9,216 Physical Qubits Protect 4,612 Logical Qubits: Quantum Error Correction Overhead Rewritten
On September 9, 2026, the journal *Quantum* published *Breaking the Orthogonality Barrier in Quantum LDPC Codes* by Associate Professor Kenta Kasai of Institute of Science Tokyo, DOI 10.22331/q-2026-09-09-2205. The code he constructed is written as [[9216,4612,d]]: 9,216 physical qubits protecting 4,612 logical qubits. That works out to roughly 2 physical qubits per logical qubit. For comparison, the surface code's overhead grows roughly with the square of the code distance; at distance 48, counting as [[d²,1,d]], it takes 2,304 physical qubits to protect a single logical qubit.
Classical LDPC Has a Full Design Theory; Quantum Adds One More Constraint
Low-density parity-check (LDPC) codes are mature technology in classical communications. Designers have a complete toolkit: choose the check degree per bit, preserve the right amount of randomness, avoid short cycles — and pursue two things at once: a large minimum distance, and a "threshold phenomenon" where decoding failure drops off a cliff once noise falls below a predictable level.
Quantum LDPC codes must carry one additional constraint: the two families of error checks (X-type and Z-type) must be mutually orthogonal, or their measurements will interfere with each other.
Here is the problem. Enforcing this orthogonality constraint throughout the entire design produces short cycles and weak structure, and both classical LDPC advantages are lost.
Kasai's Approach: Orthogonality Only Where It Matters
Kasai asked whether classical LDPC design theory could be carried over into quantum codes while preserving both distance and threshold behavior.
His construction uses affine permutation matrices, and the key step is "selective orthogonality": the orthogonality constraint is imposed only on the rows actually used for error correction (active rows); the complementary latent parent rows keep their randomness, so degree distribution and short cycles can be designed according to classical LDPC principles.
In Kasai's words, the quantum constraint is necessary, but it need not dominate every part of the design; imposing it only where error correction uses it preserves the classical LDPC design freedom needed to pursue large minimum distance and threshold behavior.
The Code's Specifics
The result is a (3,12)-regular code with girth 8.
| Parameter | Value | |---|---| | Code | [[9216,4612,d]] | | Regularity | (3,12)-regular | | Girth | 8 | | Distance | d ≤ 48; both X and Z distances are 48 | | Logical operators | Explicitly constructed at weight 48 | | Encoding rate | ~1 logical qubit per 2 physical qubits |
The team performed detailed searches and low-error-rate simulations and found no logical errors of lower weight. A global lower bound on the distance d remains open; current results provide strong evidence that the overall minimum distance is close to 48. Eliminating 4-cycles and 6-cycles also reduced the trapping sets that stall belief propagation (BP) decoding.
Decoding uses BP with low-complexity post-processing. Under 4% depolarizing noise, the frame error rate reaches 10⁻⁸ — roughly one failure per hundred million attempts.
The Waterfall Curve Is the Paper's Real Claim
Beyond the numbers, Kasai emphasizes a methodological point.
The code's decoding waterfall — the cliff where failure rates plunge — lands near the density-evolution prediction of the corresponding classical ensemble. That classical benchmark is p ≈ 0.05702, from the corresponding non-orthogonal random (3,12)-regular LDPC ensemble. The paper explicitly states this is a reference, not a measured threshold for the proposed quantum code.
Kasai's interpretation: classical LDPC coding spent decades learning to predict decoding limits from degree distributions and design toward them; seeing a quantum code's waterfall approach the same benchmark suggests this methodology can guide quantum LDPC design.
Threshold, minimum distance, short cycles, and hard error patterns can now be considered together in a single framework.
The Price of High Rate: Checks Are No Longer Local
There is no free lunch. qLDPC codes save physical qubits through high encoding rates, but pay on the other end: their checks are no longer local — one stabilizer measurement may connect physically distant qubits, and the decoder must handle a more complex graph.
The surface code's appeal has always been locality: checks involve only adjacent qubits on a grid, implementable in a planar layout, naturally suited to superconducting processors. The cost is overhead growing with d².
So the two routes suit different conditions, depending on whether hardware can do non-local connections cheaply. Harvard, MIT, and QuEra are porting this construction to reconfigurable neutral-atom machines — one reason being that neutral atoms can be moved with acousto-optic deflectors, so long-range connections do not mean long-range wiring. Cornucopia emphasizes "regular structures suited to moving atoms collectively," and GALA builds "AOD-compatible atom movement" directly into code design — evidence of the same direction.
In Eight Months, Four Groups Picked Up the Construction
The timeline itself says more about its standing than the paper does.
- January 13 — arXiv preprint posted.
- April 17 — Harvard / MIT / QuEra adapt it to neutral atoms while the paper was still under review, designing codes and error-detection flows around atom rearrangement with acousto-optic deflectors (AOD), calling the family "Kasai codes." QuEra's official post was titled "Kasai Code Breakthrough in Quantum Error Correction."
- August 3 — *Quantum* accepts the paper; a team including Tsinghua's Quantum Information Center posts Cornucopia codes, emphasizing regular structures suited to moving atoms collectively.
- August 7 — Willers Yang et al. post GALA codes, a general framework that subsumes the earlier rate-1/2 Kasai construction as a special case and incorporates AOD-compatible atom movement and logical operations into code design.
- September 9 — Formal publication.
The Decoder Side Is Accelerating in Parallel
Beyond code design, decoders are the other half of the overhead equation.
A team at Universidad Complutense de Madrid built a two-stage framework combining belief propagation with the Tesseract decoder: BP quickly handles easy errors, and hard cases go to Tesseract, which guarantees finding the most likely solution. Computational complexity drops at least 5-fold, with Tesseract node expansion reduced by up to ~15× on a [[126,12,d]] code (arXiv:2609.10056).
Another line, from Li Ying's group at the Graduate School of China Academy of Engineering Physics with Nobuyuki Yoshioka and Kento Tsubouchi of the University of Tokyo, proposes a post-selection decoding method based on candidate error reweighting: decode with the original error model to get a candidate, perturb the error model to lower the candidate's relative likelihood and re-decode, and keep the run if the candidate remains stable. It adds no physical qubits and modifies no quantum circuits, and combines with multiple mainstream decoders and code families. Published as *Physical Review Letters* 137, 110601 (2026).
What's Still Missing
The paper's own caveats are clear: the results are theoretical design and numerical experiments; performance on actual quantum processors must be evaluated separately.
Three things await verification. First, a global lower bound on the distance d remains open — 48 is strong evidence, not a proof. Second, actually deploying the construction requires that AOD atom rearrangement on the neutral-atom route achieve the collective movement Cornucopia and GALA envision. Third, the 10⁻⁸ frame error rate is a simulation value at 4% depolarizing noise; real devices have structured, not uniform, noise.
Worth watching: whether QuEra publishes measured data for Kasai codes on neutral-atom hardware, and whether third-party replications appear at the YITP and CWI workshops in October.
References
1. Kenta Kasai, *Breaking the Orthogonality Barrier in Quantum LDPC Codes*, Quantum, September 9, 2026, DOI 10.22331/q-2026-09-09-2205. 2. Institute of Science Tokyo press release, *Bringing classical LDPC code design theory to quantum computers* (EurekAlert!, 2026-09-09): [[9216,4612,d]] specs, selective orthogonality, FER 10⁻⁸ @ 4%, caveat on the classical density-evolution benchmark p≈0.05702. 3. Harvard / MIT / QuEra Computing preprint on adapting Kasai codes to reconfigurable neutral-atom quantum computers (2026-04-17) and QuEra's official post. 4. Cornucopia codes (2026-08-03, incl. Tsinghua Quantum Information Center) and GALA codes (2026-08-07, Willers Yang et al.). 5. Quantum Zeitgeist coverage of the Madrid BP + Tesseract two-stage framework (arXiv:2609.10056); China Academy of Engineering Physics Graduate School announcement of the qLDPC post-selection decoding result, PRL 137, 110601 (2026).