The search minimizes input count at fixed distance
A magic-state factory consumes many noisy input magic states and produces fewer, higher-quality ones. The paper writes a factory as [[n, k, d]]: n inputs, k outputs, and distance d, the smallest number of faulty inputs that pass every check yet still corrupt the output. Its width N = k + r counts the logical wires it acts on, where r is the number of parity checks.
For pure-T outputs the field uses the overhead exponent γ = log(n/k) / log d. Reaching an output error ϵ by concatenating a factory with itself costs O(log^γ(1/ϵ)) inputs per output, so for a fixed output target and required distance the goal is to minimize the input count n. That is the quantity the searches in this paper push down, not a full implementation cost.
Distance three reduces to distinct syndromes
The paper's central observation is that each input flips a set of checks, its syndrome. When no two inputs touch the same wires, distance at least three is equivalent to the syndromes being nonzero and pairwise distinct. This separates the choice of syndromes, which the authors call the parent, from the search for compatible output gates.
That separation is what makes the regime tractable. The parent spans the X stabilizers in triorthogonal-code language, and the output rows are the logical operators. For each parent, enumerating compatible output rows determines the factories it supports. When exhaustive enumeration is no longer feasible, group symmetry restricts the parent search instead.
One binary matrix underlies codes and circuits
The authors present a unified binary-matrix formulation that covers both triorthogonal-code searches and direct circuit searches. Read as a code, the matrix is a triorthogonal code or one of its generalizations. Read as a circuit, it is the list of parity rotations of a circuit construction. The paper does not commit to either reading and works with the matrix itself.
A theorem in the paper states that the two readings describe one object. Two selections implement the same gate up to a diagonal Clifford and global phase exactly when their column-count parities agree; the matrix is the generalized triorthogonal matrix whose prescribed moments are the target; and solutions form a single coset of the kernel of the parity map. A factory found by deleting from an identity and the same factory found by a triorthogonal solver are therefore the same point of that coset.
Language-model agents propose, deterministic solvers verify
The search workflow pairs a proposing agent with an exact engine. The proposing agent emits geometry specifications and nothing else: a value of k, an ordered list of check blocks, and a permutation group acting on each block. A fixed harness runs each specification through the slot solver, expands any solution into explicit columns, and audits it with an independent verifier.
The paper reports one concrete campaign trace. A proposal for k = 3, blocks [S3, S3, S3], target CCZ, d_target = 4, and cap 65 was recorded with the rationale that the record was 66 and the cap was set to 65 so that a solution would be a new record and an UNSAT would be a no-go inside that ansatz. The verdict was FEASIBLE at n = 64, d = 4, ok = True, marked NEW-RECORD after 123 seconds, where ok = True means the verifier re-derived the gate, the width, and the distance from the returned columns.
The authors also provide a context directory of search briefs and campaign notes so readers can train their own agents and tailor the search to their requirements.
699 classes, 564 of them new
The searches produced 699 factory classes, where a class is a quadruple (n, k, d, output gate) with the gate taken up to an invertible change of the output basis, the group GL(k, 2), together with diagonal Clifford corrections. The authors identify 564 as new because they lie outside the classified ranges and published protocol parameters. The new classes span n = 55 to 1715, with 320 at n ≤ 127.
By distance, the catalog contains 493 classes at d = 3, 88 at d = 4, 69 at d = 5, 42 at d = 6, 5 at d = 7, 1 at d = 8, and 1 at d = 9. The corresponding new counts are 382, 72, 65, 39, 4, 1, and 1. The catalog includes factories for pure-T states and factories with entangled outputs combining T, CS, and CCZ magic states.
Every class is a postselected diagonal CSS protocol: level-three diagonal rotations with independent Z faults, checks of pure X type, and rejection on any nonzero outcome. Catalytic and Hadamard-assisted protocols lie outside this class, so every comparison in the paper is within the class.
Lower exponents at every tabulated distance
The natural comparison for the wide pure-T end of the catalog is the punctured Reed–Muller triorthogonal family tabulated by Haah and Hastings, which gives explicit codes at each distance from three to seven. The paper compares its results at the same distances. At distance three, the reference value is γ = 1.528 for [[863, 161, 3]] and this work reaches γ = 1.159 for [[750, 210, 3]]. At distance four, 1.260 for [[872, 152, 4]] versus 1.185 for [[429, 83, 4]]. At distance five, 1.161 for [[887, 137, 5]] versus 1.120 for [[879, 145, 5]]. At distance six, 1.170 for [[912, 112, 6]] versus 1.057 for [[850, 128, 6]].
The [[850, 128, 6]] factory at γ = 1.057 is described as the lowest exponent the authors know of for an explicit pure-T factory at n ≤ 1000, and the distance-six one is the lowest known at any distance at n ≤ 1000. The [[1715, 287, 6]] factory at γ = 0.998 is the smallest known pure-T factory with an exponent below one.
The improvement is not confined to the largest codes. At distance three, the [[127, 23, 3]] factory uses approximately 5.52 inputs per output, compared with 5.74 for the reference [[109, 19, 3]]. The [[63, 11, 3]] factory has a similar ratio of 5.73 inputs per output while consuming 63 inputs rather than 109, and its exponent γ = 1.589 is the lowest the authors know among pure-T factories with n ≤ 100.
T-count puts different output gates on one scale
To place outputs of different kinds on one scale, the paper uses the minimum unitary T-count: the minimum number of T gates required in any circuit of CNOT and T gates to reach the output gate, computed exactly by Reed–Muller coset decoding. The computation is used up to k = 6. The T-count of a pure-T^⊗k output is k, a CCZ output is 7, and a CS output is 3.
The authors count over CNOT+T circuits because a native output replaces as many separately distilled T states as its T-count, the states conventional synthesis would spend on that gate. This metric describes the synthesis cost within CNOT+T circuits; the paper states that it does not determine a factory's full implementation cost.
Selected factories in the paper include [[55, 7, 3]] for T^⊗7 with width 14, [[63, 11, 3]] for T^⊗11 with width 17, [[100, 16, 3]] for T^⊗16 with width 23, [[127, 23, 3]] for T^⊗23 with width 30, [[850, 128, 6]] for T^⊗128 with width 166, [[63, 6, 3]] for CS^⊗3 with width 12, [[63, 6, 3]] for CCZ^⊗2 with width 12, and [[100, 12, 4]] for CCZ^⊗4 with width 20. Each listed factory has the smallest input count the authors know of for its output at its stated distance among postselected diagonal CSS protocols.
The comparison holds only inside a defined class
Codes with transversal T gates reach γ → 0, but only as the block length grows, and the paper notes that their performance at lengths a machine could afford has not been established. The authors therefore compare only explicit constructions with n ≤ 1000. Catalytic and Hadamard-assisted protocols lie outside the postselected diagonal CSS class and could reach parameters the catalog does not, so the comparisons above are within the class.
Some entries carry proved lower bounds rather than measured distances: a distance written as ≥ d is a proved lower bound. Nine output polynomials have dozens to thousands of factors, the largest 3587, and those rows record the number of factors instead of an exact minimization, with the gate itself in the database. The T-count metric is also limited to k = 6, and entangled outputs too wide for that exact minimization are marked as such in the figures.
The paper also states that its entries were found by instantiating the frameworks of Refs. [31, 65] with the puncture searches of Appendix B, and that every factory it reports is an explicit circuit available in its public database.
The n ≤ 1000 explicit-construction boundary is the real filter
For a reader deciding what to take from this work, the decisive split is not the total class count but the boundary the authors themselves use: explicit constructions with n ≤ 1000, compared at matched distance against the punctured Reed–Muller tables. Inside that boundary, the paper's claim is that its searches reach a lower overhead exponent at every distance in the table, and that [[850, 128, 6]] at γ = 1.057 is the lowest known explicit pure-T factory at n ≤ 1000.
The second split is scale. If the constraint is a small input budget, [[63, 11, 3]] at γ = 1.589 is the relevant entry, the lowest exponent the authors know among pure-T factories with n ≤ 100. If the constraint is crossing below γ = 1, the relevant entry is [[1715, 287, 6]] at γ = 0.998, the smallest known pure-T factory with an exponent below one. Those two thresholds, not the aggregate of 699 classes, are what a reader should carry away from this catalog.
Sources used in this answer
Symmetry and AI-assisted discovery of magic-state factories
Symmetry- and AI-assisted searches produced 699 magic-state factory classes, 564 of them new, including pure-T factories [[63, 11, 3]] at γ = 1.589, [[850, 128, 6]] at γ = 1.057, and [[1715, 287, 6]] at γ = 0.998.