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utils.py
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utils.py
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"""
This file contains all the necessary code for implementing the algorithm
"""
from pathlib import Path
import numpy as np
import scipy as sc
import networkx as nx
# Global path variables
DATA = Path('./data/')
RES = Path('./results/')
"""
General utilities
"""
def read_txt(path):
"""
Reads text file and returns numpy array containing all graphs
Parameters
----------
path : pathlib.Path
Path to data
Returns
-------
numpy.ndarray
"""
lns = path.read_text().splitlines()
first_line = lns[0]
dim = int(first_line[first_line.index("dim=") + 4:first_line.index("dim=") + 6])
# deg = int(first_line[first_line.index("degree=") + 7])
# lmd = int(first_line[first_line.index("lambda=") + 7])
# mu = int(first_line[first_line.index("mu=") + 3])
long_text = "TOTAL NUMBER OF STRONGLY REGULAR GRAPHS = "
NUM = 0
_1D = []
for l in lns:
if l.startswith("0") or l.startswith("1"):
l2 = ','.join(l)
_1D.append(np.fromstring(l2, sep=','))
if long_text in l:
NUM = int(l[len(long_text):])
all_graphs = np.c_[_1D]
return all_graphs.reshape(NUM, dim, dim)
def read_graph6(path):
"""
Reads graph6 file and returns adjacency matrices as numpy arrays
Parameters
----------
path : pathlib.Path
Path to data
Returns
-------
numpy.ndarray
"""
Gs = nx.read_graph6(path)
if not isinstance(Gs, list):
G = nx.to_numpy_array(Gs)
return G.reshape(1, *G.shape)
Gs_np = []
for G in Gs:
Gs_np.append(nx.to_numpy_array(G))
return np.c_[Gs_np]
def read_data(path):
"""
General data-reader
Parameters
----------
path : pathlib.Path
Path to data
Returns
-------
numpy.ndarray
"""
path = Path(path)
if path.suffix not in [".g6", ".txt"]:
print(f"Error: {path.suffix} not supported. Check file path. Only .g6 and .txt files are supported. Exiting...")
return None
if path.suffix == ".g6":
return read_graph6(path)
return read_txt(path)
def draw_graph(adj):
"""
Draws graph using adjacency matrix
Parameters
----------
adj : numpy.ndarray
Adjacency matrix of the graph
"""
G = nx.from_numpy_matrix(adj)
nx.draw(G)
def mat_exp(M, ord=20):
"""
Naive algorithm, using Taylor series to approximate matrix exponentiation
DONTUSE / Prefer scipy.linalg.expm (same results at higher order, but scipy's is faster)
"""
mp = np.linalg.matrix_power
fac = np.math.factorial
sum = 0
for i in range(ord):
sum += mp(M, i) / fac(i)
return sum
def tensor_prod(n=1):
"""
Returns a function that calculates the matrix Tensor product (Kronecker product or \otimes) via recursive function composition
Parameters
----------
n : int
Number of times to take the Kronecker product
Equivalent to ``M^\otimes n``
n == 1 => Identity or no tensor product
For usage, see `GI_quantum_test()` below.
Returns
-------
callable
Function that calculates the Tensor product
n == 1 => Identity or no tensor product
"""
if n == 1:
return lambda x, y: x # Identity
elif n == 2:
return np.kron # M \otimes M
# Recursively calculate M \otimes ... \otimes M | "Normal ordering" (left to right)
return lambda M1, M2: np.kron(M1, tensor_prod(n - 1)(M1, M2))
"""
Classical Graph Isomorphism test
"""
def GI_classical_test(adj_G1, adj_G2):
"""
Tests if two graphs are isomorphic using the classical VF2 algorithm as implemented in NetworkX
Parameters
----------
adj_G1 : numpy.ndarray
Adjacency matrix of Graph 1
adj_G2 : numpy.ndarray
Adjacency matrix of Graph 2
Returns
-------
bool
Boolean result of the test
"""
G1 = nx.from_numpy_matrix(adj_G1)
G2 = nx.from_numpy_matrix(adj_G2)
GM = nx.algorithms.isomorphism.GraphMatcher(G1, G2)
return GM.is_isomorphic() #, GM.mapping
def classical_test(graphs, family):
"""
Tests if all graphs in the given array (SRG family)are isomorphic
using the classical VF2 algorithm as implemented in NetworkX
Saves the results in a csv file
Parameters
----------
graphs : numpy.ndarray
Array of adjacency matrices
family : str
SRG family name
Returns
-------
numpy.ndarray
Contains the indices & result of the test (0 - False / Non-isomorphic | 1 - True / Isomorphic)
"""
pairs = np.transpose(np.triu_indices(graphs.shape[0], k=1))
class_res = np.empty((0, 3), int)
for p in pairs:
class_res = np.append(class_res, [[p[0], p[1], GI_classical_test(graphs[p[0]], graphs[p[1]])]], axis=0)
save_path = RES / f"{family}"
if not save_path.exists():
save_path.mkdir(parents=True, exist_ok=True)
with open(save_path / "classical_test.csv", 'w') as f:
np.savetxt(f, class_res, delimiter=',', fmt='%d')
return class_res
"""
Quantum Graph Isomorphism test using non-interacting CTQW
"""
def GI_quantum_algo(adj_G1, adj_G2, p=3, ptype="bos", time=1.):
"""
Returns list distance (`\Delta`) between graph signatures for two graphs, using the CTQW algorithm as
outlined first in Gamble et al (2010) [1]_ & then in Rudinger et al (2012) [2]_.
Equation numbers are from [2]_.
Parameters
----------
adj_G1 : numpy.ndarray
Adjacency matrix of Graph 1
adj_G2 : numpy.ndarray
Adjacency matrix of Graph 2
p : int
Number of particles in the QW
Defaults to 3
ptype : str
"ferm" for ferminonic walk & "bos" for bosonic walk
Defaults to "bos"
time : float
Time at which U is to be calculated
Defaults to 1 (From Rudinger et al [2]_)
Optional, but small values recommended to avoid numerical issues with matrix exponentiation
Returns
-------
delta : float
Distance between the lists
delta = 0 => Isomorphic & delta != 0 => Non-isomorphic graphs
References
----------
.. [1] Gamble, J., Friesen, M., Zhou, D., Joynt, R., & Coppersmith, S. (2010). Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A, 81, 052313.
.. [2] Rudinger, K., Gamble, J., Wellons, M., Bach, E., Friesen, M., Joynt, R., & Coppersmith, S. (2012). Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs. Phys. Rev. A, 86, 022334.
"""
# Matrix exponentiation (Padé approximation)
ex = sc.linalg.expm
# Settings proper coefficient depending on particle-type
if ptype not in ["bos", "ferm"]:
print("Error: Incorrect particle type. Input 'bos' for bosonic walk and 'ferm' for fermionic walk. Exiting...")
return None
fac = 1 if ptype == "bos" else -1
# 1. Calculate complex evolution matrices for the two graphs
# p = 1 | Single particle evolution matrix (Eq. 8)
U1 = ex(fac * 1j * adj_G1 * time)
U2 = ex(fac * 1j * adj_G2 * time)
# Taking tensor product to get multi-particle evolution matrices (Eq. 10)
U1 = tensor_prod(n=p)(U1, U1)
U2 = tensor_prod(n=p)(U2, U2)
# 2. Take the magnitude of each element
U1 = np.abs(U1)
U2 = np.abs(U2)
# 3. Collect real entries in lists
U1 = U1.flatten()
U2 = U2.flatten()
# 4. Sort the lists | Also see: Wang et al - https://dx.doi.org/10.1088/1751-8113/48/11/115302
U1.sort()
U2.sort()
# 5. Calculate list distance
delta = np.sum(np.abs(U1 - U2))
return delta
def GI_quantum_test(adj_G1, adj_G2, p=3, ptype="bos", time=1., tol=1e-10):
"""
Tests if two graphs are isomorphic using the algorithm in `GI_quantum_algo()` above
Parameters
----------
adj_G1 : numpy.ndarray
Adjacency matrix of Graph 1
adj_G2 : numpy.ndarray
Adjacency matrix of Graph 2
p : int
Number of particles in the QW
Defaults to 3
ptype : str
"ferm" for ferminonic walk & "bos" for bosonic walk
Defaults to "bos"
time : float
Time at which U is to be calculated
Defaults to 1. (From Rudinger et al)
Optional, but small values recommended to avoid numerical issues with matrix exponentiation
tol : float
Tolerance for the comparing list distance with 0
Defaults to 1e-10
Returns
-------
bool
Boolean result of the test
"""
delta = GI_quantum_algo(adj_G1, adj_G2, p=p, ptype=ptype, time=time)
# Checking if delta is 0 (close to 0, closeness specified by `tol` value)
res = np.allclose(delta, 0., rtol=tol, atol=tol)
return res #, delta
def quantum_test(graphs, family, p=3, ptype="bos", time=1., tol=1e-10):
"""
Tests if all graphs in the given array (SRG family) are isomorphic
using the algorithm in `GI_quantum_algo()` above
Saves the results in a csv file
Parameters
----------
graphs : numpy.ndarray
Array of adjacency matrices
family : str
SRG family name
p : int
Number of particles in the QW
Defaults to 3
ptype : str
"ferm" for ferminonic walk & "bos" for bosonic walk
Defaults to "bos"
time : float
Time at which U is to be calculated
Defaults to 1 (From Rudinger et al)
Optional, but small values recommended to avoid numerical issues with matrix exponentiation
tol : float
Tolerance for the comparing list distance with 0
Defaults to 1e-10
Returns
-------
numpy.ndarray
Contains the indices & result of the test (0 - False / Non-isomorphic | 1 - True / Isomorphic)
"""
pairs = np.transpose(np.triu_indices(graphs.shape[0], k=1))
class_res = np.empty((0, 3), int)
for pa in pairs:
class_res = np.append(
class_res,
[[pa[0], pa[1], GI_quantum_test(graphs[pa[0]], graphs[pa[1]], p=p, ptype=ptype, time=time)]],
axis=0
)
save_path = RES / f"{family}"
if not save_path.exists():
save_path.mkdir(parents=True, exist_ok=True)
with open(save_path / f"{p}-{ptype}_quantum_test.csv", 'w') as f:
np.savetxt(f, class_res, delimiter=',', fmt='%d')
return class_res