LiDMaS: Architecture-Level Modeling of Fault-Tolerant Magic-State Injection in GKP Photonic Qubits

TL;DR

This work presents LiDMaS, an architecture-level modeling framework that maps finite squeezing to logical dephasing $p_Z(s)$ and photon loss to heralded erasure $p_E$ to study fault-tolerant magic-state injection for $GKP$ photonic qubits. By integrating a repeat-until-success $T$-gate injection with a surface-code-inspired outer code, the authors quantify how squeezing, loss, and code distance shape success probability, injection overhead, and logical fidelity, without CV wavefunction simulations. The results show that RUS performance remains near unity overhead and that logical fidelity after outer coding is primarily limited by squeezing, while loss becomes largely inconsequential once erasures are heralded and corrected, with phase boundaries mapping minimum squeezing requirements across loss and distance. This framework provides actionable, quantitative guidance for co-designing photonic hardware and fault-tolerant architectures toward scalable quantum computation.

Abstract

Fault-tolerant quantum computation in photonic architectures relies on the efficient preparation of high-fidelity logical magic states under realistic constraints imposed by finite squeezing and photon loss. In this work, we study logical T-gate magic-state preparation in GKP-encoded photonic qubits using a repeat-until-success injection protocol combined with outer surface-code protection. We develop an architecture-level modeling framework based on a lightweight density-matrix simulator implemented with standard numerical linear algebra. Finite squeezing is mapped to effective logical dephasing, depolarizing noise is included at the logical level, and photon loss is treated as a heralded erasure process. This approach avoids explicit continuous-variable wavefunction simulation, hardware-specific photonic models, and quantum software frameworks, enabling transparent and computationally efficient exploration of architectural trade-offs. We perform systematic parameter sweeps over squeezing values from 8 to 16 dB, baseline loss probabilities between 0.01 and 0.03, and surface-code distances d = 1, 3, 5, and 7. Across this regime, we evaluate repeat-until-success probability, average injection overhead, and logical magic-state fidelity. We find that success probabilities exceed 0.94 across all studied parameters, with an average overhead close to unity. After outer-code protection, logical fidelities reach approximately 0.77 to 0.80 and show weak sensitivity to moderate photon loss but a strong dependence on squeezing. Phase-boundary analysis identifies minimum squeezing requirements needed to simultaneously achieve high success probability and logical fidelity. These results provide quantitative design guidance for scalable photonic fault-tolerant quantum architectures.

Figures (7)

• Figure 1: Architecture-level logical noise model used throughout this work. Encoded qubits are represented as $2\times2$ density matrices and evolve under effective logical noise channels. Finite GKP squeezing induces Pauli-$Z$ dephasing with rate $p_Z(s)$, while residual imperfections are captured by a logical depolarizing channel with rate $p_{\mathrm{depol}}$. Photon loss is treated as a heralded erasure process with probability $p_E$, aborting the current repeat-until-success injection attempt. When no erasure occurs, the composite completely positive trace-preserving map $\mathcal{E}(\rho)$ is applied. This abstraction avoids explicit continuous-variable simulation while preserving the dominant fault-tolerant error mechanisms.
• Figure 2: Schematic of the logical $T$-gate magic-state injection protocol used in this work. A data qubit encoded at the logical level interacts with an ancilla prepared in the magic state $|A\rangle = T|+\rangle$ via a CNOT gate, followed by an $X$-basis measurement of the ancilla. Conditioned on the measurement outcome, a Clifford feedforward operation ($S$ or $S^\dagger$) is applied to the data qubit. One measurement branch implements the desired logical $T$ gate (up to Clifford equivalence), while the other branch requires repetition, yielding a repeat-until-success (RUS) protocol. Photon loss events are treated as heralded erasures that abort the current attempt.
• Figure 3: RUS success probability as a function of the squeezing proxy for different baseline loss values and surface-code distances. Higher squeezing consistently improves the probability of successful magic-state injection, while the dependence on code distance remains weak.
• Figure 4: Average number of RUS rounds required for successful magic-state injection as a function of squeezing. The overhead remains close to unity across loss regimes and code distances, indicating efficient heralding and fast convergence of the RUS protocol.
• Figure 5: Average logical magic-state fidelity conditioned on successful injection, as a function of squeezing. Increasing the surface-code distance systematically improves logical fidelity, while moderate loss variations have a weak effect once error correction is applied.