Reliable quantum memory is essential for scalable quantum networks and fault-tolerant photonic quantum computing. We present a quantitative analysis of an all-optical quantum memory architecture in which a Gottesman-Kitaev-Preskill (GKP) encoded qubit is stored in a fibre loop and periodically stabilized using teleportation-based error correction. By modelling fibre propagation as a pure-loss channel and representing each correction round as an effective logical map acting on the Bloch vector, we obtain a compact description of the full multi-round memory channel. We show that syndrome decoder optimization plays a crucial role in the experimentally relevant finite-squeezing regime. The optimal decoder deviates from standard square-grid GKP decoder in both tile-size and tile-shape, leading to significant improved logical performance. Using this optimized decoding strategy, we identify a squeezing-dependent optimal spacing between correction nodes that maximizes the memory lifetime. Remarkably, this optimal segment length is largely independent of the desired storage time, providing a simple and practical design rule for fibre-loop quantum memory. We further find a squeezing threshold of approximately 6.7 dB below which intermediate error correction becomes counterproductive, while above threshold the achievable storage time increases approximately exponentially with squeezing. For example, at 17 dB squeezing, storage times exceeding 400 ms can be achieved with logical infidelity below 1%. These results establish clear performance benchmarks and reveal the fundamental trade-off between photon loss, squeezing, and correction frequency in continuous-variable architectures. Our findings provide actionable design principles for near-term photonic quantum memory and clarify the path toward scalable all-optical fault-tolerant quantum storage.
Reliable quantum memory is essential for scalable quantum networks and fault-tolerant photonic quantum computing. We present a quantitative analysis of an all-optical quantum memory architecture in which a Gottesman-Kitaev-Preskill (GKP) encoded qubit is stored in a fibre loop and periodically stabilized using teleportation-based error correction. By modelling fibre propagation as a pure-loss channel and representing each correction round as an effective logical map acting on the Bloch vector, we obtain a compact description of the full multi-round memory channel. We show that syndrome decoder optimization plays a crucial role in the experimentally relevant finite-squeezing regime. The optimal decoder deviates from standard square-grid GKP decoder in both tile-size and tile-shape, leading to significant improved logical performance. Using this optimized decoding strategy, we identify a squeezing-dependent optimal spacing between correction nodes that maximizes the memory lifetime. Remarkably, this optimal segment length is largely independent of the desired storage time, providing a simple and practical design rule for fibre-loop quantum memory. We further find a squeezing threshold of approximately 6.7 dB below which intermediate error correction becomes counterproductive, while above threshold the achievable storage time increases approximately exponentially with squeezing. For example, at 17 dB squeezing, storage times exceeding 400 ms can be achieved with logical infidelity below 1%. These results establish clear performance benchmarks and reveal the fundamental trade-off between photon loss, squeezing, and correction frequency in continuous-variable architectures. Our findings provide actionable design principles for near-term photonic quantum memory and clarify the path toward scalable all-optical fault-tolerant quantum storage.
Photonic quantum networks require error-correction architectures that remain useful with finite-energy bosonic states, pure-loss fiber transmission, and explicit resource accounting. In this light, we study a concatenated architecture in which each physical rail is a finitely squeezed Gottesman--Kitaev--Preskill (GKP) qubit transmitted through a pure-loss fiber segment, corrected by teleportation-based GKP error correction with finitely squeezed ancillae, and decoded by an outer quantum parity code (QPC). The GKP layer converts continuous homodyne syndromes into effective rail-level Pauli marginals, while the QPC layer suppresses the residual qubit-level errors. For the concatenated code family considered here, we find a finite-squeezing threshold of 5.06dB at zero propagation loss. In the memory setting, the QPC layer lowers the squeezing at which repeated error correction becomes beneficial from 6.7dB for bare GKP correction to 5.2dB for QPC(3,3) and 4.3dB for QPC(5,5), and improves the average-fidelity ratio by up to 75--90% in the relevant intermediate-noise regime. In the repeater setting, avoiding pre-amplification gives larger secret-key fractions at moderate squeezing, but also produces an optimal squeezing because highly squeezed GKP peaks become sensitive to loss-induced inward displacement. Resource-normalized rates show that QPC concatenation can exceed the repeaterless PLOB benchmark by orders of magnitude and extend the communication reach, at short repeater spacing, to distances of order 104km with 14dB squeezing. However, QPC concatenation becomes detrimental when each elementary hop is too lossy. These results provide quantitative design rules for finite-squeezing GKP--QPC quantum memories and repeaters.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate pL<10−7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p=0.36%). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing ≳12dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
Gottesman-Kitaev-Preskill (GKP) encoding holds promise for continuous-variable fault-tolerant quantum computing. While an ideal GKP encoding is abstract and impractical due to its nonphysical nature, approximate versions provide viable alternatives. Conventional approximate GKP codewords are superpositions of multiple large-amplitude squeezed coherent states. This feature ensures correctability against single-photon loss and dephasing at short times, but also increases the difficulty of preparing the codewords. To minimize this tradeoff, we utilize a neural network to generate optimal approximate GKP states, allowing effective error correction with just a few squeezed coherent states. We find that such optimized GKP codes outperform the best conventional ones, requiring fewer squeezed coherent states, while maintaining simple and generalized stabilizer operators. Specifically, the former outperform the latter with just one-third of the number of squeezed coherent states at a squeezing level of 9.55 dB. This optimization drastically decreases the complexity of codewords while improving error correctability.
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.
Fault-tolerant quantum computation in photonic architectures requires efficient preparation of high-fidelity logical magic states under realistic constraints of finite squeezing and photon loss. We present LiDMaS (Lightweight Density-Matrix Simulator), an architecture-level study of logical T-gate magic-state preparation in Gottesman--Kitaev--Preskill (GKP)-encoded photonic qubits using a repeat-until-success (RUS) injection protocol combined with outer surface-code protection. A density-matrix simulator based on standard numerical linear algebra is employed, mapping finite squeezing to effective logical dephasing, incorporating logical depolarizing noise, and treating photon loss as a heralded erasure process. Parameter sweeps are performed over squeezing values from 8 to 16~dB, baseline loss probabilities between 0.005 and 0.03, and surface-code distances d=1,3,5, and 7. Across this regime, RUS success probabilities range from about 0.90 to 0.99, with average injection overhead between 1.15 and 1.21 rounds per successful attempt. After outer-code protection, logical fidelities reach Flog≈0.765–0.796, exhibiting weak sensitivity to moderate photon loss but strong monotonic dependence on squeezing. Sensitivity analysis identifies finite squeezing as the dominant continuous error source, while loss primarily impacts heralded failure rates. Phase-boundary diagrams determine minimum squeezing requirements to achieve success probability ≥0.95 and logical fidelity ≥0.79 as a function of code distance, providing quantitative design guidance for scalable photonic fault-tolerant architectures.