Sage: Graph Codes

This is an implementation of the algorithm described in “Quantum-error-correcting codes using qudit graph states”, Shiang Yong Looi, Li Yu, Vlad Gheorghiu, and Robert B. Griffiths, Phys. Rev. A 78, 042303 (2008).

>>> from qitensor import * 
>>> import numpy as np 
>>> # use Sage's nice array formatting
>>> set_qitensor_printoptions(str_use_sage=True) 

>>> #graph = graphs.CycleGraph(4)
>>> #graph.allow_multiple_edges(True)
>>> #graph.add_edge((0, 1))

>>> graph = graphs.CubeGraph(3) 
>>> graph.relabel() 

>>> D = 2 
>>> n = graph.order() 
>>> (D, n) 
(2, 8)

>>> # Create the graph state
>>> hilb = [ qudit('H_%d' % i, D, dtype=complex) for i in range(n) ] 
>>> c0 = np.product([h.x_plus() for h in hilb]) 
>>> for (l, m, _label) in graph.edges(): 
...       c0 = cphase(hilb[l], hilb[m]) * c0
...
>>> c0.space 
|H_0,H_1,H_2,H_3,H_4,H_5,H_6,H_7>

>>> # Create the graph basis
>>> import itertools 
>>> nS = D^n 
>>> hS = qudit('S', nS) 
>>> cv_arr = (np.product(hilb) * hS.H).array() 
>>> for (i, a) in enumerate(itertools.product(range(D), repeat=n)): 
...       cv_i = c0
...       for j in range(n):
...           cv_i = hilb[j].pauliZ(a[j]) * cv_i
...       cv_arr[{ hS.H: i }] = cv_i
...
>>> cv_arr.space 
|H_0,H_1,H_2,H_3,H_4,H_5,H_6,H_7><S|

>>> # Create the size-1 Pauli operators
>>> Q_list = [] 
>>> for H in hilb: 
...     for i in range(D):
...         for j in range(D):
...             if i>0 or j>0:
...                 Q_list.append(H.pauliX(i) * H.pauliZ(j))
...
>>> len(Q_list) 
24

>>> # Compute the Pauli distance between graph basis states.
>>> # Because of Eq. 17 this could be made a lot faster.  But for
>>> # simplicity this is not taken advantage of.
>>> # Also, using the X-Z rule should give an exponential speedup.
>>> cv_Q_list = [cv_arr.H * Q for Q in Q_list] 
>>> delta_matrix = np.zeros((nS, nS), dtype=int) 
>>> delta_matrix[:] = 3 
>>> for (i1, Q1) in enumerate(Q_list): 
...     Q1_cv = Q1 * cv_arr
...     for (i2, cv_Q2) in enumerate(cv_Q_list):
...         if i1 != i2:
...             coeffs = cv_Q2 * Q1_cv
...             nonzero = abs(coeffs.nparray) > 1e-14
...             delta_matrix[nonzero] = 2
...
>>> for cv_Q in cv_Q_list: 
...     coeffs = cv_Q * cv_arr
...     nonzero = abs(coeffs.nparray) > 1e-14
...     delta_matrix[nonzero] = 1
...

>>> delta_matrix.shape 
(256, 256)

>>> delta_matrix 
array([[3, 1, 1, ..., 2, 2, 2],
       [1, 3, 2, ..., 2, 2, 2],
       [1, 2, 3, ..., 2, 2, 2],
       ...,
       [2, 2, 2, ..., 3, 2, 1],
       [2, 2, 2, ..., 2, 3, 1],
       [2, 2, 2, ..., 1, 1, 3]])

>>> # Create the S graph for delta=2, and find the maximum
>>> # clique size.
>>> S_matrix = np.where(delta_matrix >= 2, 1, 0) 
>>> S_graph = Graph(S_matrix) 
>>> len(S_graph.clique_maximum()) 
64

>>> # Create the S graph for delta=3, and find the maximum
>>> # clique size.
>>> S_matrix = np.where(delta_matrix >= 3, 1, 0) 
>>> S_graph = Graph(S_matrix) 
>>> len(S_graph.clique_maximum()) 
8