[\![401,\,1,\,6]\!] factory

As gates: \mathrm{T}_{0}

As monomials: 0 (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.

None. With every rotation a \mathrm{T}=\mathrm{diag}(1,e^{i\pi/4}), the circuit's action is exactly the gate above, and an error-free run passes every check with certainty.

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/401.1.6.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-401.1.6.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

6\le d\le 13. No fault of weight below 6 is both undetectable and damaging. None of weight 6 has been found, and one of weight 13 is, so the distance lies between the two.

A damaging fault of weight 13 sits on columns [0,65,130,177,224,265,306,329,352,369,386,393,400].

Its source certifies d\ge 13, as a lower bound. That certificate is not re-checked here, but rankings on this site may use it. With the damaging fault of weight 13 above, it makes the distance exactly d=13. Source: A. Gong (github.com/gongaa/RM127, triply-even/): Gong's doubly even self-dual [[65,1,13]] code doubled onto [[271,1,11]]; the doubling theorem (Jain and Albert, arXiv:2408.12752, Sec. III) gives d >= min(d_sd, d_tri + 2) = 13; the distances of Gong's self-dual [[41,1,9]] and [[65,1,13]] codes are proved here by enumerating every word of their stabiliser spaces.

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, "On some K=1 codes admitting strongly transversal T," note in the RM127 repository, triply-even/ (2025). github.com/gongaa/RM127/tree/main/triply-even

a code of the triply even family of A. Gong (note in github.com/gongaa/RM127, triply-even/; transversal_t_codes/): her doubly even self-dual [[41,1,9]] and [[65,1,13]] codes and quantum QR codes doubled onto [[95,1,7]], rebuilt here from her self-dual codes, where T on every qubit is a logical T or T-dagger; a verified witness, not a maximum

Found by: Gong doubling: triply even family

Discovery: pre-existing

[
 {
  "regime": "Gong doubling: triply even family",
  "file": "transversal_t_codes/factories.json",
  "label": "G-401",
  "N": 104,
  "d": 6,
  "d_is_exact": false,
  "origin": "A. Gong, \"On some K=1 codes admitting strongly transversal T\", note in github.com/gongaa/RM127, triply-even/ (2025)",
  "provenance": "Gong's doubly even self-dual [[65,1,13]] code doubled onto [[271,1,11]] (Gong's note)"
 }
]

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/401.1.6.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[1019].