2024)] and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multimode rotation-symmetric bosonic codes by making a parallel between oscillator Fock states and rotor states with fixed non-negative angular momentum. This family of homological number-phase codes protects against dephasing and changes in occupation number. Encoding and decoding circuits for these codes can be derived from the corresponding rotor Clifford operations. As a result of independent interest, we show how to nondestructively measure the oscillator phase using conditional occupation-number addition and postselection. We also outline several rotor and oscillator varieties of the Gottesman-Kitaev-Preskill-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020).].
We introduce a family of bosonic quantum error-correcting codes built as a rotation-symmetric superposition of squeezed vacuum states, which promise protection against both loss and dephasing noise channels. The robustness of these "squeezed-vacuum codes" arises from being arranged at evenly spaced angles in phase-space, and simultaneously in evenly spaced photon-number support n≡2k(mod2m). We present simple preparation circuits: a two-legged code using a Hadamard-conditional-squeezing-Hadamard sequence on an ancilla qubit, and for general "m-legged" codewords using sequences of conditional rotations. The performance of these codes is evaluated against loss and dephasing noises using the Knill-Laflamme violation function and benchmarked against cat codes. As the number m of squeezed-vacuum states in a code increases, the code exhibits improved loss tolerance at the cost of higher dephasing sensitivity. We outline implementations in circuit QED and trapped-ion platforms, where high-fidelity Gaussian operations and conditional controls are available or under active development. These results help establish squeezed-vacuum codes as practical, hardware-ready, members of the bosonic codes class.
Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for 4≤n≤25 over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of SU(2) and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number nd=⌈(3d2+1)/4⌉ and construct each corresponding code through distance d=10. To our knowledge, the constructed (n,d)=(28,6),(49,8),(76,10) codes give the smallest reported n for their respective distances; all members through d=9 attain the fixed-n LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.