The circuit
The 16\times9212 matrix of this circuit is not drawn: the catalogue stores it as the rule that builds it, below, and every number on this page is derived from the columns that rule rebuilds.
This circuit has 16 rows and 9,212 columns, 147,392 cells. The catalogue stores it as the two-group construction with k=4, N=16, s_{\mathrm{total}}=4, s_{O}=2, which rebuilds every column; download the matrix as a CSV or copy the columns (see Download).
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 n9212Noisy T states per run.
outputs k4
distance d2Exact, proved and witnessed here.
error coefficient A_{2}12,282Faults of weight 2 in the circuit shown that pass every check and damage the output. The output error is A_{2}\,p^{2} to leading order.
wires N16
checks r12
V_{\mathrm{ex}}4T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d11.1693Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d11.1693Overhead per output qubit.
\gamma_T = \log(n/T)/\log d11.1693Overhead per unit of T-count.
T-count4Minimal T-count of the output gate.
phase-polynomial degree1Minimised over changes of output basis.
effective width4Rank of the output rows modulo the check rows.
Output gate
4 T0 CS0 CCZon 4 output wires
As gates: \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}
As monomials: 0+1+2+3 (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 4 single-wire gates: \mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S} on output 2, \mathrm{S} on output 3. 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 every rotation as \mathrm{T}^\dagger. 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.
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/9212.4.2.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-9212.4.2.a.txt", "w") as out:
out.write("\n".join(lines) + "\n")
Distance
d=2, exact. No fault of weight below 2 is both undetectable and damaging. One of weight 2 is.
Exactly 12,282 faults of weight 2 are undetectable and damaging, so A_{2}=12{,}282. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{2}\,p^{2}+O(p^{3}).
A damaging fault of weight 2 sits on columns [0,1056].
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
S. Singh, C. Gidney, and C. Jones, "Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search," arXiv:2606.28518 (2026). arxiv.org/abs/2606.28518
Provenance
found by the two-group borrowed-identity search of Singh, Gidney and Jones (arXiv:2606.28518; borrowed_identities/): an identity circuit symmetric within its output and check blocks, with its output-only gates removed; a verified witness, not a maximum
Found by: borrowed-identity search: two-group
Discovery: pre-existing
1 recorded source
[
{
"regime": "borrowed-identity search: two-group",
"file": "borrowed_identities/circuits/unified_l3_large.json",
"label": "lsc-0749",
"N": 16,
"d": 2,
"d_is_exact": true,
"origin": "https://github.com/shraggy/Magic_state_factory_search at 4627014, outputs/large_sweep.csv line 749",
"provenance": "unified search l=3, two-group s_total=4,s_O=2"
}
]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/9212.4.2.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")
Raw record
Show the JSON this page is built from
Source: master_catalog_large.json, factories[281] (stored as its construction; the columns here are rebuilt from it).