Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code utilizes is helpful. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer g apart, and the gap g imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting g-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, g-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing and amplitude damping errors.
Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code utilizes is helpful. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer g apart, and the gap g imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting g-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, g-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing and amplitude damping errors.
Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code utilizes is helpful. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer g apart, and the gap g imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting g-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, g-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing and amplitude damping errors.
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 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.
Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error-correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on the concatenation of number-phase codes and Bacon-Shor subsystem codes.
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.
Bosonic quantum error correction (QEC) relies on non-Gaussian bosonic encodings whose preparation cost is a central practical constraint. In this work, we use stellar rank as a resource measure to design and benchmark bosonic codes under finite non-Gaussian resources. For fixed cat and Gottesman--Kitaev--Preskill (GKP) code families, we show that finite stellar rank creates a trade-off among state approximability, energy, and logical protection under photon loss and photon-number dephasing, evaluated with optimal recovery. This trade-off implies that codewords with better ideal error-correction properties need not be optimal once finite-rank preparation constraints are imposed. Going beyond fixed-target codewords, we directly optimize bosonic encodings at fixed stellar rank, revealing noise-adapted code structures and concrete resource thresholds. Grid-like encodings emerge under photon loss, whereas approximately rotation-symmetric encodings arise under dephasing. In the optimized search, stellar rank k=2 suffices to surpass break-even for all dephasing strengths considered, while under photon loss the required rank increases with the loss rate. These results establish stellar rank as an operationally meaningful resource measure for bosonic QEC under practical state-preparation constraints.