A Lower Bound on Energy Cost per Logical-Qubit Operation in Surface-Code Quantum Error Correction

In the paper's words

Surface-code quantum error correction translates one logical qubit into a two-dimensional array of physical qubits, and the energy cost of that translation is rarely quantified at the level of a logical operation. This paper specifies a minimal, convention-explicit energy model for a rotated surface-code patch of distance d and computes it in full. The model states three conventions up front: the patch uses N_c = 2d^2 - 1 physical qubits (d^2 data, d^2 - 1 ancilla); one logical operation costs N_g = 8 d^3 physical gates (roughly 2d^2 stabilizer measurements of four CNOT gates each per syndrome round, d rounds per logical operation); and the energy decomposes as E = (E_g N_g + E_c N_c)/N_l with gate energy E_g and per-round control-and-measurement energy E_c per physical qubit. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. All quantitative predictions are computed by a deposited verification script: the Landauer erasure floor $k_{B} T \ln 2$ per bit at 300 K, 77 K, 4 K, and 20 mK; the full energy table for $d = 3..31$; the exact cubic scaling check $N_{g}(15)/N_{g}(9) = (15/9)^{3}$ reproduced to four decimal places; and the observation that realistic cryogenic control dissipates roughly 2.6 x 10^10 times the 4 K Landauer floor, so the binding constraint on joules per logical operation is the efficiency of the control electronics, not thermodynamics.

   from Abstract

2. At $d = 11$, the erasure floor at 4 K is 4.076e-19 J per logical operation, while a realistic gate energy of 1 pJ yields 1.065e-08 J, a ratio of 2.6 x 10^10.

   from 5. Results

3. Because the Landauer ratio is 2.6 x 10^10, every joule saved in the control electronics translates directly into logical-operation energy; thermodynamic arguments against scaling are misdirected at current dissipation levels.

   from 7. What a practitioner can do with this result

4. This is the standard rotated-lattice count; a distance-3 patch has 17 qubits, a distance-11 patch 241.

   from 3. Model and conventions

5. For realistic $E_{g}$ of order 1 pJ and $E_{c}$ of order 0.1 pJ per round, the cubic term dominates for all d above 3, so the model reduces to E approximately $E_{g} 8 d^{3}$ in the relevant regime.

   from 4. Analysis

6. The paper defines JPCUB, joules per logical-qubit operation, as the normalization that isolates algorithmic cost from error-correction overhead, and tabulates the overhead factor for distances 3 to 21.

   from Abstract

How it is done

1. The thermodynamic and informational bottlenecks of scalable fault-tolerant quantum computation are examined in a companion QNFO record [4], which argues that energy, erasure, and wiring constraints bound fault-tolerant machines more tightly than gate fidelity alone.

   from 2. Prior work

2. The model treats one logical qubit in a rotated surface-code patch of distance d.

   from 3. Model and conventions

3. The logical error rate of a surface-code patch scales as $p_{L}$ proportional to $(p/p_{th})^{d/2}$ for physical error rate p well below threshold $p_{th}$.

   from 4. Analysis

4. Third, the constant-overhead routes of [3] bound the model's regime.

   from 6. Discussion

What would falsify it

1. Both $E_{g}$ and $E_{c}$ are parameters to be supplied by device measurements; the model predicts the functional dependence on d, not the absolute dissipation of any particular controller.

   from 3. Model and conventions

2. A convention-explicit energy model for the rotated surface code, with $N_{c} = 2 d^{2} - 1$ physical qubits, $N_{g} = 8 d^{3}$ physical gates per logical operation, and the decomposition $E = (E_{g} N_{g} + E_{c} N_{c})/N_{l}$, predicts that the energy per logical operation grows cubically with code distance, that realistic controllers operate ten orders of magnitude above the Landauer floor, and that the JPCUB normalization isolates the overhead factor $8 d^{3}$ that a benchmark must carry.

   from 8. Conclusion

Where the claim stops

1. Each round measures approximately $2 d^{2}$ stabilizers (X- and Z-type), and each stabilizer measurement is implemented with four CNOT gates plus preparation and measurement.

   from 3. Model and conventions

2. The four-CNOT estimate therefore gives approximately $8 d^{2}$ physical gates per round, and $N_{g} = 8 d^{3}$ physical gates per logical operation.

   from 3. Model and conventions

3. For realistic $E_{g}$ of order 1 pJ and $E_{c}$ of order 0.1 pJ per round, the cubic term dominates for all d above 3, so the model reduces to E approximately $E_{g} 8 d^{3}$ in the relevant regime.

   from 4. Analysis

Where to read

1. Abstract314 words · 1 min
2. 1. Introduction247 words · 1 min
3. 2. Prior work227 words · 1 min
4. 3. Model and conventions272 words · 1 min
5. 4. Analysis246 words · 1 min
6. 5. Results237 words · 1 min
7. 6. Discussion222 words · 1 min
8. 7. What a practitioner can do with this result165 words · 1 min
9. 8. Conclusion117 words · 1 min
10. Changelog39 words · 1 min
11. Verification73 words · 1 min