Magic State Factory Catalog

[\![43,\,4,\,3]\!] factory

\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}

distance, proved here
d=3
error coefficient
A_{3}=117
\gamma_\rho
—
V_{\mathrm{ex}}
—
wires
N=11
Clifford correction
S, CZ
found by
pre-existing

The circuit

The 11\times43 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight.

  • output wires
  • check wires, measured and postselected
wire
0 10 20 30 40
out 0 (7)·········1······1··1·····1····1··1···1·····
out 1 (18)1111··1·1······111·11·1·1·11···1·1···1·····
out 2 (18)··11··11···11·1··11·1··1·1111·1··1···1·····
out 3 (14)11··11··1·1··1·1·····111··111··············
check 4 (24)11··1·11·111·1··111·11·1··111·11·11···1··1·
check 5 (24)1·1··11·1·111·1·11111···1111··11··11··1···1
check 6 (20)·11···1··11·1··11·1···1··11·1···1···111·111
check 7 (20)···1111····11····11··11····11··11··11··1111
check 8 (20)·······111111······1111······1111····111111
check 9 (20)·············1111111111··········1111111111
check 10 (20)·······················11111111111111111111

The gate is read off the output rows. An output row of odd weight gives a T, a pair of output rows with odd overlap gives a CS, and an odd triple overlap gives a CCZ. The circuit is a valid factory because every such parity that touches a check row is even.

Figures of merit

inputs n43Noisy T states per run.
outputs k4
distance d3Exact, proved and witnessed here.
error coefficient A_{3}117Faults of weight 3 in the circuit shown that pass every check and damage the output. The output error is A_{3}\,p^{3} to leading order.
wires N11
checks r7
V_{\mathrm{ex}}not definedUndefined, because the gate's terms overlap.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log dnot definedOverhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d2.1617Overhead per output qubit.
\gamma_T = \log(n/T)/\log d1.9586Overhead per unit of T-count.
T-count5Minimal T-count of the output gate.
phase-polynomial degree2Minimised over changes of output basis.
effective width4Rank of the output rows modulo the check rows.

Output gate

1 T2 CS0 CCZon 4 output wires

As gates: \mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}

As monomials: 0+12+13 (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.

Clifford correction

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 9 single-wire gates and 32 \mathrm{CZ} gates: \mathrm{S} on output 0, \mathrm{S}^\dagger on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S} on output 3, \mathrm{Z} on check 6, \mathrm{Z} on check 7, \mathrm{Z} on check 8, \mathrm{Z} on check 9, \mathrm{Z} on check 10, \mathrm{CZ} on output 1, output 2, \mathrm{CZ} on output 1, check 4, \mathrm{CZ} on output 1, check 7, \mathrm{CZ} on output 1, check 8, \mathrm{CZ} on output 1, check 10, \mathrm{CZ} on output 2, check 8, \mathrm{CZ} on output 2, check 9, \mathrm{CZ} on output 3, check 4, \mathrm{CZ} on output 3, check 5, \mathrm{CZ} on output 3, check 6, \mathrm{CZ} on output 3, check 7, \mathrm{CZ} on check 4, check 5, \mathrm{CZ} on check 4, check 6, \mathrm{CZ} on check 4, check 7, \mathrm{CZ} on check 4, check 8 and 17 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

All 41 correction gates

\mathrm{S} on output 0, \mathrm{S}^\dagger on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S} on output 3, \mathrm{Z} on check 6, \mathrm{Z} on check 7, \mathrm{Z} on check 8, \mathrm{Z} on check 9, \mathrm{Z} on check 10, \mathrm{CZ} on output 1, output 2, \mathrm{CZ} on output 1, check 4, \mathrm{CZ} on output 1, check 7, \mathrm{CZ} on output 1, check 8, \mathrm{CZ} on output 1, check 10, \mathrm{CZ} on output 2, check 8, \mathrm{CZ} on output 2, check 9, \mathrm{CZ} on output 3, check 4, \mathrm{CZ} on output 3, check 5, \mathrm{CZ} on output 3, check 6, \mathrm{CZ} on output 3, check 7, \mathrm{CZ} on check 4, check 5, \mathrm{CZ} on check 4, check 6, \mathrm{CZ} on check 4, check 7, \mathrm{CZ} on check 4, check 8, \mathrm{CZ} on check 4, check 9, \mathrm{CZ} on check 4, check 10, \mathrm{CZ} on check 5, check 6, \mathrm{CZ} on check 5, check 7, \mathrm{CZ} on check 5, check 8, \mathrm{CZ} on check 5, check 9, \mathrm{CZ} on check 5, check 10, \mathrm{CZ} on check 6, check 7, \mathrm{CZ} on check 6, check 8, \mathrm{CZ} on check 6, check 9, \mathrm{CZ} on check 6, check 10, \mathrm{CZ} on check 7, check 8, \mathrm{CZ} on check 7, check 9, \mathrm{CZ} on check 7, check 10, \mathrm{CZ} on check 8, check 9, \mathrm{CZ} on check 8, check 10, \mathrm{CZ} on check 9, check 10

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.

Convert the output with CNOT and S

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.

Or list every gate it reaches. The list is built in your browser and saved as a text file, one gate per line.

Python code that writes the same list 
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/43.4.3.e.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-43.4.3.e.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

Distance

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

Exactly 117 faults of weight 3 are undetectable and damaging, so A_{3}=117. 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 [0,1,2].

A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.

Original source

  • 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
  • S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535

Provenance

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) symmetry-SAT search AI search

Discovery: pre-existing

58 recorded sources
[
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  "label": "0+12+13",
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  "d": 3,
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  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
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 {
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  "label": "0+1+2+3+01+02+03+12+012+013+023+123",
  "N": 11,
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  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
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 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
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  "label": "0+1+2+3+01+02+03+12+13+012+013+023",
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 {
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 {
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 {
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  "provenance": "RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-07); high_tcount_subframes_REPS.json"
 },
 {
  "regime": "symmetry-SAT search",
  "file": "symmetry_sat_search/catalog/factories.json",
  "label": "T0 . CS01 . CS02 . CS12 . CS13 . CS23 . CCZ012 . CCZ013 . CCZ023 . CCZ123",
  "N": 11,
  "d": 3,
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  "origin": null,
  "provenance": "explicit circuit from ansatz-free SAT; archived result: t5_43_4_3.json"
 },
 {
  "regime": "AI search",
  "file": "master_catalog/catalogue_new",
  "label": "T0.CS01.CS02.CS12.CS13.CS23.CCZ012.CCZ013.CCZ023.CCZ123",
  "N": 11,
  "d": 3,
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  "origin": "frontier_campaign",
  "provenance": "frontier_campaign \u00b7 catalogue.jsonl:124"
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 {
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 {
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 {
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 {
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 {
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 {
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 {
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 {
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 {
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 {
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 {
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 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
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 {
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 {
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 {
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 {
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 {
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 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+3+01+12+23+012+023",
  "N": 11,
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  "d_is_exact": true,
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  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
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  "label": "0+2+3+01+12+012+023",
  "N": 11,
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  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-13); high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+2+01+23+012+023",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/census_r7.json",
  "label": "0+3+01+012+023",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 3430, origin 3; high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/census_r7.json",
  "label": "0+2+01+03+12+13+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-14); high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/census_r7.json",
  "label": "0+3+01+03+12+13+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-15); high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+2+01+03+13+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+2+01+03+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+3+01+03+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+01+12+13+012",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+01+13+012+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+01+02+03",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+2+3+01+02+03+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+01+02+12+13+23+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+01+02+13+23",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+2+01+02+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/census_r7.json",
  "label": "0+1+01+02+23",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/census_r7.json",
  "label": "0+1+01+23+023",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+12+13+23+012+013+023+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+1+2+012+013+023+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+12+13+012+013",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+2+3+012+013+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+12+012+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+12+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/legacy/rank7_census/catalog/sk_classes_r7.json",
  "label": "0+12+13+23+123",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": null,
  "provenance": "RM(3,7) orbit class 2936, origin 3; census_shard_06.json"
 },
 {
  "regime": "exhaustive classification n<=54 (Pareto point)",
  "file": "classification/length54/pareto_frontier.json",
  "label": "protocol 22",
  "N": 11,
  "d": 3,
  "d_is_exact": true,
  "origin": "exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)",
  "provenance": "Pareto point 22 of the length-54 classification: output class Q4_00000000000000e0 (T tensor (CS12CS13)), (n, S, d_Z) = (43, 11, 3), leading error coefficient 125; the outputs are its 4 logical rows"
 }
]

Use it in code

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/43.4.3.e.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")

Raw record

Show the JSON this page is built from

    
Source: master_catalog.json, factories[171].