[\![53,\,1,\,5]\!] 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 6 single-wire gates and 8 \mathrm{CZ} gates: \mathrm{Z} on output 0, \mathrm{Z} on check 1, \mathrm{Z} on check 2, \mathrm{S}^\dagger on check 6, \mathrm{S}^\dagger on check 7, \mathrm{Z} on check 9, \mathrm{CZ} on output 0, check 6, \mathrm{CZ} on output 0, check 7, \mathrm{CZ} on check 1, check 6, \mathrm{CZ} on check 1, check 7, \mathrm{CZ} on check 2, check 6, \mathrm{CZ} on check 2, check 7, \mathrm{CZ} on check 6, check 9, \mathrm{CZ} on check 7, check 9. 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 24, 26, 42, 43, 45, 48, 49, 50, 51, 52; \mathrm{T}^5 on rotations 4, 5, 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/53.1.5.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-53.1.5.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

d=5, exact. No fault of weight below 5 is both undetectable and damaging. One of weight 5 is.

Exactly 3,149 faults of weight 5 are undetectable and damaging, so A_{5}=3{,}149. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{5}\,p^{5}+O(p^{6}).

A damaging fault of weight 5 sits on columns [20,24,26,27,36].

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": "gamma_frontier_release/protocols/classified_direct_protocol_03_53_1_5.json",
  "label": "classified_direct_protocol_03_53_1_5",
  "N": 11,
  "d": 5,
  "d_is_exact": true,
  "origin": "Product-T basis recovered from supplied generalized triorthogonal classification",
  "provenance": "gamma frontier release record classified_direct_protocol_03_53_1_5 (source catalogue id classified_direct_protocol_03_53_1_5); construction: Product-T basis recovered from supplied generalized triorthogonal classification; matrix 9699d55ab25d526c...; the release states d=5 (exact), gamma=2.4668810662894636 (exact)",
  "notes": "matrix gamma_frontier_release/matrices/9699d55ab25d526c3115f2d98a46c9ddbf65b6095b91bde5d8e8687c2722b783.matrix"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/length54/pareto_frontier.json",
  "label": "protocol 3",
  "N": 11,
  "d": 5,
  "d_is_exact": true,
  "origin": "exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)",
  "provenance": "Pareto point 3 of the length-54 classification: output class Q1_0000000000000001 (T), (n, S, d_Z) = (53, 11, 5), leading error coefficient 3149; the outputs are its 1 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/53.1.5.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[308].