Bosonic Codes as the Native Encoding: Resource-Commensurable Comparison of Cat, GKP, Binomial, and Surface Codes

In the paper's words

X3.3 — Resource-commensurable comparison of bosonic codes vs surface codes using photons/logical-qubit metric at p_L=10^-6. Bosonic codes require 5-40x fewer photons and ~100x fewer modes. Novel claim: HO as QM IR attractor implies bosonic codes are the native encoding.

The abstract, verbatim; every line below is quoted from the text.

What it claims

The harmonic oscillator as QM's universal IR attractor implies bosonic codes are the structurally native encoding for QEC.

Numbers it reports

1. The 12 dB required for $p_{L} = 10^{-6}$ is a factor of ~1.6× improvement in $\Delta$ (from ~0.32 to ~0.25), which is within engineering reach. [projected]

   from 5.4 Caveats

2. Code Family Mean photons per logical qubit Bosonic modes per logical qubit Dominant limitation Status Cat (4-comp) — Scenario A (pure) 5–8 1–2 Dephasing $\kappa _{\varphi }$ must be ≤10⁻⁷/gate [projected] [projected beyond current κ_φ] Cat (4-comp) — Scenario B (concat.) ~28 (8 cat + ~20 outer) ~41 Outer code overhead [derived — Chamberland 2022] GKP (square) 7–8 2 (GKP + ancilla) 12 dB squeezing required [derived — projected] Binomial (S=2–3) 10–20 2–3 Dephasing floor; no exp. demo [derived — projected] Surface code (baseline) ~100 ($\approx 200 \,\mathrm{qubits} \times 0.5$) ~200 Physical qubit fabrication & control [derived — Fowler 2012; Google 2023]

   from 7.1 Central Estimates

3. The most honest representation is as an order-of-magnitude band rather than a point estimate: Code Family Photon band (pessimistic–central–optimistic) Confidence Cat (pure, ultra-low $\kappa _{\varphi }$) 8 – 6 – 4 Low ($\kappa _{\varphi }$ not demonstrated) Cat (concatenated) 50 – 28 – 15 Medium (Chamberland model) GKP 15 – 8 – 5 Medium (12 dB squeezing is credible) Binomial 40 – 15 – 8 Low (dephasing floor; no experiment) Surface code 200 – 100 – 50 High (established scaling) The robust conclusion across all uncertainty bands: bosonic codes require approximately 5–40× fewer photons and ~100× fewer modes than surface codes to achieve the same logical error rate.

   from 7.3 Honest Uncertainty Band

4. Demonstration of 12 dB GKP squeezing with logical error rate measurement.

   from 8.3 What Would Strengthen the Thesis

5. The ~100× photon cost is the penalty for not using the native encoding.

   from 9.1 Why the Resource Advantage Exists

6. The resource advantage is approximately 10–40× in photon count and ~200× in mode count .

   from Abstract

How it is done

1. Here we test this claim quantitatively: we define a resource-commensurable metric (mean photons per logical qubit to reach logical error rate $p_{L} = 10^{-6}$), derive honest scaling relationships for each code family from published literature, and compare against the surface code as the standard qubit-based baseline.

   from Abstract

2. The "photon count" across all physical qubits provides a lower bound on the electromagnetic energy stored.

   from 1.2 Why This Comparison Has Not Been Made

3. We define the resource cost R of a QEC code as: R = (total mean photon number in all physical bosonic modes) / (number of logical qubits encoded) to achieve a logical error rate $p_{L} = 10^{-6}$, assuming a physical error probability per elementary operation of $p_{\mathrm{phys}} = 10^{-3}$.

   from 2.1 Definition

4. It is intended as a lower bound on the physical resource and an upper bound on the effective conversion efficiency of surface codes.

   from 2.2 Conversion: Physical Qubits to Photon-Equivalents

What would falsify it

1. The native-encoding thesis makes a falsifiable prediction: Falsification condition.

   from 8.2 Falsifiability

2. If any non-bosonic QEC code achieves $p_{L} \leq 10^{-6}$ with a lower photon-per-logical-qubit resource cost than the best bosonic code on the same physical platform (same material, same temperature, same frequency band), the thesis that bosonic codes are the native encoding is disconfirmed.

   from 8.2 Falsifiability

3. The claim is falsifiable: if any non-bosonic QEC code achieves $p_{L} = 10^{-6}$ with fewer photons per logical qubit than bosonic codes, the native-encoding thesis is disconfirmed.

   from Abstract

Where the claim stops

1. The above analysis assumes gate errors are subdominant — an assumption that requires the gate error rate to be below the idle error rate.

   from 5.4 Caveats

2. This is approachable with current technology [projected] but not yet demonstrated.

   from 5.4 Caveats

3. The above assumes a single-layer GKP code.

   from 5.4 Caveats

Where to read

1. Abstract259 words · 1 min
2. 1. Introduction: From Diagnosis to Derivation492 words · 2 min
3. 2. The Resource Metric: Photons per Logical Qubit349 words · 2 min
4. 3. Surface Code Baseline278 words · 1 min
5. 4. Cat Code741 words · 3 min
6. 5. GKP Code458 words · 2 min
7. 6. Binomial Code522 words · 2 min
8. 7. Resource-Commensurable Comparison306 words · 1 min
9. 8. Caveats and Cross-Platform Limitations314 words · 1 min
10. 9. Connection to the RG/Tree Framework (X3.1, X3.2)362 words · 2 min
11. 10. Conclusion276 words · 1 min