Aggregating logical quantum information encoded in continuous-variable phase space is essential for distributed quantum computing. However, passive linear optics fail for non-Gaussian Gottesman-Kitaev-Preskill (GKP) codes due to symplectic lattice compression and entanglement-induced decoherence. We present an active, measurement-based framework for the homomorphic aggregation of multi-node GKP states. Utilizing GKP Bell states and homodyne feed-forward, we construct a completely positive trace-preserving map that computes the logical sum of distributed states while preserving the logical code space geometry up to correctable finite-squeezing deformations. We prove this protocol operates as an approximate quantum non-demolition measurement, bound its cryptographic leakage for continuous one-time pads, and derive analytical logical fidelity limits under finite-squeezing constraints.
Aggregating logical quantum information encoded in continuous-variable phase space is essential for distributed quantum computing. However, passive linear optics fail for non-Gaussian Gottesman-Kitaev-Preskill (GKP) codes due to symplectic lattice compression and entanglement-induced decoherence. We present an active, measurement-based framework for the homomorphic aggregation of multi-node GKP states. Utilizing GKP Bell states and homodyne feed-forward, we construct a completely positive trace-preserving map that computes the logical sum of distributed states while preserving the logical code space geometry up to correctable finite-squeezing deformations. We prove this protocol operates as an approximate quantum non-demolition measurement, bound its cryptographic leakage for continuous one-time pads, and derive analytical logical fidelity limits under finite-squeezing constraints.
Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.
Bell tests based on homodyne detection are strongly constrained in continuous-variable systems. Can Gottesman-Kitaev-Preskill (GKP) encoding turn homodyne detection into a practical tool for revealing Bell nonlocality? We consider a physically motivated model in which each party performs homodyne detection and digitizes the continuous outcome via a fixed periodic binning, corresponding to logical Pauli measurements. Within this framework, we derive a bipartite no-go: CHSH cannot be violated for Bell-pair states. Moving beyond two parties, we show that finitely squeezed GKP-encoded GHZ and W states nevertheless exhibit strong multipartite nonlocality, violating multipartite Bell inequalities with homodyne-only readout. We quantify the required squeezing thresholds and robustness to loss, providing a route toward homodyne-based Bell tests in continuous-variable systems.
The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors, but when the displacement magnitude becomes large, lattice-crossing events arise beyond the correctable range of the GKP state. To address this issue, we concatenate a Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Unlike conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against both large and small displacement errors, enabling CV error correction within the continuous encoding space and contrasting with earlier approaches that concatenate GKP states with repetition codes for discrete qubit or qudit encodings. Analytical results show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors across all qumodes by up to 50 percent while enabling unbiased correction of lattice-crossing events, with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture continues to provide Gaussian-error suppression together with lattice-crossing correction, and the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides new insight into the design of concatenated CV error-correcting 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.
Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest. Published by the American Physical Society 2024
We consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman--Kitaev--Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.