Photon-loss tolerance of the surface-GKP code with anti-photon-loss decoding

Technical Summary: Photon-loss Tolerance of the Surface-GKP Code with Anti-photon-loss Decoding

Problem Statement
Photon loss is a primary obstacle to fault-tolerant photonic quantum computing, affecting both linear photonic platforms and bosonic modes in superconducting circuits. While the Gottesman–Kitaev–Preskill (GKP) code effectively corrects random displacement errors in continuous-variable (CV) systems, and the surface code provides topological fault tolerance for discrete-variable systems, their concatenation (the surface-GKP code) faces a specific challenge regarding photon loss. Unlike a simple stochastic displacement channel, pure photon loss induces two distinct physical effects on finite-energy GKP states: (1) a contraction of the phase-space lattice toward the origin, and (2) the introduction of vacuum noise that broadens the peaks. Standard decoding strategies, such as nearest-lattice-point decoding, often model loss as a zero-mean random displacement, failing to account for the systematic inward shift of peak positions. This mismatch between the contracted lattice and the fixed decoding grid increases the probability of logical errors.

Methodology
The authors investigate the photon-loss tolerance of the surface-GKP code by modeling the pure-loss channel directly, rather than converting it into an additive Gaussian displacement channel via amplification. The study employs the following methodological framework:

1. Noise Modeling: The pure-loss channel is modeled as a beam-splitter interaction with transmissivity $\eta = 1 - \gamma$, coupling the system to an environmental vacuum mode. This explicitly accounts for both attenuation (scaling quadratures by $\sqrt{1-\gamma}$) and the addition of vacuum noise.
2. State Preparation: Finite-energy GKP states are generated using an envelope operator $\hat{E}(\epsilon) = e^{-\epsilon\hat{n}}$, which regularizes ideal GKP states. Two quality levels are tested: $\epsilon = 0.01$ (approx. 20 dB squeezing) and $\epsilon = 0.05$ (approx. 13 dB squeezing).
3. Decoding Strategies:
   • Nearest-Lattice-Point (NLP) Decoding: The standard approach where quadrature outcomes are assigned to the nearest logical grid point.
   • Anti-Photon-Loss (APL) Decoding: A proposed inner decoder that applies a quadrature-dependent counter-squeezing operation ($\hat{S}(r_{comp})$) before homodyne readout. This operation compensates for the deterministic contraction of the measured quadrature, realigning the peak positions with the fixed decoding grid. The compensation parameter is set to $r_{comp} = -\frac{1}{2}\ln(1-\gamma)$.
4. Simulation Workflow: The authors perform a two-stage simulation:
   • Inner Layer: Calculate the quadrature probability densities for the GKP states after loss (with and without APL) and integrate over incorrect decision regions to determine effective bit ($p_x$) and phase ($p_z$) error probabilities.
   • Outer Layer: Feed these error probabilities into a Monte Carlo simulation of the rotated surface code (distances $d=3, 5, 7, 9$). Syndrome measurement rounds are simulated, and Minimum-Weight Perfect Matching (MWPM) is used to decode the logical failure probability ($p_L$).

Key Contributions

• Direct Modeling of Pure Loss: The paper avoids the approximation of converting pure loss into a displacement channel via amplification, instead modeling the direct contraction and noise addition effects on finite-energy GKP states.
• APL Inner Decoder: The authors introduce and evaluate an anti-photon-loss inner decoder that compensates for the loss-induced lattice contraction. This approach restores the scale of the measured quadrature to match the decoding grid, addressing a source of error distinct from peak broadening.
• Threshold Estimation: The study provides quantitative estimates of photon-loss thresholds for the surface-GKP code under both standard and APL decoding schemes for different GKP state qualities.

Results
The Monte Carlo simulations yield the following photon-loss threshold estimates ($\gamma^*$), defined as the loss fraction below which increasing the code distance suppresses logical failures:

• High-Quality States ($\epsilon = 0.01$, ~20 dB):
  • Without APL: $\gamma^* \approx 5.5\%$ (0.24 dB).
  • With APL: $\gamma^* \approx 9.8\%$ (0.45 dB).
  • Observation: The APL decoder increases the threshold by approximately 79% relative to the uncompensated case. Notably, the uncompensated threshold for high-quality states is slightly lower than for moderate-quality states, attributed to the larger inward displacement of distant peaks in the extended envelope of high-quality states.
• Moderate-Quality States ($\epsilon = 0.05$, ~13 dB):
  • Without APL: $\gamma^* \approx 5.9\%$ (0.26 dB).
  • With APL: $\gamma^* \approx 6.3\%$ (0.28 dB).
  • Observation: APL provides a relative increase of approximately 7% in the threshold.

Additionally, the authors analyze the required GKP squeezing threshold ($s^*_{GKP}$) as a function of loss. They find that while APL compensates for lattice mismatch, the residual vacuum noise (which scales with loss) necessitates slightly higher initial squeezing to maintain low logical error rates as loss increases.

Significance and Claims
The paper claims that accounting for loss-induced lattice contraction in inner decoding is critical for maximizing the fault-tolerance of surface-GKP codes. The results demonstrate that quadrature compensation (APL) significantly enhances photon-loss tolerance, particularly for high-quality GKP states. The authors emphasize that this approach provides a route toward more robust optical quantum computing by improving the tolerance of concatenated codes to pure loss. They conclude that the APL decoder represents a directly testable decoding choice for future models incorporating realistic operations, highlighting that the benefit of realigning peak positions outweighs the cost of amplified vacuum noise within the tested regimes.