Bonsai Algorithm#
The Bonsai Algorithm provides a method to create a TernaryTree encoding with the same toplogy as a target quantum computer.
First we’ll define a RustworkX graph representing the topology of the device (this can be obtained from most quantum SDKs.)
Just like the original paper, we’ll use a heavy-hex layout with 36 qubits.
import rustworkx as rx
from rustworkx import visualization as rxv
import matplotlib.pyplot as plt
import numpy as np
import ferrmion as fr
graph = rx.PyGraph()
graph.add_nodes_from(range(37))
graph.add_edges_from_no_data(
[
(0, 1),
(0, 2),
(0, 3),
(1, 4),
(2, 5),
(3, 6),
(4, 7),
(4, 8),
(5, 9),
(5, 10),
(6, 11),
(6, 12),
(7, 13),
(8, 14),
(9, 15),
(10, 16),
(11, 17),
(12, 18),
(13, 19),
(13, 20),
(14, 21),
(14, 22),
(15, 23),
(15, 24),
(16, 25),
(16, 26),
(17, 27),
(17, 28),
(18, 29),
(18, 30),
(22, 31),
(26, 32),
(30, 33),
(31, 34),
(32, 35),
(33, 36),
]
)
EdgeIndices[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]
Let’s plot a diagram of the device so we can see it’s right.
(This bit looks complicated but it’s just to plot a nice-looking diagram so you can safely itnore it)
r = np.array([0, -1]) * np.sqrt(1 / graph.num_nodes())
s = np.array([1 * np.sin(45), np.cos(45)]) * np.sqrt(1 / graph.num_nodes())
q = np.array([-1 * np.sin(45), np.cos(45)]) * np.sqrt(1 / graph.num_nodes())
pos = {
0: [0, 0],
1: q,
2: s,
3: r,
4: 2 * q,
5: 2 * s,
6: 2.0 * r,
7: 2 * q - s,
8: 2 * q - r,
9: 2 * s - r,
10: 2 * s - q,
11: 2 * r - q,
12: 2 * r - s,
13: 2 * (q - s),
14: 2 * q - 2 * r,
15: 2 * s - 2 * r,
16: 2 * s - 2 * q,
17: 2 * (r - q),
18: 2 * r - 2 * s,
19: 3 * q - 2 * s,
20: 2 * (q - s) + r,
21: 3 * q - 2 * r,
22: 2 * (q - r) + s,
23: 2 * (s - r) + s,
24: 2 * (s - r) + q,
25: 3 * s - 2 * q,
26: 2 * (s - q) + r,
27: 3 * r - 2 * q,
28: 2 * r - 2 * q + s,
29: 3 * r - 2 * s,
30: 2 * (r - s) + q,
31: 2 * q - 2 * r + 2 * s,
32: 2 * (s - q + r),
33: 2 * (r - s + q),
34: 2 * q - 3 * r + 2 * s,
35: 2 * (s - q + r) - q,
36: 2 * (r - s + q) - s,
}
pos = rx.spring_layout(graph, pos=pos, fixed={*range(graph.num_nodes())})
rxv.mpl_draw(graph, with_labels=True, pos=pos, node_color="orange", node_size=400)
Root node#
The root node is set as the node for which its longest path is the shortest of all the nodes’ longest paths.
distances = rx.distance_matrix(graph)
plt.bar([*range(distances.shape[1])], np.max(distances, axis=1))
plt.title("Longest Paths")
plt.xlabel("Start Node")
plt.ylabel("Path Length")
plt.show()
So for this graph, node 0 has the shortest path length and is therefore the root of the tree.
To implement the algorithm, we’ll build the graph out of Ternary Tree Nodes from ferrmion. By doing this, making an encoding from the graph is as simple as passing the root node to a ferrmion.TernaryTree.
from ferrmion.encode.ternary_tree_node import TTNode
from enum import Enum
import numpy as np
# First we find the root
root_index = np.argmin(np.max(distances, axis=1))
node_queue = [root_index]
used_indices = {root_index}
# Set values for the root node.
nodes = [TTNode(parent=None) for _ in range(graph.num_nodes())]
nodes[root_index].root_path = ""
nodes[root_index].qubit_label = root_index
Labelling Scheme#
At this point it’s useful to define a preference for which branch we want to assign new nodes to.
In the original paper there are two methods, which assign each of X, Y and Z in order of prefernce:
Homogeneous: [X, Y, Z]
Heterogenous: [Z, X, Y]
class LabellingTypes(Enum):
HOM = "homogenous"
HET = "heterogenous"
labelling = LabellingTypes("homogenous")
Building the Tree#
To build the tree, we follow a bredth-first search, adding child nodes to each parent in the order defined by the Labelling type we have used.
while len(node_queue) > 0:
node = node_queue.pop(0)
parent = nodes[node]
match labelling:
case LabellingTypes.HOM:
chars = ["x", "y", "z"]
case LabellingTypes.HET:
chars = ["z", "x", "y"]
neighbors = list(set(graph.neighbors(node)).difference(used_indices))
for child in neighbors[:3]:
node_queue.append(child)
used_indices.add(child)
n_neighbors = len(neighbors)
for neighbor, char in zip(neighbors[:3], chars[:n_neighbors]):
parent.add_child(
char,
child_node=nodes[neighbor],
root_path=f"{parent.root_path}{char}",
qubit_label=neighbor,
)
nodes[neighbor].parent = parent
Orphaned Nodes#
At this point we need to check if the tree covers all the nodes of our original graph (i.e. the tree spans the graph).
If not, then we attach orphaned nodes to the node closest to it which has space for another child.
if len(used_indices) == graph.num_nodes():
print("Tree spans the graph!")
else:
print("Tree does not span the graph!")
unused_indices = set(range(graph.num_nodes())).difference(used_indices)
for unused in unused_indices:
closest = np.argsort(distances[unused])
for used in closest:
used_node: TTNode = nodes[used]
for child_branch in chars:
if getattr(used_node, child_branch) is None:
used_node.add_child(
child_branch,
child_node=nodes[unused],
root_path=f"{used_node.root_path}{child_branch}",
qubit_label=unused,
)
unused_indices.remove(used)
break
if len(unused_indices) > 0:
print("Error, not all qubits assigned to nodes.")
Tree spans the graph!
Creating an Encoding#
Now we can create a TernaryTree encoding from the root node.
bonsai = fr.TernaryTree(graph.num_nodes(), root_node=nodes[root_index])
bonsai.enumeration_scheme = bonsai.default_enumeration_scheme()
Let’s see what this encoding looks like
from ferrmion.visualise import draw_tt
draw_tt(bonsai, enumeration_scheme=bonsai.enumeration_scheme)
Inbuilt Function#
There is an in-built function in ferrmion, we only need to pass the initial graph.
from ferrmion.optimize import bonsai_algorithm
bonsai = bonsai_algorithm(graph=graph, homogenous=True)
draw_tt(bonsai, enumeration_scheme=bonsai.enumeration_scheme)
or the heterogenous labelling, and optionally a maximum number of nodes
bonsai = bonsai_algorithm(graph=graph, homogenous=False, max_nodes=14)
draw_tt(bonsai)
Encoding a Hamiltonian#
We can encode a hamiltonian in the usual way, using the TernaryTree encoding we just created.
Note that the number of qubits in the encoding is equal to the number of qubits in the original graph, even when you input a value for max_nodes.
import json
from pathlib import Path
folder = Path.cwd().joinpath(Path("../../../python/tests/"))
with open(folder.joinpath("./data/h2o_sto-3g.json"), "rb") as file:
data = json.load(file)
ones = np.array(data["ones"])
twos = np.array(data["twos"])
fham = fr.molecular_hamiltonian(ones, twos, 0.0)
bonsai.encode(fham)
{'ZIYYYIIZZIIIIXIIIIIIIIIIIIIIIIIIIIIII': (1.640297548266427e-08+0j),
'ZIZZIXYIZIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-3.686346771541274e-08+0j),
'ZIZXIYIIIIIIIZIIIIIIIIIIIIIIIIIIIIIII': (1.919499197929779e-07+0j),
'ZIYXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIII': (-1.1999918999715813e-08+0j),
'IYIIYIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-3.1898032780365845e-08+0j),
'XZIZXIXXZIIIZYIIIIIIIIIIIIIIIIIIIIIII': (-9.887471522063672e-08+0j),
'XXIZIIXIIXIIYIIIIIIIIIIIIIIIIIIIIIIII': (0.008126858431383244+0j),
'XXXZIXYIIIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-3.0252015407492863e-09+0j),
'ZIZZIZZIIZIYIIIIIIIIIIIIIIIIIIIIIIIII': (-0.07170028934810821+0j),
'YXYIZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0033451847586839474+0j),
'IIIXXIZXIIIXIIIIIIIIIIIIIIIIIIIIIIIII': (9.982206229658626e-09+0j),
'YZZIYZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.04260474193554964+0j),
'XZIXIIIZYIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-5.991892318599132e-08+0j),
'IYXIYXIIIIZIIIIIIIIIIIIIIIIIIIIIIIIII': (-3.0252015407492863e-09+0j),
'IIIIIIIZIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.14977690192751728+0j),
'ZIZXIYIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (4.8146866424820886e-09+0j),
'XIIZXIZIZIIZIIIIIIIIIIIIIIIIIIIIIIIII': (-0.003950528800448665+0j),
'ZIZXYXIYXIYIIIIIIIIIIIIIIIIIIIIIIIIII': (7.728518362924502e-08+0j),
'IYIIXIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIII': -2.7755575615628914e-16j,
'XZIZZZZZIIXXIYIIIIIIIIIIIIIIIIIIIIIII': (-0.011304778122555207+0j),
'YXZIZXIIYIZIIIIIIIIIIIIIIIIIIIIIIIIII': (3.3006513049228165e-08+0j),
'XYIZYIZYXIIXIIIIIIIIIIIIIIIIIIIIIIIII': (-2.7710446468664605e-08+0j),
'IIXIIXIIIZXIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.001276799856755393+0j),
'ZYYZZIYZIIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-0.007600926421688087+0j),
'ZIZYIXIIZIZIIIIIIIIIIIIIIIIIIIIIIIIII': (1.697359468547683e-07+0j),
'XZIZZIYZIIIIZXIIIIIIIIIIIIIIIIIIIIIII': (-0.041748033270122595+0j),
'YZZYZYYZIIIIIXIIIIIIIIIIIIIIIIIIIIIII': (6.545167376159874e-08+0j),
'YZZIZXIYIIZIZIIIIIIIIIIIIIIIIIIIIIIII': (-0.04767397029989949+0j),
'XZIZXIZIZIIYIZIIIIIIIIIIIIIIIIIIIIIII': (-0.010563285704591748+0j),
'ZIZZIIXIIXXIYIIIIIIIIIIIIIIIIIIIIIIII': (1.0612387392739576e-08+0j),
'ZZZZIZZIIZIYIIIIIIIIIIIIIIIIIIIIIIIII': (-0.00020318282248258035+0j),
'XZIXZXIYIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.0022519417697686554+0j),
'XIIZZIZXIIIYIIIIIIIIIIIIIIIIIIIIIIIII': (2.4121078272479885e-08+0j),
'IXIIXXIIXZIIIIIIIIIIIIIIIIIIIIIIIIIII': (9.821165340814609e-09+0j),
'XZIZZIZZIIYYIYIIIIIIIIIIIIIIIIIIIIIII': (-0.011304778122555207+0j),
'YZZIZYZXIIIIZIIIIIIIIIIIIIIIIIIIIIIII': (0.022536585911798693+0j),
'YZYYZIXYIIIIZIIIIIIIIIIIIIIIIIIIIIIII': (0.0034860696019765654+0j),
'IIIIIZIIIIZYIIIIIIIIIIIIIIIIIIIIIIIII': (0.0007153171629876367+0j),
'XZIZXXXIXXIIYIIIIIIIIIIIIIIIIIIIIIIII': (2.9037494294204896e-08+0j),
'ZIZZXXXXIIYIZIIIIIIIIIIIIIIIIIIIIIIII': (0.00024874569931095396+0j),
'XZIZZIIXIIIZZIIIIIIIIIIIIIIIIIIIIIIII': (-5.88758398825461e-09+0j),
'YZXIZIIZIIIIZYIIIIIIIIIIIIIIIIIIIIIII': (-2.2756168494170135e-08+0j),
'XYIZIZXIIIZIZIIIIIIIIIIIIIIIIIIIIIIII': (1.720435219498683e-08+0j),
'IIIXIYZIIZZYIIIIIIIIIIIIIIIIIIIIIIIII': (2.493105283292244e-08+0j),
'YZZIZYIXIIZIIIIIIIIIIIIIIIIIIIIIIIIII': (0.03962027835876848+0j),
'ZIZIIXXIIIYIZIIIIIIIIIIIIIIIIIIIIIIII': (2.5857859202230354e-08+0j),
'ZYZZZXYZIIXIIYIIIIIIIIIIIIIIIIIIIIIII': (-0.01075440524521853+0j),
'IIIXYIZYZIIXIIIIIIIIIIIIIIIIIIIIIIIII': (-7.384054213595088e-09+0j),
'YXZIXXIXIIYIIIIIIIIIIIIIIIIIIIIIIIIII': (-5.3662020597575315e-09+0j),
'ZIZZZZZIZZIZIIIIIIIIIIIIIIIIIIIIIIIII': (0.24174347916141453+0j),
'ZIYYIYIIIZXIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0056398152645194535+0j),
'XZZZZIZYIIIXIIIIIIIIIIIIIIIIIIIIIIIII': (3.608549427980128e-08+0j),
'ZYZZXXZIZIYZIIIIIIIIIIIIIIIIIIIIIIIII': (0.00470053558938752+0j),
'IIIIIIIIIIIZIZIIIIIIIIIIIIIIIIIIIIIII': (0.14133466753116217+0j),
'YZZIXZYIYXIYXIIIIIIIIIIIIIIIIIIIIIIII': (-0.009559977200109288+0j),
'ZIXXIIXIIIIYIIIIIIIIIIIIIIIIIIIIIIIII': (-9.821165340814609e-09+0j),
'XZXZYXXIIIZIZIIIIIIIIIIIIIIIIIIIIIIII': (-0.0008471127705963529+0j),
'YZXIXIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIII': -1.6653345369377348e-16j,
'IIXIIYIYIIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-6.758033844461568e-09+0j),
'XZIZZIXZIXIIXYIIIIIIIIIIIIIIIIIIIIIII': (1.6428420971060635e-08+0j),
'IIIIIYXIIZXZIIIIIIIIIIIIIIIIIIIIIIIII': (-7.347201655634372e-08+0j),
'ZIZXIZIIIYIIXIIIIIIIIIIIIIIIIIIIIIIII': (1.7485180976190498e-08+0j),
'ZIZXXXIZIIYIIXIIIIIIIIIIIIIIIIIIIIIII': (-5.3662020597575315e-09+0j),
'ZYYZXIZIZIIYIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0005906897714898403+0j),
'IZIYIIZIIIIZIIIIIIIIIIIIIIIIIIIIIIIII': (0.0032825324569077186+0j),
'ZXZZZZXYIXIIXIIIIIIIIIIIIIIIIIIIIIIII': (-6.26362927343161e-08+0j),
'XZXZZXZXIIXZIIIIIIIIIIIIIIIIIIIIIIIII': (3.71881326193215e-09+0j),
'IIXXIXZIIIXXIIIIIIIIIIIIIIIIIIIIIIIII': (0.004664385121536082+0j),
'ZIZZYXZZZIYZIXIIIIIIIIIIIIIIIIIIIIIII': (4.097051117877952e-09+0j),
'IYIIZIIYIIXIIIIIIIIIIIIIIIIIIIIIIIIII': (0.01920048641000208+0j),
'YZZYYXXIIIYIZIIIIIIIIIIIIIIIIIIIIIIII': (3.1898032780365845e-08+0j),
'IIIIYIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIII': (1.4168683990934354e-07+0j),
'YYZYIXZIIIXZIIIIIIIIIIIIIIIIIIIIIIIII': (0.0022886102997761895+0j),
'IIIYIIZIIIIYIIIIIIIIIIIIIIIIIIIIIIIII': (-0.28158077115505264+0j),
'XYYYIZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.014588413013127716+0j),
'XZIZZIYZZIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-0.02571022834443164+0j),
'ZXZZZYXXIIIIZIIIIIIIIIIIIIIIIIIIIIIII': (-6.545167376159874e-08+0j),
'ZXXYZIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIII': (6.5698468974556556e-09+0j),
'XZYZZZZZIZIZIYIIIIIIIIIIIIIIIIIIIIIII': (-2.4974529554956165e-07+0j),
'YZZXXZXIYZIIZIIIIIIIIIIIIIIIIIIIIIIII': (7.349942199732871e-09+0j),
'ZIZZYYZZXIIXIYIIIIIIIIIIIIIIIIIIIIIII': (-0.01609883405788545+0j),
'IIIIZYIIYZXIIIIIIIIIIIIIIIIIIIIIIIIII': (0.0005352864175997575+0j),
'YZXIYIXIIIIXIIIIIIIIIIIIIIIIIIIIIIIII': (8.885015986546761e-09+0j),
'IIIIIIIIIXIIXIIIIIIIIIIIIIIIIIIIIIIII': (5.2096403297715135e-09+0j),
'XZIYZIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIII': -1.7928828757176555e-16j,
'XYIZIIXIIIIIZIIIIIIIIIIIIIIIIIIIIIIII': (8.533051967376144e-08+0j),
'YZZIZYIZXIIIIXIIIIIIIIIIIIIIIIIIIIIII': (0.012224502972950158+0j),
'YIZIXXIIYIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.01901799263023132+0j),
'YZYIZIIZZIIIIXIIIIIIIIIIIIIIIIIIIIIII': (1.505432923495098e-08+0j),
'XZIYXIIYXIIIIYIIIIIIIIIIIIIIIIIIIIIII': (0.0024102072388789866+0j),
'IIYIIZYIIXIZYIIIIIIIIIIIIIIIIIIIIIIII': (-0.006509359475489988+0j),
'IIIYIZYIIIXIIIIIIIIIIIIIIIIIIIIIIIIII': (0.00516966115931119+0j),
'ZYYYXIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.0022886102997761895+0j),
'YZYIXIIIXIIZIIIIIIIIIIIIIIIIIIIIIIIII': (0.0021316548314588226+0j),
'IIZIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.27883453983573275+0j),
'IIIIXIIXIZIIIIIIIIIIIIIIIIIIIIIIIIIII': (7.812829627745813e-08+0j),
'IIYIIZXIIZIYIIIIIIIIIIIIIIIIIIIIIIIII': (1.7153035605304745e-08+0j),
'IIIIIXIIIZIIIZIIIIIIIIIIIIIIIIIIIIIII': (2.028476402288274e-08+0j),
'ZIXYIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.005108425151220365+0j),
'XZIZZYXYIXXIYIIIIIIIIIIIIIIIIIIIIIIII': (7.268409746192926e-09+0j),
'IIIIIZXIIIXXIIIIIIIIIIIIIIIIIIIIIIIII': (-0.007326677936559049+0j),
'XZIYXIIIZIZIIIIIIIIIIIIIIIIIIIIIIIIII': (0.026007683665143607+0j),
'XZZZXIXIYIIIZIIIIIIIIIIIIIIIIIIIIIIII': (3.336460876894293e-07+0j),
'IIIXIIXIXIIIZIIIIIIIIIIIIIIIIIIIIIIII': (6.830796729644614e-08+0j),
'ZIYZIIZIYIIYIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0001247385335499872+0j),
'YXIIIXIIIIYIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.02707115378387988+0j),
'XZIZXIZIZIZZIIIIIIIIIIIIIIIIIIIIIIIII': (0.0027832027120741603+0j),
'ZIZZIXZYIIYZIXIIIIIIIIIIIIIIIIIIIIIII': (0.0032643069539711215+0j),
'YXZIYYIXYIIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.019038338034766407+0j),
'YZXYZIYYIIIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0034860696019765654+0j),
'ZYZZZZXXIYIIYIIIIIIIIIIIIIIIIIIIIIIII': (6.26362927343161e-08+0j),
'XZIXZIYZIIIYZYIIIIIIIIIIIIIIIIIIIIIII': (0.019038338034766407+0j),
'ZIZZYZYYXZIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.0009533430025949679+0j),
'XZIXYIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIII': (-0.003534651529570849+0j),
'XZIXXIIXXIIIIYIIIIIIIIIIIIIIIIIIIIIII': (0.0024102072388789866+0j),
'XZIZZYZZIZYZIXIIIIIIIIIIIIIIIIIIIIIII': (-1.4026196029089865e-07+0j),
'ZIZZYXZZZIXYIYIIIIIIIIIIIIIIIIIIIIIII': (2.268847683767937e-08+0j),
'IXIZXIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.03557775846404743+0j),
'XZIZZYYYIZXIIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0012155829277210994+0j),
'ZYYZXIZIZIIZIIIIIIIIIIIIIIIIIIIIIIIII': (-0.0056398152645194535+0j),
'XZIZZIZXIIIYIZIIIIIIIIIIIIIIIIIIIIIII': (1.7307763482067758e-07+0j),
'YZYIZIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-1.9517876574944395e-08+0j),
'IYIIIIIZZIIIIYIIIIIIIIIIIIIIIIIIIIIII': (1.1270484178015994e-07+0j),
'ZIZZYXYZXIXIIYIIIIIIIIIIIIIIIIIIIIIII': (-0.0009197248503949491+0j),
'IIIIYIYYYIIXZIIIIIIIIIIIIIIIIIIIIIIII': (0.01609883405788545+0j),
'ZIYIIIXIIIIIZIIIIIIIIIIIIIIIIIIIIIIII': (7.924300902775944e-09+0j),
'YZZYYXYIIIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-3.1898032780365845e-08+0j),
'XZIZXXYIXZIIIIIIIIIIIIIIIIIIIIIIIIIII': (0.0007820302480193845+0j),
'IIIIZIIIXIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-2.6857132619785154e-08+0j),
'ZYZXZYIZIIIIIYIIIIIIIIIIIIIIIIIIIIIII': (0.010343698462014462+0j),
'XZIXZZIZIIXIIYIIIIIIIIIIIIIIIIIIIIIII': (0.01926468575777221+0j),
'IIIXIIYYIIIIIXIIIIIIIIIIIIIIIIIIIIIII': (-1.148275881106232e-07+0j),
'IIXIYXIYZIXIIIIIIIIIIIIIIIIIIIIIIIIII': (-4.097051117877952e-09+0j),
'XXIZYIZXYIIZIIIIIIIIIIIIIIIIIIIIIIIII': (1.159901719433355e-08+0j),
'ZIZZIXZIYIYYIIIIIIIIIIIIIIIIIIIIIIIII': (0.017779208140147496+0j),
'IYIIXIIIYIIZIIIIIIIIIIIIIIIIIIIIIIIII': (-0.02421430214238626+0j),
'YZZIZZZYIXIIXIIIIIIIIIIIIIIIIIIIIIIII': (-0.005925937029776886+0j),
'XZIZXIZYXIIZIYIIIIIIIIIIIIIIIIIIIIIII': (-0.002993118809594667+0j),
'ZYZZXYZIYIIXIIIIIIIIIIIIIIIIIIIIIIIII': (1.0091648988092029e-07+0j),
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