[\![575,\,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 36 single-wire gates and 33 \mathrm{CZ} gates: \mathrm{S} on output 0, \mathrm{Z} on check 51, \mathrm{Z} on check 52, \mathrm{Z} on check 53, \mathrm{Z} on check 54, \mathrm{Z} on check 55, \mathrm{Z} on check 56, \mathrm{Z} on check 57, \mathrm{Z} on check 58, \mathrm{Z} on check 59, \mathrm{Z} on check 60, \mathrm{Z} on check 61, \mathrm{Z} on check 62, \mathrm{Z} on check 63, \mathrm{Z} on check 64, \mathrm{Z} on check 65, \mathrm{Z} on check 66, \mathrm{Z} on check 67, \mathrm{Z} on check 68, \mathrm{Z} on check 69, \mathrm{Z} on check 70, \mathrm{Z} on check 71, \mathrm{Z} on check 72, \mathrm{Z} on check 73 and 45 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 82, 85, 86, 87, 88, 89, 91, 94, 96, 101, 102, 107, 111, 112, 113, 116, 117, 119, 120, 121, 122, 123, 124, 125 and 144 more (listed below); \mathrm{T}^\dagger on rotations 0, 1, 2, 4, 5, 11, 13, 14, 16, 18, 19, 20, 21, 24, 25, 26, 27, 29, 30, 31, 35, 36, 39, 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/575.1.6.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-575.1.6.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

6\le d\le 15. No fault of weight below 6 is both undetectable and damaging. None of weight 6 has been found, and one of weight 15 is, so the distance lies between the two.

A damaging fault of weight 15 sits on columns [0,79,158,227,296,343,390,435,480,503,526,543,560,567,574].

Its source certifies d\ge 15, 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 15 above, it makes the distance exactly d=15. Source: Jain and Albert (arXiv:2408.12752), Table I: [[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[417,1,13]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 15, 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-575",
  "N": 148,
  "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": "[[79,1,15]] quantum QR code (extended QR [80,40,16]) doubled onto [[417,1,13]] (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/575.1.6.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[1149].