Nineteen qubits are necessary and sufficient for symmetric quantum codes correcting two errors

Yuxuan Zhang · 10 November 2026 · Version 1.0

A logical qubit encoded in the fully symmetric subspace needs at least nineteen physical qubits to correct every error on two qubits. A nineteen-qubit construction is already known, so the lower bound determines the minimum length:

The theorem answers the original QIQC Open Problems catalogue question. This is a complete computer-assisted proof candidate, reviewed by a separate AI agent in a fresh context. External expert review is outstanding.

Full paper (PDF) · Source, certificates and review (ZIP) · Version and SHA-256 metadata

What is being minimized

Every state in the code must be unchanged when the physical qubits are permuted. Each logical basis vector is therefore a linear combination of Dicke states. The coefficients may be complex, and neither parity of the length nor any extra symmetry of the logical basis is assumed.

The code must correct arbitrary errors supported on at most two qubits, including their linear combinations. Equivalently, it detects all Pauli operators of weight at most four. The theorem includes degenerate codes. It concerns the fully symmetric physical subspace; it is not a lower bound for general quantum error-correcting codes.

Why eighteen qubits are the key case

Aydin, Alekseyev and Barg constructed a nineteen-qubit code in 2024 using the real Pollatsek–Ruskai family. Their Section 7 reports interval certification of solutions to the corresponding polynomial equations. We use this published existence result for the upper bound.

The scalar single-spin linear-programming framework goes back to Bumgardner. Okada’s dissertation, Table 3.1, reports distance-five dimension bounds of 1 through seventeen qubits, 2.019 at eighteen and 2.096 at nineteen. The latter bounds still allow a two-dimensional code. The text discusses agreement with exact rational solvers for smaller cases, without giving a cutoff for the table.

The decisive new exclusion here is eighteen qubits. We also supply exact certificates for every length from nine through seventeen, so that the finite exclusion is self-contained. Those earlier bounds are credited to the literature, not counted again as new results. Bond and collaborators conjecture the more general formula \(3t^2+3t+1\) from numerical searches. This result settles its two-error case, not the formula for arbitrary numbers of errors. The targeted primary-source review found no prior proof of the unrestricted eighteen-qubit exclusion; its source comparison is included in the package.

How the lower bound works

A putative code projector gives a positive third moment after averaging over collective spin rotations. Error detection becomes a set of linear constraints on this moment. Permutation symmetry and positivity after partial time reversal impose further constraints.

Two details keep the relaxation valid for every code. Complex conjugation is averaged at the level of moments, so complex code coefficients remain covered. The standard representation of the three-copy permutation group is retained alongside the symmetric component; discarding it would incorrectly exclude valid projectors.

Angular-momentum identities then express the constraints in rational coordinates. For each length from nine to eighteen, an exact dual certificate proves that this necessary relaxation is infeasible. The checker regenerates all coefficients, verifies the dual identity, and checks positive matrix pivots and a strictly negative rational right-hand side. It does not accept a numerical solver’s infeasibility status as a proof. The quantum Singleton bound excludes lengths at most eight.

The scalar enumerator and time-reversal ingredients have prior single-spin formulations, including Okada’s work and the more recent intrinsic MacWilliams identities. The contribution is the three-copy relaxation with partial time reversal and its exact exclusion at eighteen qubits.

Evidence and review

The ten certificates contain 435 nonnegative scalar multipliers and 2,496 positive rational matrix pivots. The release runner checks file hashes and verifies every certificate in a fresh temporary directory without an optimizer or cached coefficient matrices.

The independent model review examined the analytic necessity of the constraints, not only their computational replay. Its controls include generic complex rank-two projectors, independently constructed partial transposes, exact Clebsch–Gordan contractions, and a known twenty-one-qubit distance-five code. It also checks the source distinction between an unrestricted code and the real Pollatsek–Ruskai family.

The package records both the proof review and its limits. The nineteen-qubit upper bound is inherited from the published construction, rather than recertified by this package. The calculations support the finite certificates and the formulas they use; the written argument establishes why all admissible codes enter the relaxation.

The author directed the research and workflow. AI systems assisted with derivation, source comparison, computation and drafting. Review by a separate model context is not external expert or proof-assistant verification. This website release is separate from journal review, archival submission and catalogue acceptance.