[\![248,\,8,\,6]\!] factory

As gates: \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\cdot \mathrm{T}_{6}\cdot \mathrm{T}_{7}

As monomials: 0+1+2+3+4+5+6+7 (output wires per term, separated by +)

This row stands for every gate that a CNOT circuit and S, Z and CZ gates on the outputs can reach from this one. They all prepare the same magic state. The wires are shown in the S_k-canonical frame.

With every rotation a \mathrm{T}=\mathrm{diag}(1,e^{i\pi/4}), the rotations deposit the gate above times a diagonal Clifford. Undo it after the rotations and before the checks are measured, with 8 single-wire gates: \mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S} on output 2, \mathrm{S} on output 3, \mathrm{S} on output 4, \mathrm{S} on output 5, \mathrm{S} on output 6, \mathrm{S} on output 7. The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

Or no Clifford gate at all: run every rotation as \mathrm{T}^\dagger. A power of \mathrm{T} costs one magic state, like \mathrm{T} itself. For a code this is a transversal \mathrm{T} gate, \mathrm{T}^j on each qubit.

Rotations are numbered by matrix column, from 0. The correction comes from the columns by the weight expansion of Bravyi and Haah (2012). verify_catalog.py re-derives it, checks the corrected circuit's phase input by input against the gate, and on small circuits simulates it gate by gate.

A CNOT circuit on the outputs, followed by S, Z and CZ gates, turns this factory's gate into other gates that prepare the same magic state. Type a target gate to check whether it is reachable. If it is, you get the circuit, checked on every basis state.

import json, urllib.request

NAMES = {1: "T", 2: "CS", 3: "CCZ"}


def equivalent_gates(k, terms, limit=1_000_000):
    """Every gate a CNOT circuit and S, Z, CZ gates reach from this one.

    A gate is its set of terms mod Clifford: {a} is T, {a,b} is CS and
    {a,b,c} is CCZ on those outputs. CNOT c->t sends x_t to
    x_c + x_t - 2 x_c x_t, so a term on t but not c adds the term with t
    replaced by c, and, below CCZ, the term with c added.
    """
    start = frozenset(frozenset(m) for m in terms)
    seen, queue = {start}, [start]
    for gate in queue:
        for c in range(k):
            for t in range(k):
                if c == t:
                    continue
                new = set(gate)
                for m in gate:
                    if t in m and c not in m:
                        new ^= {(m - {t}) | {c}}
                        if len(m) < 3:
                            new ^= {m | {c}}
                new = frozenset(new)
                if new not in seen:
                    seen.add(new)
                    queue.append(new)
                    if len(seen) > limit:
                        raise RuntimeError(f"more than {limit} equivalent gates")
    return seen


def name(gate, k):
    sep = "," if k > 10 else ""
    terms = sorted((sorted(m) for m in gate), key=lambda m: (len(m), m))
    return "·".join(NAMES[len(m)] + sep.join(map(str, m)) for m in terms) or "I"


url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/248.8.6.a.json"
record = json.load(urllib.request.urlopen(url))
k = record["parameters"]["k"]
gates = equivalent_gates(k, record["gate"]["monomials"])
lines = sorted(name(g, k) for g in gates)
print(len(lines), "equivalent gates")
with open("equivalent-gates-248.8.6.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

d=6, exact. No fault of weight below 6 is both undetectable and damaging. One of weight 6 is.

Exactly 4,340 faults of weight 6 are undetectable and damaging, so A_{6}=4{,}340. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{6}\,p^{6}+O(p^{7}).

A damaging fault of weight 6 sits on columns [23,50,66,102,165,237].

A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.

S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535

cap witnesses, NOT records. Each ATTAINS a proved maximum puncture count for its ambient size at d=6, making that bound exact rather than an upper bound: m=8 gives [[248,8,6]] and m=9 gives [[496,16,6]], both with n/t = 31 exactly, hence gamma = log(31)/log(6) = 1.9165 independent of m. Filed as evidence that these cells are closed, not as frontier points -- the d=6 pure-T bar is gamma 1.063146.

Found by: AI search: pure-T puncture caps

Discovery: AI search

[
 {
  "regime": "AI search: pure-T puncture caps",
  "file": "magic-states-AI/campaigns/2026-09-15_m8_m9_d6/found/n248_T8_d6_CAP_WITNESS_not_a_record.json",
  "label": "n248_T8_d6_CAP_WITNESS_not_a_record",
  "N": 53,
  "d": 6,
  "d_is_exact": true,
  "origin": null,
  "provenance": "m = 8 maximal parent class 2 (|T| = 16 intersecting triples, dim 53). Campaign 2026-09-15_m8_m9_d6 proves t <= 8 on that parent at d = 6 (and t <= 16 over ALL 15 maximal m = 8 parents, closing the [[223,33,6]] cell); this ATTAINS the cap, so the bound is tight. Pure disjoint T^8, V_ex = 8, rho = n/V_ex = 31.000, gamma = log(248/8)/log 6 = 1.916545. NOT a record and not near one: the external d=6 bar is gamma 1.086033, and this is dominated many times over by direct sums of small catalogued factories (CLAUDE.md rule 6). Filed as a tightness witness for a closure, not as an improvement."
 }
]

Load this factory and re-derive every number from its circuit with the catalogue's own verifier.

# git clone https://github.com/shubhamj810/magic-state-factory-repo
# cd magic-state-factory-repo/master_catalog
import json, urllib.request
import verify_catalog as VC

url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/248.8.6.a.json"
record = json.load(urllib.request.urlopen(url))
columns = record["circuit"]["columns"]
k, N = record["parameters"]["k"], record["parameters"]["N"]

facts, problems = VC.derive(columns, k, N)    # gate, T-count, degree, ...
distance = VC.measure_distance(columns, k, N)  # exhaustive fault search
print(facts["gate_human"], distance["d_exact"] or distance["d_at_least"],
      problems or "verified")

Source: master_catalog.json, factories[891].