LiDMaS RUS: Magic-State Injection in GKP Photonic Qubits

• The paper introduces LiDMaS, an architecture-level protocol that injects high-fidelity magic states into GKP photonic qubits via a repeat-until-success circuit.
• The protocol integrates GKP stabilizer error correction, heralded erasure, and surface-code protection to transparently map photonic noise into logical error metrics.
• Design guidelines specify optimal squeezing levels and photon loss tolerances to achieve high success probability with minimal overhead in scalable quantum systems.

LiDMaS RUS Magic-State Injection is an architecture-level protocol for the efficient and fault-tolerant realization of non-Clifford gate resources—specifically T-gate magic states—in GKP-encoded photonic qubits. The LiDMaS framework models the interplay of finite squeezing, photon loss (heralded erasure), and logical error correction, providing a transparent link between photonic physics, logical noise, injection performance, and scalable design trade-offs in a surface-code-protected architecture (Wayo, 22 Jan 2026). The protocol aggregates GKP-stabilizer error correction, repeat-until-success (RUS) logic, and surface-code outer protection to supply high-fidelity logical T-gate states, with explicit density-matrix noise mapping and detailed analytic and numerical design prescriptions.

1. RUS Magic-State Injection Circuit for GKP Photonic Qubits

The core RUS injection routine consists of sequential rounds combining GKP error correction, Clifford entangling gates, ancilla measurement, and conditional feedforward. Each round proceeds as follows:

• Initialization: Data qubit is encoded in a GKP code ($\rho_\mathrm{data}$). Ancilla mode is prepared in the approximate GKP T-type magic state $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$.
• Error Correction: GKP-stabilizer readout is performed on both data and ancilla, correcting small displacement errors and projecting each into the code space (“resetting” the logical degrees of freedom).
• Entanglement and Measurement: Logical CNOT is applied with the data as control, ancilla as target.
• Measurement Branching: Ancilla is measured in the X basis via GKP-corrected homodyne. The measurement outcome $m = \pm1$ determines the Clifford-class feedforward: $S^\dagger$ if $m = +1$; $S$ if $m = -1$.
• Success Criterion and Aborts: The “successful” branch (typically $m = +1$) implements the logical T-gate (up to known Cliffords). The “failure” branch or any detected photon loss (heralded by auxiliary detectors) aborts the round and triggers restart with fresh ancilla and rebasing of the circuit.

The process is repeated until success or until a maximum round cap (generally much larger than the mean required rounds) is reached. Each GKP-stabilizer measurement before Clifford gates ensures displacement errors are mapped into discrete Pauli errors, preserving the logic and avoiding continuous-variable simulation (Wayo, 22 Jan 2026).

2. Effective Logical Noise Modeling

LiDMaS proceeds entirely at the logical $2\times2$ density-matrix level, with all continuous-variable details abstracted away. Three sequential noise channels describe each physical round:

• Finite-Squeezing Dephasing: Modeled as a Pauli-Z channel,

$\mathcal{D}_Z(\rho) = (1-p_Z)\rho + p_Z Z \rho Z$

where $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$0 with coefficients $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$1 fit from CV simulations and $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$2 the squeezing parameter (dB).

• Logical Depolarizing Noise: Uniform

$$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$3

• Heralded Erasure (Photon Loss): With probability $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$4, the process is immediately aborted; no error propagates to the logical layer.

The density-matrix map on $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$5 is:

• “Erasure” with probability $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$6 triggers restart.
• Otherwise, sequential application of dephasing and depolarizing noise completes the logical transformation.

This abstraction enables efficient parameter studies and direct mapping between photonic device errors and logical operation outcomes (Wayo, 22 Jan 2026).

3. RUS Success, Overhead, and Design Figures of Merit

Key performance metrics include the probability of successful injection within a round cap, the expected overhead in terms of RUS repetitions, and the logical fidelity after error-correction.

• Success Probability: The probability per attempt is $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$7 (where 1/2 reflects the symmetric measurement branch at the Clifford point). Total success after $$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$8 rounds:

$$|A\rangle = T|+\rangle = (1/\sqrt{2})(|0\rangle + e^{i\pi/4}|1\rangle)$$9

For $m = \pm1$0, $m = \pm1$1 for $m = \pm1$2 in [8,16] dB and $m = \pm1$3 in [0.01, 0.03]; at $m = \pm1$4 dB, $m = \pm1$5 exceeds 0.98.

• Injection Overhead: Mean number of rounds per success ($m = \pm1$6) is approximately 1.15–1.20, essentially independent of code distance or moderate photon loss, due to rapid abort on failure branches.
• Logical Magic-State Fidelity: After injection and surface-code outer protection (distance $m = \pm1$7), the fidelity is

$m = \pm1$8

Typical values: For $m = \pm1$9 dB, $S^\dagger$0, $S^\dagger$1; after code correction, $S^\dagger$2 for $S^\dagger$3 and $S^\dagger$4 dB.

Table: Example Metrics Across Squeezing and Loss

Squeezing $S^\dagger$5 (dB)	$S^\dagger$6	Distance $S^\dagger$7	$S^\dagger$8	Overhead	$S^\dagger$9
8	0.01	7	≈0.94	≈1.15	≈0.77
12	0.02	5	≈0.97	≈1.18	≈0.79
16	0.03	3	>0.98	≈1.20	≈0.80

4. Phase-Boundary and Trade-off Analysis

Simultaneous constraints on high success probability and logical fidelity enable determination of minimum required squeezing as a function of photon loss and code distance. The phase boundary for squeezing is

$m = +1$0

with $m = +1$1 decreasing ∼0.5 dB per increment of code distance $m = +1$2. For example, $m = +1$3 dB with $m = +1$4 dB; $m = +1$5 dB with $m = +1$6 dB. Loss enters only logarithmically, indicating heralded erasures have a quantitatively minor impact compared to squeezing. This analysis enables real-time navigation of (s, $m = +1$7, d) design space (Wayo, 22 Jan 2026).

5. Comparison and Integration with Parity-Checker Protocols

Distinct from parity-checker and multi-step distillation schemes (Campbell et al., 2017), LiDMaS RUS is optimized for continuous-variable photonic platforms and GKP encoding. Parity-checker protocols utilize a two-step structure: pre-distillation of a multi-qubit resource (e.g., CCZ chains from noisy T-states), followed by non-Pauli parity checks with pivotal rotations, yielding high-fidelity $m = +1$8 states via quadratic error suppression. RUS/LiDMaS-style injection (e.g., Duclos-Cianci–Poulin, 10 → 2 protocols) commit all resources at once, consuming more T-count but permitting low-latency, direct injection by aborting failed rounds early. In the LiDMaS context, the success branch is determined by GKP-corrected measurement and heralded erasure rather than parity-check in the W(θ) basis. The separation of roles—heralded loss determining overhead, squeezing determining fidelity—is a characteristic design feature (Wayo, 22 Jan 2026, Campbell et al., 2017).

6. Practical Design Guidelines and Experimental Implications

LiDMaS analysis yields concrete architectural prescriptions for scalable photonic quantum computation:

• Squeezing: End-to-end $m = +1$9 dB achieves $S$0, $S$1 for photon loss up to 2%.
• Code Distance: If squeezing is limited (8–10 dB), outer code distance $S$2 compensates to reach $S$3.
• Photon Loss: Loss rates up to 3% marginally increase overhead but do not strongly affect fidelity, emphasizing that focusing on raising squeezing and minimizing depolarizing noise is more effective.
• Latency: With mean RUS rounds ≲1.2, overall latency is set by GKP-correction and Clifford gate depth, not by repeated RUS cycling.
• Optimization Tools: Phase-boundary diagrams mapping $S$4 allow real-time architectural optimization for specific hardware constraints.

These results provide a transparent framework for balancing photonic hardware capabilities with error-correction overhead, enabling clear resource allocation to maximize scalable logical gate performance (Wayo, 22 Jan 2026).

7. Implications for Scalable Photonic Fault-Tolerance and Future Directions

The LiDMaS RUS magic-state injection framework establishes rigorous, quantitative benchmarks for photonic architectures using GKP encoding under realistic noise: logical T-gate injection can robustly achieve $S$5, overhead ≈1.15, and fidelity ≈0.79–0.80 even with order-percent loss and modest squeezing. The division of operational roles—heralded erasure for overhead, squeezing for fidelity—clarifies experimental priorities and the most leverageable approaches to device and code optimization. A plausible implication is that future improvements to GKP state preparation and squeezing, coupled with advances in depolarizing noise reduction and surface-code integration, will systematically extend the scalability and efficiency of photonic quantum computing platforms (Wayo, 22 Jan 2026).