[\![52,\,6,\,3]\!] factory

As gates: \mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{45}\cdot \mathrm{CCZ}_{123}\cdot \mathrm{CCZ}_{145}

As monomials: 0+01+23+45+123+145 (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 and 34 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{Z} on output 1, \mathrm{S} on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S}^\dagger on output 4, \mathrm{S} on output 5, \mathrm{Z} on check 8, \mathrm{Z} on check 10, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 0, output 4, \mathrm{CZ} on output 0, output 5, \mathrm{CZ} on output 0, check 6, \mathrm{CZ} on output 0, check 7, \mathrm{CZ} on output 0, check 12, \mathrm{CZ} on output 1, output 2, \mathrm{CZ} on output 1, output 4, \mathrm{CZ} on output 1, check 6, \mathrm{CZ} on output 1, check 8, \mathrm{CZ} on output 2, output 3, \mathrm{CZ} on output 2, output 4, \mathrm{CZ} on output 2, output 5, \mathrm{CZ} on output 2, check 6, \mathrm{CZ} on output 2, check 8 and 18 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

These Clifford gates cannot be avoided. No choice of \mathrm{T}, \mathrm{T}^3, \mathrm{T}^5 or \mathrm{T}^\dagger for each rotation removes them; the catalogue proves this by showing that the linear system over \mathbb{Z}_4 the powers would have to solve has no solution.

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/52.6.3.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-52.6.3.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

d=3, exact. No fault of weight below 3 is both undetectable and damaging. One of weight 3 is.

Exactly 176 faults of weight 3 are undetectable and damaging, so A_{3}=176. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{3}\,p^{3}+O(p^{4}).

A damaging fault of weight 3 sits on columns [0,5,8].

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. Wills, S. P. Jain, and S. Singh, "Classification of Generalised Triorthogonal Codes through Length 54," arXiv:2609.30860 (2026). arxiv.org/abs/2609.30860

a Pareto point of the exhaustive classification of generalised triorthogonal protocols through length 54 (classification/length54/): no protocol with n <= 54, the same CNOT+S output class and the same exact distance has n and S both no larger and one of them smaller, where S counts every matrix row -- the N of this table

Found by: exhaustive classification n<=54 (Pareto point)

Discovery: pre-existing

[
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/length54/pareto_frontier.json",
  "label": "protocol 62",
  "N": 13,
  "d": 3,
  "d_is_exact": true,
  "origin": "exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)",
  "provenance": "Pareto point 62 of the length-54 classification: output class Q6_000000000a08a400 (T tensor (T5CS14CS23CCZ145CCZ235)), (n, S, d_Z) = (52, 13, 3), leading error coefficient 176; the outputs are its 6 logical rows"
 }
]

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/52.6.3.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[305].