[\![417,\,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.

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 35 single-wire gates and 33 \mathrm{CZ} gates: \mathrm{Z} on check 12, \mathrm{Z} on check 13, \mathrm{Z} on check 14, \mathrm{Z} on check 15, \mathrm{Z} on check 16, \mathrm{Z} on check 17, \mathrm{Z} on check 18, \mathrm{Z} on check 19, \mathrm{Z} on check 20, \mathrm{Z} on check 21, \mathrm{Z} on check 22, \mathrm{Z} on check 23, \mathrm{Z} on check 24, \mathrm{Z} on check 25, \mathrm{Z} on check 26, \mathrm{Z} on check 27, \mathrm{Z} on check 28, \mathrm{Z} on check 29, \mathrm{Z} on check 30, \mathrm{Z} on check 31, \mathrm{Z} on check 32, \mathrm{Z} on check 33, \mathrm{Z} on check 34, \mathrm{Z} on check 58 and 44 more (listed below). 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 2, 3, 4, 5, 6, 7, 11, 13, 14, 17, 18, 23, 25, 27, 28, 29, 33, 34, 71, 72, 73, 74, 75, 76 and 88 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/417.1.6.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-417.1.6.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 [0,69,138,185,232,277,322,345,368,385,402,409,416].

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: Jain and Albert (arXiv:2408.12752), Table I: [[69,1,13]] code of the self-dual [70,35,12] code SD70 doubled onto [[279,1,11]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 13, with d_sd at least the classical distance less one (their Lemma 2.4); the [[69,1,13]] input's distance is proved here by enumerating every word of SD70 through the punctured coordinate.

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 and V. V. Albert, "Transversal Clifford and T-gate codes of short length and high distance," IEEE J. Sel. Areas Inf. Theory 6, 127-137 (2025); arXiv:2408.12752. doi.org/10.1109/JSAIT.2025.3570832

a code of the triorthogonal family of Jain and Albert (arXiv:2408.12752, Table I; transversal_t_codes/), self-dual CSS codes doubled onto [[95,1,7]], rebuilt here from the paper's construction; a verified witness, not a maximum

Found by: Jain-Albert doubling: triorthogonal family

Discovery: pre-existing

[
 {
  "regime": "Jain-Albert doubling: triorthogonal family",
  "file": "transversal_t_codes/factories.json",
  "label": "JA-417",
  "N": 108,
  "d": 6,
  "d_is_exact": false,
  "origin": "S. P. Jain and V. V. Albert, \"Transversal Clifford and T-gate codes of short length and high distance\", IEEE J. Sel. Areas Inf. Theory 6, 127 (2025), arXiv:2408.12752",
  "provenance": "[[69,1,13]] code of the self-dual [70,35,12] code SD70 doubled onto [[279,1,11]] (Table I)",
  "notes": "its [[69,1,13]] input is the self-dual [70,35,12] code found by transversal_t_codes/search_sd70.py, whose words through the punctured coordinate all weigh at least 14, in place of the paper's [70,35,14] code, which is only formally self-dual"
 }
]

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/417.1.6.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[1042].