We realize a hardware-native time--frequency Gottesman--Kitaev--Preskill (GKP) logical qubit encoded in the continuous phase space of single photons, establishing a propagating photonic implementation of bosonic grid encoding. Finite-energy grid states are generated deterministically using coherently driven entangled nonlinear biphoton sources that produce single-photon frequency-comb supermodes. An optical-frequency-comb reference anchors the time--frequency phase space and enforces commuting displacement stabilizers directly at the hardware level, continuously defining the logical subspace. Timing jitter, spectral drift, and phase noise map naturally onto Gaussian displacement errors within this lattice, yielding intrinsic correctability inside a stabilizer cell. Logical operations correspond to experimentally accessible phase and delay controls, enabling deterministic state preparation and manipulation. Building on the modal time--frequency GKP framework, we identify a concrete pathway toward active syndrome extraction and deterministic displacement recovery using ancillary grid states and interferometric time--frequency measurements. These primitives establish a hardware-compatible route for integrating the time--frequency GKP logical layer into erasure-aware and fusion-based fault-tolerant photonic architectures.
We realize a hardware-native time--frequency Gottesman--Kitaev--Preskill (GKP) logical qubit encoded in the continuous phase space of single photons, establishing a propagating photonic implementation of bosonic grid encoding. Finite-energy grid states are generated deterministically using coherently driven entangled nonlinear biphoton sources that produce single-photon frequency-comb supermodes. An optical-frequency-comb reference anchors the time--frequency phase space and enforces commuting displacement stabilizers directly at the hardware level, continuously defining the logical subspace. Timing jitter, spectral drift, and phase noise map naturally onto Gaussian displacement errors within this lattice, yielding intrinsic correctability inside a stabilizer cell. Logical operations correspond to experimentally accessible phase and delay controls, enabling deterministic state preparation and manipulation. Building on the modal time--frequency GKP framework, we identify a concrete pathway toward active syndrome extraction and deterministic displacement recovery using ancillary grid states and interferometric time--frequency measurements. These primitives establish a hardware-compatible route for integrating the time--frequency GKP logical layer into erasure-aware and fusion-based fault-tolerant photonic architectures.
Bosonic quantum error correction enables hardware-efficient protection of quantum information by encoding logical qubits in harmonic oscillators. Bosonic grid states, such as Gottesman-Kitaev-Preskill (GKP) states, are particularly promising due to their potential to correct small displacements and boson loss. However, their generation remains challenging, typically relying on probabilistic protocols or auxiliary qubit systems. Here, we propose deterministic protocols for generating bosonic grid states using programmable nonlinear bosonic circuits composed solely of squeezing, displacement, and Kerr operations. We show that aiming to enforce GKP symmetries in the output of these circuits yields states with competitive performance with respect to current realizations, but whose quality saturates with increasing circuit depth due to imperfect symmetry restoration. Instead, we find that these bosonic circuits naturally give rise to a distinct class of states, that we label as phased-comb states, which are unitarily related to standard grid states but feature an intrinsic phase structure. We demonstrate that these states define a scalable bosonic quantum error-correcting code with near-optimal performance under boson loss comparable to that of approximate GKP states. We further analyze their logical operations and show how to implement a universal gate set for them. Our results establish programmable nonlinear bosonic circuits as a viable route towards the generation of scalable bosonic quantum error-correcting states beyond standard GKP encodings.
GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous variable logical qubits that can be corrected for errors caused by phase space displacements. Their experimental realization is challenging, in particular, using propagating fields, where quantum information is encoded in the quadratures of the electromagnetic field. However, traveling photons are essential in many applications of GKP codes involving the long-distance transmission of quantum information. We introduce a new method for encoding GKP states in propagating fields using single photons, each occupying a distinct auxiliary mode given by the propagation direction. The GKP states are defined as highly correlated states described by collective continuous modes, as time and frequency. We analyze how the error detection and correction protocol scales with the total photon number and the spectral width. We show that the obtained code can be corrected for displacements in time-frequency phase space, which correspond to dephasing, or rotations, in the quadrature phase space and to photon losses. Most importantly, we show that generating two-photon GKP states is relatively simple, and that such states are currently produced and manipulated in several photonic platforms where frequency and time-bin biphoton entangled states can be engineered.
Bosonic quantum error correction encodes a logical qubit in an oscillator, avoiding the hardware overhead of large qubit arrays. Among such encodings, Gottesman-Kitaev-Preskill (GKP) states are paticularly powerful because their phase-space grid structure protects against small displacement errors simultaneously in both conjugate quadratures. Here we provide the first protocol for preparing magnonic GKP states, which involves an ellipsoidal magnetic crystal effectively coupled to a superconducting qubit via a microwave cavity. The geometric anisotropy intrinsically squeezes the magnon mode, while the cavity-mediated qubit control realizes an effective conditional-displacement interaction. We show that two rounds of a conditional-displacement interaction and a qubit projective measurement yield three- and four-component magnonic GKP-like states. We also show how to realize single logical qubit gate operations, such as Pauli, Hadamard and phase gates, completing the logical Pauli basis of the approximate GKP code. Our results establish hybrid magnon-qubit systems as a promising platform for preparing bosonic code states, with applications in magnonic fault-tolerant quantum computation and quantum sensing.
Building a useful photonic quantum computer requires robust techniques to synthesize optical states that can encode qubits. Gottesman–Kitaev–Preskill (GKP) states1 offer one of the most attractive classes of such qubit encodings, as they enable the implementation of universal gate sets with straightforward, deterministic and room temperature-compatible Gaussian operations2. Existing pioneering demonstrations generating optical GKP states3 and other complex non-Gaussian states4–11 have relied on free-space optical components, hindering the scaling eventually required for a utility-scale system. Here we use an ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, to generate GKP qubit states. These states show critical mode-level features necessary for fault tolerance, including at least four resolvable peaks in both p and q quadratures, and a clear lattice structure of negative Wigner function regions, in this case a 3 × 3 grid. We also show that our GKP states show sufficient structure to indicate that the devices used to make them could, after further reduction in optical losses, yield states for the fault-tolerant regime. This experiment validates a key pillar of bosonic architectures for photonic quantum computing2,12, paving the way for arrays of GKP sources that will supply future fault-tolerant machines. An ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, is used to generate Gottesman–Kitaev–Preskill qubit states.
To be useful, quantum computers will be required to successfully correct errors occurring at the hardware level. Bosonic codes provide a hardware-efficient option for error correction, but fault-tolerance further requires that the available gate interactions be compatible with the code. A promising bosonic code is the Gottesman-Kitaev-Preskill (GKP) code, for which a linear beamsplitter-like coupling between two bosonic modes is fault-tolerant, making this a key primitive for building larger systems. Here, using two motional modes of a trapped ion, we demonstrate the generation of entangled states of GKP qubits by interfering two qunaught states, which have a grid structure but carry no logical information, on a beamsplitter. We generate all four Bell states with an average fidelity of 69%, and subsequently demonstrate an extension of the entangled state lifetime through the use of quantum error correction. These results complete the set of Gaussian operations required for quantum computing with GKP codes and enable explorations of multi-mode bosonic encodings as well as fundamental tests of information channels.
Quantum computing holds the promise of solving classically intractable problems. Enabling this requires scalable and hardware-efficient quantum processors with vanishing error rates. This perspective manuscript describes how bosonic codes, particularly grid state encodings, offer a pathway to scalable fault-tolerant quantum computing in superconducting circuits. By leveraging the large Hilbert space of bosonic modes, quantum error correction can operate at the single physical unit level, therefore reducing drastically the hardware requirements to bring fault-tolerant quantum computing to scale. Going beyond the well-known Gottesman-Kitaev-Preskill (GKP) code, we discuss how using multiple bosonic modes to encode a single qubit offers increased protection against control errors and enhances its overall error-correcting capabilities. Given recent successful demonstrations of critical components of this architecture, we argue that it offers the shortest path to achieving fault tolerance in gate-based quantum computing processors with a MHz logical clock rate.