The circuit
The 811\times3211 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.
This matrix has 811 rows and 3,211 columns, 2,604,121 cells, too many to draw here. Download it 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 n3211Noisy T states per run.
outputs k1
distance d≥ 3A lower bound proved here. Its source certifies d\ge 31, and a fault of weight 31 is shown below, so d=31.
error coefficient A_{3}not definedUndefined while the distance is only a lower bound.
wires N811
checks r810
V_{\mathrm{ex}}1T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d7.3496Overhead per extractable T state. Lower is better. At the certified d=31: 2.3513.
\gamma = \log(n/k)/\log d7.3496Overhead per output qubit. At the certified d=31: 2.3513.
\gamma_T = \log(n/T)/\log d7.3496Overhead per unit of T-count. At the certified d=31: 2.3513.
T-count1Minimal T-count of the output gate.
phase-polynomial degree1Minimised over changes of output basis.
effective width1Rank of the output rows modulo the check rows.
Output gate
1 T0 CS0 CCZon 1 output wire
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.
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 64 single-wire gates and 67 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{Z} on check 606, \mathrm{Z} on check 607, \mathrm{Z} on check 608, \mathrm{Z} on check 609, \mathrm{Z} on check 613, \mathrm{Z} on check 615, \mathrm{Z} on check 617, \mathrm{Z} on check 621, \mathrm{Z} on check 622, \mathrm{Z} on check 626, \mathrm{Z} on check 631, \mathrm{Z} on check 632, \mathrm{Z} on check 633, \mathrm{Z} on check 638, \mathrm{Z} on check 640, \mathrm{Z} on check 641, \mathrm{Z} on check 642, \mathrm{Z} on check 643, \mathrm{Z} on check 647, \mathrm{Z} on check 650, \mathrm{Z} on check 653, \mathrm{Z} on check 655, \mathrm{Z} on check 706 and 107 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 131 correction gates
\mathrm{S}^\dagger on output 0, \mathrm{Z} on check 606, \mathrm{Z} on check 607, \mathrm{Z} on check 608, \mathrm{Z} on check 609, \mathrm{Z} on check 613, \mathrm{Z} on check 615, \mathrm{Z} on check 617, \mathrm{Z} on check 621, \mathrm{Z} on check 622, \mathrm{Z} on check 626, \mathrm{Z} on check 631, \mathrm{Z} on check 632, \mathrm{Z} on check 633, \mathrm{Z} on check 638, \mathrm{Z} on check 640, \mathrm{Z} on check 641, \mathrm{Z} on check 642, \mathrm{Z} on check 643, \mathrm{Z} on check 647, \mathrm{Z} on check 650, \mathrm{Z} on check 653, \mathrm{Z} on check 655, \mathrm{Z} on check 706, \mathrm{Z} on check 707, \mathrm{Z} on check 708, \mathrm{Z} on check 709, \mathrm{Z} on check 710, \mathrm{Z} on check 711, \mathrm{Z} on check 712, \mathrm{Z} on check 713, \mathrm{Z} on check 714, \mathrm{Z} on check 715, \mathrm{Z} on check 716, \mathrm{Z} on check 717, \mathrm{Z} on check 718, \mathrm{Z} on check 719, \mathrm{Z} on check 720, \mathrm{Z} on check 721, \mathrm{Z} on check 722, \mathrm{Z} on check 723, \mathrm{Z} on check 724, \mathrm{Z} on check 725, \mathrm{Z} on check 726, \mathrm{Z} on check 727, \mathrm{Z} on check 728, \mathrm{Z} on check 752, \mathrm{Z} on check 755, \mathrm{Z} on check 756, \mathrm{Z} on check 759, \mathrm{Z} on check 760, \mathrm{Z} on check 762, \mathrm{Z} on check 763, \mathrm{Z} on check 766, \mathrm{Z} on check 769, \mathrm{Z} on check 771, \mathrm{Z} on check 799, \mathrm{Z} on check 801, \mathrm{Z} on check 803, \mathrm{S} on check 805, \mathrm{Z} on check 806, \mathrm{S}^\dagger on check 807, \mathrm{S} on check 809, \mathrm{Z} on check 810, \mathrm{CZ} on output 0, check 805, \mathrm{CZ} on output 0, check 807, \mathrm{CZ} on output 0, check 809, \mathrm{CZ} on check 606, check 803, \mathrm{CZ} on check 607, check 803, \mathrm{CZ} on check 608, check 803, \mathrm{CZ} on check 609, check 803, \mathrm{CZ} on check 613, check 803, \mathrm{CZ} on check 615, check 803, \mathrm{CZ} on check 617, check 803, \mathrm{CZ} on check 621, check 803, \mathrm{CZ} on check 622, check 803, \mathrm{CZ} on check 626, check 803, \mathrm{CZ} on check 631, check 803, \mathrm{CZ} on check 632, check 803, \mathrm{CZ} on check 633, check 803, \mathrm{CZ} on check 638, check 803, \mathrm{CZ} on check 640, check 803, \mathrm{CZ} on check 641, check 803, \mathrm{CZ} on check 642, check 803, \mathrm{CZ} on check 643, check 803, \mathrm{CZ} on check 647, check 803, \mathrm{CZ} on check 650, check 803, \mathrm{CZ} on check 653, check 803, \mathrm{CZ} on check 655, check 803, \mathrm{CZ} on check 706, check 801, \mathrm{CZ} on check 707, check 801, \mathrm{CZ} on check 708, check 801, \mathrm{CZ} on check 709, check 801, \mathrm{CZ} on check 710, check 801, \mathrm{CZ} on check 711, check 801, \mathrm{CZ} on check 712, check 801, \mathrm{CZ} on check 713, check 801, \mathrm{CZ} on check 714, check 801, \mathrm{CZ} on check 715, check 801, \mathrm{CZ} on check 716, check 801, \mathrm{CZ} on check 717, check 801, \mathrm{CZ} on check 718, check 801, \mathrm{CZ} on check 719, check 801, \mathrm{CZ} on check 720, check 801, \mathrm{CZ} on check 721, check 801, \mathrm{CZ} on check 722, check 801, \mathrm{CZ} on check 723, check 801, \mathrm{CZ} on check 724, check 801, \mathrm{CZ} on check 725, check 801, \mathrm{CZ} on check 726, check 801, \mathrm{CZ} on check 727, check 801, \mathrm{CZ} on check 728, check 801, \mathrm{CZ} on check 752, check 799, \mathrm{CZ} on check 755, check 799, \mathrm{CZ} on check 756, check 799, \mathrm{CZ} on check 759, check 799, \mathrm{CZ} on check 760, check 799, \mathrm{CZ} on check 762, check 799, \mathrm{CZ} on check 763, check 799, \mathrm{CZ} on check 766, check 799, \mathrm{CZ} on check 769, check 799, \mathrm{CZ} on check 771, check 799, \mathrm{CZ} on check 805, check 806, \mathrm{CZ} on check 805, check 807, \mathrm{CZ} on check 805, check 808, \mathrm{CZ} on check 805, check 809, \mathrm{CZ} on check 805, check 810, \mathrm{CZ} on check 807, check 808, \mathrm{CZ} on check 807, check 809, \mathrm{CZ} on check 807, check 810, \mathrm{CZ} on check 809, check 810
Or no Clifford gate at all: run \mathrm{T}^3 on rotations 201, 203, 206, 208, 210, 212, 214, 215, 217, 218, 219, 223, 224, 225, 227, 230, 233, 235, 237, 238, 243, 245, 246, 249 and 799 more (listed below); \mathrm{T}^5 on rotations 1369, 1894, 2061; \mathrm{T}^\dagger on rotations 0, 1, 3, 5, 6, 8, 10, 12, 14, 17, 21, 22, 23, 27, 29, 30, 32, 33, 35, 37, 40, 41, 42, 43 and 30 more (listed below), 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.
All 823 rotations at T³
201, 203, 206, 208, 210, 212, 214, 215, 217, 218, 219, 223, 224, 225, 227, 230, 233, 235, 237, 238, 243, 245, 246, 249, 250, 252, 257, 259, 260, 261, 262, 263, 264, 267, 269, 271, 272, 273, 277, 279, 280, 281, 283, 284, 285, 291, 292, 294, 297, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 400, 402, 403, 405, 407, 409, 411, 414, 418, 419, 420, 424, 426, 427, 429, 430, 432, 434, 437, 438, 439, 440, 442, 445, 446, 449, 451, 452, 453, 454, 456, 463, 464, 466, 468, 472, 473, 474, 476, 480, 484, 485, 486, 487, 488, 491, 493, 494, 496, 596, 599, 601, 602, 604, 606, 608, 610, 613, 617, 618, 619, 623, 625, 626, 628, 629, 631, 633, 636, 637, 638, 639, 641, 644, 645, 648, 650, 651, 652, 653, 655, 662, 663, 665, 667, 671, 672, 673, 675, 679, 683, 684, 685, 686, 687, 690, 692, 693, 695, 795, 1179, 1180, 1181, 1182, 1183, 1188, 1189, 1190, 1193, 1194, 1196, 1197, 1199, 1202, 1204, 1208, 1209, 1210, 1214, 1217, 1219, 1221, 1222, 1224, 1225, 1226, 1227, 1231, 1232, 1234, 1236, 1237, 1238, 1241, 1242, 1243, 1245, 1246, 1249, 1253, 1254, 1256, 1261, 1265, 1272, 1368, 1375, 1376, 1377, 1378, 1382, 1383, 1386, 1389, 1391, 1392, 1394, 1396, 1397, 1398, 1402, 1403, 1404, 1406, 1407, 1409, 1411, 1414, 1419, 1420, 1421, 1424, 1426, 1430, 1431, 1435, 1438, 1439, 1441, 1442, 1443, 1446, 1448, 1449, 1450, 1451, 1453, 1454, 1455, 1457, 1458, 1459, 1460, 1461, 1462, 1464, 1465, 1466, 1467, 1468, 1469, 1470, 1471, 1472, 1473, 1474, 1475, 1476, 1477, 1478, 1479, 1480, 1481, 1482, 1483, 1484, 1485, 1486, 1487, 1488, 1489, 1490, 1491, 1492, 1493, 1494, 1495, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520, 1521, 1522, 1523, 1524, 1525, 1526, 1527, 1528, 1529, 1530, 1531, 1532, 1533, 1534, 1535, 1536, 1537, 1538, 1539, 1540, 1541, 1542, 1543, 1544, 1545, 1546, 1547, 1548, 1549, 1550, 1551, 1552, 1553, 1554, 1555, 1556, 1557, 1558, 1560, 1563, 1565, 1567, 1569, 1571, 1572, 1576, 1578, 1582, 1583, 1584, 1585, 1586, 1588, 1590, 1591, 1592, 1594, 1595, 1599, 1601, 1604, 1605, 1606, 1608, 1609, 1611, 1612, 1615, 1618, 1619, 1621, 1623, 1626, 1629, 1630, 1632, 1635, 1637, 1640, 1642, 1726, 1727, 1730, 1732, 1734, 1736, 1738, 1739, 1743, 1745, 1749, 1750, 1751, 1752, 1753, 1755, 1757, 1758, 1759, 1761, 1762, 1766, 1768, 1771, 1772, 1773, 1775, 1776, 1778, 1779, 1782, 1785, 1786, 1788, 1790, 1793, 1796, 1797, 1799, 1802, 1804, 1807, 1809, 1893, 2228, 2230, 2235, 2236, 2239, 2240, 2241, 2242, 2244, 2245, 2246, 2247, 2253, 2255, 2256, 2257, 2258, 2259, 2262, 2264, 2269, 2270, 2271, 2275, 2276, 2278, 2330, 2331, 2333, 2338, 2339, 2342, 2343, 2344, 2345, 2347, 2348, 2349, 2350, 2356, 2358, 2359, 2360, 2361, 2362, 2365, 2367, 2372, 2373, 2374, 2378, 2379, 2381, 2433, 2434, 2436, 2437, 2439, 2441, 2444, 2445, 2447, 2453, 2455, 2459, 2460, 2461, 2462, 2463, 2464, 2465, 2470, 2471, 2472, 2475, 2476, 2477, 2478, 2482, 2485, 2531, 2535, 2537, 2538, 2540, 2542, 2545, 2546, 2548, 2554, 2556, 2560, 2561, 2562, 2563, 2564, 2565, 2566, 2571, 2572, 2573, 2576, 2577, 2578, 2579, 2583, 2586, 2632, 2636, 2637, 2639, 2640, 2646, 2648, 2649, 2651, 2653, 2654, 2655, 2656, 2659, 2660, 2661, 2662, 2664, 2665, 2666, 2670, 2671, 2674, 2714, 2717, 2720, 2721, 2722, 2723, 2724, 2726, 2729, 2731, 2736, 2737, 2742, 2746, 2747, 2748, 2751, 2752, 2754, 2755, 2756, 2757, 2758, 2759, 2760, 2761, 2762, 2763, 2764, 2765, 2766, 2767, 2768, 2769, 2770, 2771, 2772, 2773, 2774, 2775, 2776, 2777, 2778, 2779, 2780, 2781, 2782, 2783, 2784, 2785, 2786, 2787, 2788, 2789, 2790, 2791, 2792, 2796, 2797, 2798, 2799, 2800, 2801, 2805, 2807, 2808, 2811, 2812, 2817, 2819, 2821, 2822, 2823, 2827, 2828, 2865, 2866, 2867, 2868, 2869, 2870, 2874, 2876, 2877, 2880, 2881, 2886, 2888, 2890, 2891, 2892, 2896, 2897, 3027, 3030, 3031, 3037, 3038, 3039, 3040, 3041, 3042, 3044, 3045, 3046, 3049, 3070, 3071, 3073, 3074, 3077, 3078, 3079, 3080, 3081, 3088, 3092, 3093, 3095, 3096, 3097, 3098, 3099, 3100, 3101, 3102, 3103, 3104, 3105, 3106, 3107, 3108, 3109, 3110, 3111, 3112, 3113, 3114, 3116, 3120, 3121, 3122, 3124, 3126, 3138, 3139, 3143, 3144, 3145, 3147, 3149, 3161, 3163, 3164, 3166, 3172, 3178, 3179, 3182, 3184, 3185, 3186, 3187, 3188, 3190, 3191, 3192, 3193, 3194
All 54 rotations at T†
0, 1, 3, 5, 6, 8, 10, 12, 14, 17, 21, 22, 23, 27, 29, 30, 32, 33, 35, 37, 40, 41, 42, 43, 45, 48, 49, 52, 54, 55, 56, 57, 59, 66, 67, 69, 71, 75, 76, 77, 79, 83, 87, 88, 89, 90, 91, 94, 96, 97, 99, 398, 597, 1178
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/3211.1.3.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-3211.1.3.a.txt", "w") as out:
out.write("\n".join(lines) + "\n")
Distance
3\le d\le 31. No fault of weight below 3 is both undetectable and damaging. None of weight 3 has been found, and one of weight 31 is, so the distance lies between the two.
A damaging fault of weight 31 sits on columns [0,199,398,597,796,987,1178,1369,1560,1727,1894,2061,2228,2331,2434,2535,2636,2715,2794,2863,2932,2979,3026,3071,3116,3139,3162,3179,3196,3203,3210].
Its source certifies d\ge 31, 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 31 above, it makes the distance exactly d=31. Source: Jain and Albert (arXiv:2408.12752), Table I: [[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2813,1,29]]; the doubling theorem of their Sec. III gives d >= min(d_sd, d_tri + 2) = 31, 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.
Original source
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
Provenance
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
1 recorded source
[
{
"regime": "Jain-Albert doubling: triorthogonal family",
"file": "transversal_t_codes/factories.json",
"label": "JA-3211",
"N": 811,
"d": 3,
"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": "[[199,1,31]] quantum QR code (extended QR [200,100,32]) doubled onto [[2813,1,29]] (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"
}
]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/3211.1.3.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")
Raw record
Show the JSON this page is built from
Source: master_catalog.json, factories[1479].