Results
Error rates, gate fidelity, and quantum scalability on integrated photonic platforms
Reference Parameters (Representative Silicon Platform)
1 - Feff: Physical Error per Segment
The physical error per segment, quantified as 1 - Feff, is the probability that a photonic qubit suffers an error (photon loss or coupling error) when traversing a single segment of the AGE-TGI-2S3/IDS decomposition. This parameter is calculated from the component's optical loss and coupling/phasing fidelity.
For a typical platform with competitive loss per segment, the resulting physical error is at a moderate level. The QEC code reduces it exponentially: with d=13 and high ηdet, the logical error drops by several orders of magnitude.
It is important to note that the physical error has two components: passive optical loss (attenuation in waveguides and junctions) and coupling errors (deviations in the coupling ratio from the nominal value). Both are modeled as erasures in the QEC framework, allowing efficient correction.
| Component | Typical loss | Contribution |
|---|---|---|
| Waveguide (Si) | ~0.5 dB/cm | Propagation |
| Directional coupler | ~0.05 dB | Operator A |
| Phase shifter | ~0.02 dB | Operator B |
| Bending loss | ~0.01 dB | Routing |
| Total/segment | 0.103 dB | Reference |
εlogical: Logical Error of the QEC Code
| Code | Distance (d) | Failure condition | εlogical (moderate η) | εlogical (high η) | Reduction vs. no QEC |
|---|---|---|---|---|---|
| No QEC | 1 | Any error | 4.76×10-2 | 4.76×10-2 | — |
| Mini-QEC | 5 | ≥ 3 / 5 | 7.18×10-3 | ~10-4 | ~7× |
| Strong QEC | 13 | ≥ 7 / 13 | 1.4×10-6 | ~10-9 | ~105× |
Error Evolution with Distance
The repetition code for erasures shows an exponential reduction of the logical error with distance d. For erasures, the code failure probability is the probability that more than d/2 physical qubits are lost, which decreases exponentially with d as long as the physical error probability is below the code threshold.
Our simulation data confirm this exponential trend. The decay rate depends directly on ηdet: with high η, the slope is much steeper than with moderate η, explaining the large improvement observed in εlogical.
| d | εlógico (η=0.95) | εlógico (η=0.99) |
|---|---|---|
| 3 | 2.14×10-2 | ~10-3 |
| 5 | 7.18×10-3 | ~10-4 |
| 7 | 1.89×10-3 | ~10-5 |
| 9 | 4.56×10-4 | ~10-7 |
| 13 | 1.4×10-6 | ~10-9 |
GHZ State Fidelity
GHZ (Greenberger-Horne-Zeilinger) states are maximally entangled states of N qubits: |GHZ_N⟩ = (|0⟩^⊗N + |1⟩^⊗N)/√2. GHZ fidelity is a rigorous metric of quantum entanglement quality and degrades exponentially with N in the presence of physical errors.
Our results show that AGE-TGI-2S3/IDS + QEC preserves GHZ fidelity significantly better than the unprotected system. With the d=13 code, GHZ fidelity remains high for states of up to several logical qubits, while without QEC, fidelity drops rapidly.
| N (qubits) | F_GHZ no QEC | F_GHZ QEC d=5 | F_GHZ QEC d=13 |
|---|---|---|---|
| 2 | 0.91 | 0.99 | > 0.999 |
| 3 | 0.47 | 0.92 | 0.997 |
| 5 | < 0.10 | 0.68 | 0.94 |
| 7 | < 0.01 | 0.38 | 0.85 |
| 10 | < 10-3 | 0.12 | 0.67 |
Scalability and Perspectives
The scalability of AGE-TGI-2S3/IDS + QEC depends on three critical factors: optical loss per segment (determines physical error), detection efficiency ηdet (determines QEC code effectiveness), and integration density of the chip (determines how many physical qubits fit in a given area).
Our results demonstrate that the best‑in‑class platform offers the best path to scalability, with hundreds of logical qubits (d=5) on a single chip. The combination of AGE-TGI-2S3/IDS (7 efficient segments) with moderate-distance QEC (d=13) and high-efficiency detectors (η > 0.99) represents the most promising configuration for achieving fault-tolerant quantum computing in integrated photonics.
Next steps include: (1) 3D FDTD simulations with MEEP to individually optimize optical components, (2) experimental verification of predictions on fabricated chips, and (3) extension to topological codes (surface codes) adapted to the erasure structure of photonics.
Simulation Data
Raw data available for independent replication
Repetition Code
Logical error rates for different code distances and ηdet values. CSV format.
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GHZ state fidelity for different qubit counts and QEC configurations. CSV format.
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