[\![72,\,3,\,6]\!] factory

As gates: \mathrm{CCZ}_{012}

As monomials: 012 (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 3 single-wire gates and 3 \mathrm{CZ} gates: \mathrm{Z} on output 0, \mathrm{Z} on output 1, \mathrm{Z} on output 2, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 1, output 2. 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 \mathrm{T}^3 on rotations 16, 17, 22, 23, 24, 27, 28, 31, 32, 34, 36, 38, 40, 42, 45, 47, 50, 51, 54, 55, 60, 61, 62, 63 and 4 more (listed below), and \mathrm{T} on the rest. 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/72.3.6.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-72.3.6.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 37,632 faults of weight 6 are undetectable and damaging, so A_{6}=37{,}632. 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 [3,6,8,11,40,43].

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

A. Gong, C. A. Pattison, P. Rall, and A. Wills, "Magic State Distillation via Codes over Binary Extension Fields," arXiv:2608.09727 (2026). arxiv.org/abs/2608.09727

a concatenated protocol of Gong, Pattison, Rall and Wills (arXiv:2608.09727): an F8 code fed by copies of the 8T->CCZ protocol, rebuilt from the paper's construction and written as one circuit on 18 wires; a verified witness, not a maximum

Found by: Gong et al.: F8 codes concatenated with 8T->CCZ

Discovery: pre-existing

[
 {
  "regime": "Gong et al.: F8 codes concatenated with 8T->CCZ",
  "file": "literature_imports/ccz72_d6/factories.json",
  "label": "GPRW-72T-CCZ",
  "N": 18,
  "d": 6,
  "d_is_exact": true,
  "origin": "A. Gong, C. A. Pattison, P. Rall, and A. Wills, \"Magic State Distillation via Codes over Binary Extension Fields,\" arXiv:2608.09727 (2026)",
  "provenance": "(8T -> CCZ) * (9CCZ -> CCZ) of Table 7: the F8 code of Construction 4.5, Eq. (49), binarised in the self-dual normal basis of Eq. (46), each of its nine input CCZ states made by its own 8T -> CCZ, written as one circuit with the nine inner checks as wires 9..17",
  "notes": "A_6 = 37632. The paper states 10^{4.58} p^6 (Table 7). Its Eq. (189) with its Table 5 and the 8T -> CCZ error weights (12, 12, 4) p^2 gives exactly 37632, which is 10^4.576. Clifford correction: Z on each output and CZ on each pair, since transversal U_7 over F_8 deposits (-1)^[u != 0] on the logical qudit u"
 }
]

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/72.3.6.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[513].