One generation of the Standard Model as a quantum error-correcting code

Doubling and Dirac pairing.

• Doubling combines a triorthogonal seed and a P-code using paired stabilizers and one linking row, producing distance min(dS, dT + 2) when Nκ^3 + λT = 0.The P-code coefficients must take both signs for doubling to preserve integrality within anomaly-free spectra.Source: Doubling and Dirac pairing · pp. 3–4
• A Standard Model generation has the vectorlike pairs and chiral linking row required by doubling, with its chirality residing in weak isospin.After column rescaling, the code is the doubling of a trivial seed with a [[7, 1, 3]]_p P-code; for p = 2 this is the Steane code.Source: Doubling and Dirac pairing · p. 4
• The paired stabilizers are the charges (q5 − qi)/2, spanning the Cartan algebra of SU(3)c × U(1)em, while 2T3L supplies the linking row.The first copy contains uc, dc and ec; the second contains their Dirac partners, and the neutrino ν is the seed.Source: Doubling and Dirac pairing · p. 4
• No distance-three triorthogonal qutrit code is shorter than fifteen qutrits, and every [[15, 1, 3]]_3 code is equivalent to this one.Its monomial automorphism group is S5 × Z2, with Z2 negating every qutrit.Source: Doubling and Dirac pairing · p. 4
• The qutrit code’s S5 symmetry permutes the five oscillators and has orbits 10 and 5̄, matching the generation’s symmetry up to an overall sign.The construction also extends to larger SO(4k + 2) spinors, yielding [[22k − 1, 1, 3]]_p codes for odd p; the 27 of E6 gives distance two.Source: Doubling and Dirac pairing · p. 4
• 6.25 ϵ^3 versus 35 ϵ^3: the qutrit code has lower leading output error than the fifteen-qubit code at equal input infidelity.On each of its 25 weight-three logical supports, only 2 of 8 error patterns are logical.Source: Doubling and Dirac pairing · p. 4

Supplemental Material

• The supplement proves the lattice reduction, generator-matrix and qubit results, doubling, qutrit minimality and uniqueness, and distillation performance.It also interprets minimal logical errors as four-fermion operators, treats which fermions can be removed and other representations, and describes the reproducing code.Source: Supplemental Material · p. 7

PART A: BACKGROUND FOR READERS FROM EITHER SIDE

• The background explains that cubic-phase gates are non-Clifford and that magic state distillation uses the code in reverse, detecting errors through X-stabilizer measurements.For qutrits, the gate phase can be fixed up to a Pauli Z correction; when no correction is needed, the gate is exact.Source: PART A: BACKGROUND FOR READERS FROM… · p. 7
• A cubic anomaly is the chirality-weighted sum of products of three symmetry-charge vectors, and its vanishing is the condition connecting spectra to triorthogonal codes.The triangle diagram assigns each Weyl fermion a contribution cℓxℓyℓz.Source: PART A: BACKGROUND FOR READERS FROM… · p. 8, Figure 2 · p. 8
• A single-U(1) example gives a rank-one triorthogonal lattice but only a distance-one code after removing a charge-one fermion.The background contrasts this with nonabelian groups, whose Cartan weights provide higher-rank charge lattices.Source: PART A: BACKGROUND FOR READERS FROM… · p. 8
• The paper identifies anomaly cancellation with triorthogonality as a new connection, while situating it alongside existing code–field-theory links.It distinguishes its charge-assignment and anomaly-cancellation construction from gauge-invariant codes and approximate superselection codes.Source: PART A: BACKGROUND FOR READERS FROM… · p. 9

Triorthogonal lattices and their reductions

• A lattice is triorthogonal when its cubic form vanishes; over Q this is equivalent to vanishing of the associated trilinear form.Reduction gives a triorthogonal space modulo every prime that divides none of the coefficients.Source: Triorthogonal lattices and their… · p. 9
• For all but finitely many odd primes, puncturing the reduction of a triorthogonal lattice gives a [[n − 1, 1, dQ]]_p code.Here dQ is the smallest number of other columns whose rational span contains the removed column.Source: Triorthogonal lattices and their… · p. 9
• The distance remains dQ outside a finite exceptional set because minimal-support relations persist modulo p and lower-support span relations are excluded by rank minors.The proof discards primes dividing the finitely many relevant coefficients or minors.Source: Triorthogonal lattices and their… · p. 9

The fifteen-qudit code

• The generator matrix has fifteen fields as columns, a logical row first, and four X-stabilizer rows; restoring νc makes its integer cubic form vanish.The rows span a triorthogonal lattice reducible modulo any odd prime.Source: The fifteen-qudit code · p. 10, Table III · p. 10
• For p = 2, the sign-bit code is triply even when the generation’s fermion-number linear and quadratic sums vanish modulo 16 alongside its cubic anomalies.The relevant weights, pairwise overlaps, and triple overlaps are 8, 4, and 2.Source: The fifteen-qudit code · p. 10
• Deleting the νc coordinate gives the fifteen-qubit code, on which T^⊗15 acts as T† on the encoded qubit.Each term of the encoded state acquires phase e^(−iπa/4).Source: The fifteen-qudit code · p. 10

• If Nκ^3 + λT = 0, doubling yields a triorthogonal code with logical exponent λT and distance min(dS, dT + 2).Source: Doubling and Dirac pairing · p. 11
• The right-handed sector's stabilizers span the [4] simplex code, which is the Steane code for p = 2.Source: Doubling and Dirac pairing · p. 12
• The Standard Model construction combines a right-handed sector of distance 3, its Dirac-partner copy, and a trivial neutrino seed to obtain a doubled code of distance 3.The right-handed sector has distance 3 for every p; the seed has distance 1, and the doubled-code distance follows from Proposition S2.Source: Doubling and Dirac pairing · p. 12
• Undoing the rescaling, a unimodular change of basis identifies the doubled matrix's lattice with the generation's charge lattice and relabels its logical gate as U†.Source: Doubling and Dirac pairing · p. 13
• Integral doubling requires an integral P-lattice with coefficients of both signs; with unit coefficients and λT = ±1, the condition requires |N| = 1.An integral P-lattice of distance five and |N| = 1 would provide the input to double the fifteen-qudit code to distance five.Source: Doubling and Dirac pairing · p. 13
• With unit coefficients, doubling gives an anomaly-free charge lattice; iterating it yields exact qutrit gates and triorthogonal reductions for all but finitely many primes.The reductions retain the lattice's distance over Q.Source: Doubling and Dirac pairing · p. 13

• No triorthogonal qutrit code with distance at least 3 has length at most 14; at length 15, the [[15, 1, 3]]3 code is unique up to equivalence.Source: Minimality, uniqueness and symmetry · p. 13
• The exhaustive search found 36 length-15 solutions, all equivalent to the code of Theorem 1; none has distance 4.Every [[15, 1, 3]]3 code appears among the solutions, and all 36 have distance 3.Source: Minimality, uniqueness and symmetry · p. 14
• No triorthogonal [[15, k, 3]]3 code exists for k ≥ 2.Source: Minimality, uniqueness and symmetry · pp. 13–14
• The code's monomial automorphism group is S5 × Z2.The Weyl-group stabilizer of the removed weight consists of permutations of five coordinates; the full monomial group is stated in Proposition S4.Source: Minimality, uniqueness and symmetry · pp. 13–14

• About 5.6 at equal infidelity and about 3 at equal depolarizing strength δ are the qutrit code's factors of advantage in the leading term.Source: Distillation performance · p. 14
• 6.25ϵ^3 for the qutrit code versus 35ϵ^3 for the qubit code are the leading output-error terms; in δ, they are 2.78δ^3 versus 8.75δ^3.The paper attributes the qutrit coefficient to 25 weight-three supports, with two logical patterns among eight error patterns on each support.Source: Distillation performance · p. 14
• 1 − 15ϵ is the leading-order acceptance probability for both codes at equal infidelity.Source: Distillation performance · p. 14
• The advantage is a constant factor because both codes have yield exponent log 15 / log 3 ≈ 2.46.Noisy qutrit and qubit magic states need not cost the same to prepare, and the distilled gates differ.Source: Distillation performance · p. 14

The minimal logical errors as four-fermion operators

• Each minimum-weight logical is a Cartan-neutral four-fermion operator, but need not be invariant under the full Standard Model gauge group.Its fields have zero charge under every Cartan generator of SO(10), including those of the Standard Model.Source: The minimal logical errors as… · p. 15
• There are 25 such relations, each involving three fields besides νc with coefficients ±1.Each relation has a weight-three support.Source: The minimal logical errors as… · p. 15
• Every relation has the charge structure of a tree-level exchange: a 10 scalar for all relations, and a gauge boson for the fifteen marked g.The scalar-exchange pairs share one coordinate and oppose each other in the other four; the g-marked relations split into gauge-boson-exchange pairs.Source: The minimal logical errors as… · p. 15

Which fermions are removed

• Requiring the removed fermion to be neutral under the Standard Model Cartan generators selects νc over the positron ec over Z and primes p ≥5.Modulo 3, both νc and ec are neutral, corresponding to Georgi–Glashow and flipped SU(5) embeddings, respectively.Source: Which fermions are removed · pp. 15–16
• Removing k independent fermions gives a triorthogonal code with k logical qudits, but can reduce distance.For every p, removing any two gives [[14, 2, 2]]p; for odd p, certain triples give [[13, 3, 2]]p, while other triples have d = 1.Source: Which fermions are removed · p. 16

Other representations

• Real and pseudoreal representations have vectorlike partners and d = 1, while several identical generations likewise fail to avoid d = 1.Among algebras without a cubic Casimir, only so(4k + 2) for k ≥2 and e6 have complex representations; the smallest cited candidates are the 16 and 27.Source: Other representations · p. 16
• The 27 of E6 gives d = 2 for every odd p because each weight participates in a three-weight relation.Its invariant cubic form permits relations among three weights, unlike the four-fermion logical errors in the 16.Source: Other representations · p. 16
• For odd primes, removing one weight from a spinor of so(2m), with odd m ≥5, gives a [[2^m−1 −1, 1, 3]]p code.The proof establishes the distance by showing both an upper bound of three and the absence of one- or two-weight relations.Source: Other representations · p. 17
• The spinor construction gives a [[63, 1, 3]]p code when m = 7.For m = 5 it recovers the distance in Theorem 1.Source: Other representations · p. 17

REPRODUCIBILITY

• Repository scripts reproduce every computational statement in the Letter and Supplement, and Magma independently checks minimality, uniqueness, and automorphism results.The documentation maps scripts to statements, including code parameters, logical errors, exhaustive searches, and distillation figures.Source: REPRODUCIBILITY · p. 17