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

The gist One generation of the Standard Model as a quantum error-correcting code shows that anomaly cancellation in SO(10) grand unification provides a combinatorial structure equivalent to quantum error-correcting codes used for magic state distillation.

Anomaly Cancellation and Code Definition

Anomaly cancellation is defined as the vanishing of a cubic form, which is the defining property of triorthogonal codes, the quantum error-correcting codes behind magic state distillation Anomaly cancellation is the vanishing of this form, which defines triorthogonal codes Writing each bit b as a sign (−1)b turns the sixteen fields into sixteen integer charge vectors Imposed modulo a prime p, the same condition defines triorthogonal codes, a class of quantum error-correcting codes with a transversal non-Clifford gate on which magic state distillation is built Removing the right-handed neutrino gives a [[15, 1, 3]]p code for every prime p For qutrits it is the unique shortest distance-three triorthogonal code.

Physical Meaning of Code Ingredients

Every ingredient of the code has a Standard Model meaning The physical qudits are the fifteen quarks and leptons, and the removed right-handed neutrino νc is the logical qudit The stabilizers are the symmetries under which νc is neutral Since νc is a gauge singlet, they include the gauge charges, colour, weak isospin and hypercharge, and for p ≥ 5 they are exactly these Their triorthogonality is Standard Model anomaly cancellation The logical operator is the one remaining charge of SO(10) Its mixed anomalies with the gauge charges vanish, as triorthogonality requires, and its cubic anomaly, which νc would cancel, is the logical gate.

Code Construction and Logical Gate

The code construction involves reducing a triorthogonal lattice modulo p to define a triorthogonal space W = Λ mod p Deleting coordinate 0 means removing a fermion, ψ0, and the remaining fermions are the physical qudits The X-stabilizers are then the Zp symmetries under which ψ0 is neutral, and the logical X is one under which ψ0 has charge 1 The exponent Pl̸=0 clw3l in Eq. (2) is the cubic anomaly of the symmetry w on the remaining fermions, and it is fixed because the full spectrum is anomaly free The logical gate is thus the anomaly that the removed fermion would cancel.

Distillation Performance and Uniqueness

The distance relevant for distillation is the Z-distance d, which is defined as d = minJ: 0 ∈/ J, h0 ∈ span For odd p, triorthogonality and the exact gate follow from Proposition 1 The distance is at most dQ for all but finitely many odd primes p For p = 2, the sign bits of the sixteen weights, together with 1, span the first-order Reed–Muller code RM(1, 4), which is triply even The leading output error of the qutrit code at equal input infidelity is 6.25 ε3 against 35 ε3 for the fifteen-qubit code.

Comparison and Further Applications

The qutrit code is better in both equal infidelity and equal δ, by a factor of about 5.6 in the leading term at equal ε and about 3 at equal δ The construction applies beyond SO(10), where larger spinors of SO(4k + 2) with k ≥ 3 give [[22k − 1, 1, 3]]p codes for odd p in the same way The work establishes that the identification of anomaly cancellation with triorthogonality and the minimality and uniqueness of the qutrit code have not appeared before

Minimal Logical Errors

The minimal logical errors are four-fermion operators, such as the product of Dirac masses (ννc)(eec) Each is a statement of charge conservation The shared coordinate says which component of the 10 is exchanged The removed field is either νc, with nw = 0 and 6Y = 0, or the positron e c, with nw = 2 and 6Y = 6. Over Z, or modulo any prime p ≥ 5, only νc is neutral, and this is the field we remove Modulo 3 the positron is neutral too. The minimal logical errors have d = 3 (Theorem 1) Every relation in Table V splits in this way, with νc in one of the pairs; for conjugated fields the charge vector is −q. The removed field is the singlet of an su(5) whose Cartan generators are the stabilizers and whose 10 and 5¯ are the two orbits of the remaining fields.

Symmetry and Automorphisms

The monomial automorphism group of the code of Theorem 1 is S5 × Z2 This strengthens Theorem 3 The stabilizer has order 1920/16 = S5 For p = 5, 7 and 11 the group is S5 × F×p, so again no symmetries appear beyond the Weyl group and global scalars.

Doubling and Iteration

Doubling builds a triorthogonal code from two ingredients: a P-code and a seed T Doubling preserves integrality, so it can be iterated within anomaly-free spectra, but only with P-codes whose coefficients take both signs The doubled code is triorthogonal with logical exponent 1 and distance min(3, 1 + 2) = 3 Doubling can therefore be iterated within anomaly-free spectra; every code obtained in this way has an exact qutrit gate (Proposition 1).

Comparison to Previous Work

To our knowledge, the identification of anomaly cancellation with triorthogonality, the [[15, 1, 3]]p codes for odd p, and the minimality and uniqueness of the qutrit code have not appeared before The SO(10) structure of a generation and its anomaly cancellation, triorthogonal codes and doubling, and, for p = 2, the fifteen-qubit code itself are known. General solutions of the anomaly equations have been found for a single U(1) [40–42] and for two U(1) factors [43], and the equations have also been solved for extensions of the Standard Model gauge group [44, 45]. Anomaly conditions for discrete gauge symmetries (which constrain the Zp charges that label our codewords) go back to [46, 47], with a more recent treatment in [48]. The codes are related to Gauss’s law in a lattice gauge theory

Repercussions for Other Representations

Among compact simple Lie algebras only su(m) with m ≥ 3 has a cubic Casimir [5] so every representation of every other one is anomaly free The larger spinors give d = 3, as we now show. The 27 of E6 gives d = 2 Under SO(10) × U(1) ⊂ E6 it decomposes as 161 ⊕ 10−2 ⊕ 14, so in the normalization of the main text its weights are (q, 1) for the sixteen weights q of the 16, (±2ˆei, −2) and (0, 4). No weight is a multiple of another, even modulo an odd prime. The larger spinors give d = 3 as we now show For m = 5 this is the distance in Theorem 1, and for m = 7 it gives a [[63, 1, 3]]p code.

Repercussions for Qubits

For p = 2 the generation is Bravyi-Kitaev’s famous 15-qubit code, a standard benchmark for magic state distillation The qutrit code is also particularly interesting for its potential applications to distillation The fifteen-qudit code, also the shortest distance-three qubit triorthogonal code [[15, 27]], has instead the larger group GL(4, 2) ∼= A8 The qutrit code is also particularly interesting for its potential applications to distillation.

1. AI systems can implement fault-tolerant quantum computation by using One generation of the Standard Model as a quantum error-correcting code, which leverages anomaly cancellation is the vanishing of a cubic form to define triorthogonal codes with a transversal non-Clifford gate on which magic state distillation is built.

2. AI systems can perform magic state distillation more efficiently by utilizing the derived codes, achieving an output error of 6.25 ε3 for qutrits compared to 35 ε3 for fifteen-qubit codes at equal infidelity.

3. AI systems can design and analyze novel quantum error-correcting codes by using the mathematical structure of anomaly cancellation, specifically triorthogonal spaces, which are defined by the vanishing of a cubic form modulo a prime p, applicable to both qubits and qutrits.

4. AI systems can develop high-rate quantum codes for distillation by employing Doubling techniques on existing triorthogonal codes, such as doubling the fifteen-qudit code with the ternary Golay code to obtain a [[37, 1, 5]]3 code.

5. AI systems can interpret physical processes in particle physics by mapping logical errors to specific four-fermion operators derived from the minimal weight logicals, which are described as a product of Dirac masses (ννc)(eec).