Quantum Precoded Polar Codes

Matthieu R. Bloch, Shrinivas Kudekar, Tyler Kann

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Pith reviewed 2026-05-14 19:22 UTC · model grok-4.3

classification 💻 cs.IT math.ITquant-ph

keywords CSS codespolar codesprecodingquantum error correctiondepolarizing channelsurface codegenetic algorithmlogical error rate

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The pith

Rate-1 precoded polar codes yield CSS quantum codes with logical error rates matching a much larger surface code.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a family of quantum CSS codes derived from rate-1 precoded polar codes. It demonstrates that performance gains from precoding in classical short-blocklength polar codes transfer to the quantum setting after genetic-algorithm optimization of the rate profile and precoder. The resulting [[256,2]] and [[512,2]] codes achieve logical error rates similar to the [[1201,1,25]] surface code over the depolarizing channel. This matters because it points to quantum error-correcting codes that maintain reliability at substantially shorter block lengths than conventional surface-code constructions.

Core claim

We present a new family of CSS codes obtained from rate-1 precoded polar codes. By optimizing the rate profile and precoder with a genetic algorithm, we obtain [[256,2]] and [[512,2]] codes that exhibit logical error rates comparable to the [[1201,1,25]] surface code over the depolarizing channel.

What carries the argument

Rate-1 precoded polar codes adapted into CSS quantum codes with genetic-algorithm optimization of the rate profile and precoder.

If this is right

Quantum CSS codes can reach comparable reliability at block lengths much smaller than those required by surface codes.
Classical precoding advantages apply to quantum codes after targeted parameter optimization.
Low-dimensional codes become practical for quantum error correction without sacrificing error-rate performance.
Genetic optimization allows systematic design of precoders tailored to quantum noise models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Rerunning the genetic optimization for other quantum channels could produce codes adapted to different noise statistics.
Pairing the precoded construction with improved quantum decoders may yield further reductions in logical error rates.
Implementation on quantum hardware would require verifying that the precoder structure does not complicate syndrome extraction circuits.

Load-bearing premise

The performance benefits of classical rate-1 precoding for short-blocklength polar codes transfer directly to the quantum CSS setting once the rate profile and precoder are optimized.

What would settle it

Monte Carlo simulation showing that the logical error rate of the optimized [[256,2]] precoded polar code on the depolarizing channel exceeds the rate of the [[1201,1,25]] surface code by more than a modest margin would falsify the similarity result.

Figures

Figures reproduced from arXiv: 2605.12796 by Matthieu R. Bloch, Shrinivas Kudekar, Tyler Kann.

**Figure 1.** Figure 1: A comparison between JN, 2K precoded polar codes for N = 256, 512 and J1201, 1, 25K surface code with χ = 4, 6. The J256, 2, L = 8, 32K codes are two different codes from Algorithm 1, as opposed to one code evaluated at two list sizes. 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 Physical Error Rate p 10−3 10−2 10−1 Logical Error Rate JN, 2, L = 4K N = 64, A0 N = 64, A N = 64, (T , A) N = 64, (T ∗ , A∗ ) N = 12… view at source ↗

**Figure 2.** Figure 2: The J64, 2, L = 4K code seen through the stages of Algorithm 1, and compared against the J128, 2, L = 4K polar code. A0 is the profile from [7]. at the cost of degrading the protection of its pair. B. Gates and Stabilizer weights While a hurdle for quantum polar codes becoming practical is their large weight stabilizers, we nevertheless discuss the impact of the precoding on the construction of quantum cir… view at source ↗

read the original abstract

We introduce a new family of CSS codes obtained from rate-1 precoded polar codes, which harnesses the precoding benefits obtained for classical short blocklength polar codes. We optimize the rate profile and precoder of these codes with a genetic algorithm, and present codes of dimension $ [\![256, 2 ]\!] $ and $ [\![512, 2]\!] $ that have logical error rates similar to the $ [\![1201, 1, 25 ]\!] $ surface code over the depolarizing channel.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unverified transfer of classical precoding gains to the quantum CSS setting and on the effectiveness of the genetic-algorithm search; no explicit free parameters or invented entities are named in the abstract.

free parameters (2)

rate profile
Chosen by genetic algorithm for each code length; exact values not stated in abstract.
precoder matrix
Optimized by genetic algorithm; concrete entries not provided in abstract.

axioms (1)

domain assumption Classical rate-1 precoding benefits transfer to quantum CSS codes after optimization
Implicit premise that allows the construction to be presented as advantageous.

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discussion (0)

Reference graph

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