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Grey_Box_generate_combs_revised.py
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Grey_Box_generate_combs_revised.py
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"""
This script generates different quantum combs related to Grey Box MBQC for the
graph given by
G: V = {1,2,3,4}, E = {(1,2), (1,3), (1,4), (2,4), (3,4)})
I = {1}, O = {3,4}
There are 15 gflows and the corresponding combs sigma1, ..., sigma15 are
computed. Furthermore, a variety of D_MBQC comb are generated:
D_{MBQC}^{g} = sum_{g} bigot_{1}^{m} sigma_{g}) ot P(g)|g><g|
D_{MBQC}^{mp} = sum_{mp} (sum_{g~mp} 1/5 bigot_{1}^{m} sigma_{g})
ot P(mp)|mp><mp|
D_{MBQC}^{two_mp} = sum_{mp=XY,XZ} (sum_{g~mp} 1/5 bigot_{1}^{m} sigma_{g})
ot P(mp)|mp><mp|
D_{MBQC}^{XY-gflows} = sum_{g~XY} bigot_{1}^{m} sigma_{g}) ot P(g)|g><g|
D_{MBQC}^{XY,1<2} = sum_{g=1,2,4,5} bigot_{1}^{m} sigma_{g}) ot P(g)|g><g|
D_{MBQC}^{XY,1<2,theta_1,theta_2} = sum_{g=1,2,4,5} bigot_{1}^{m} sigma_{g})
ot P(g)tilde{rho_{G}}
where tilde{rho_{G}} is a graph state acted upon by amplitude damping channels.
NOTE: the ordering above, i.e. with the state space corresponding to the random
variable last, is in order to be compatible with the function that computes
the guessing probability - see guessing_probability.py
This script primarily interacts with comb_constraints.py.
This work was conducted within the Quantum Information and Computation group at
the University of Innsbruck.
Contributors: Isaac D. Smith, Marius Krumm
This work is licensed under a Creative Commons by 4.0 license.
"""
import numpy as np
import cvxpy as cvx
from cvxpy.expressions import cvxtypes
from comb_constraints import qcomb_constraints, elementary_matrix
__all__ = ["grey_box_MBQC_meas_planes",
"grey_box_MBQC_two_meas_planes",
"grey_box_MBQC_gflows",
"grey_box_MBQC_XY_gflows",
"grey_box_MBQC_XY_1_2_partial_order",
"grey_box_MBQC_gflows_XY_1_2_noisy",
"observational_meas_small"]
# # # # # # # # # # # # # # # # Global Arguments # # # # # # # # # # # # # # # #
#DTYPE = np.cfloat
DTYPE = np.float32
# # # # # # # # # # # # # # # # Helpful Matrices # # # # # # # # # # # # # # # #
X = np.matrix([[0,1], [1,0]])
Z = np.matrix([[1,0], [0,-1]])
id_2x2 = np.identity(2)
id_4x4 = np.identity(4)
id_8x8 = np.identity(8)
id_16x16 = np.identity(16)
SWAP13 = np.matrix(
[[1,0,0,0,0,0,0,0],
[0,0,0,0,1,0,0,0],
[0,0,1,0,0,0,0,0],
[0,0,0,0,0,0,1,0],
[0,1,0,0,0,0,0,0],
[0,0,0,0,0,1,0,0],
[0,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,1]])
SWAP13_I_I_I = np.kron(SWAP13, id_8x8)
def amplitude_damping(theta=0,Kraus_op=0):
if Kraus_op == 0:
op = np.matrix([[1,0],[0,np.sqrt(1-np.sin(theta)**2)]])
elif Kraus_op == 1:
op = np.matrix([[0,np.sqrt(np.sin(theta)**2)],[0,0]])
return op
# # # # # # # # # # # # Generating the sigma_{MBQC}^g # # # # # # # # # # # # #
def generate_sigmas(dtype=DTYPE):
""" Generates matrices sigma_{MBQC}^{g} for gflows g for the given graph
G = ({1, 2, 3, 4}, {(1,2), (1,3), (1,4), (2,4), (3,4)}) with I = {1} and
O = {3,4}.
Returns:
List of 15 np.arrays, i.e. [sigma_g1, ..., sigma_g15]
"""
# Gflows for (XY, XY):
sigma_g1 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g2 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g3 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g4 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g5 = np.zeros((2**6, 2**6), dtype=dtype)
# Gflows for (XY, XZ):
sigma_g6 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g7 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g8 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g9 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g10 = np.zeros((2**6, 2**6), dtype=dtype)
# Gflows for (XY, YZ):
sigma_g11 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g12 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g13 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g14 = np.zeros((2**6, 2**6), dtype=dtype)
sigma_g15 = np.zeros((2**6, 2**6), dtype=dtype)
for i in range(2**10):
binary = format(i, 'b')
if len(binary) < 10:
binary = (10 - len(binary))*'0' + binary
a1 = int(binary[0])
a2 = int(binary[2])
a3 = int(binary[4])
a4 = int(binary[6])
b1 = int(binary[1])
b2 = int(binary[3])
b3 = int(binary[5])
b4 = int(binary[7])
c1 = int(binary[8])
c2 = int(binary[9])
# We generate the operators sigma by applying correction operators
# conditioned on c1 and c2 on computational basis states of the form
#
# |a1><b1| ot |c1><c1| ot |a2><b2| ot |c2><c2| ot |a3><b3| ot |a4><b4|
#
# Consequently the correction operators will be of the form
#
# I ot I_{c1} ot D'^{c1} ot I_{c2} ot D''^{c1,c2} ot D'''^{c1,c2}
#
# where the first identity is on the state space of |a1><b1| since it
# can receive no corrections, the operators on the state spaces
# associated with c1 and c2 recieve no corrections since they are input
# state spaces, and the operators D', D'', and D''' are products of X
# and Z operators with exponents given by some combination of c1 and c2
# determined by the specific gflow.
comp_ket = [a1, c1, a2, c2, a3, a4]
comp_bra = [b1, c1, b2, c2, b3, b4]
pre_corr_state = elementary_matrix(2, comp_ket, comp_bra)
# The following swapped state is required for three gflows (g3, g8, g12)
# which follow a different partial order.
swapped_pre_corr_state = np.matmul(np.matmul(SWAP13_I_I_I,
pre_corr_state),
SWAP13_I_I_I)
graph_state_factor = (1/16)*(-1)**(a1*a2+a1*a3+a1*a4+a2*a4+a3*a4
+b1*b2+b1*b3+b1*b4+b2*b4+b3*b4)
X_c1 = np.linalg.matrix_power(X, c1)
X_c2 = np.linalg.matrix_power(X, c2)
X_c1c2 = np.linalg.matrix_power(X, (c1 + c2) % 2)
Z_c1 = np.linalg.matrix_power(Z, c1)
Z_c2 = np.linalg.matrix_power(Z, c2)
Z_c1c2 = np.linalg.matrix_power(Z, (c1 + c2) % 2)
# g1: 1 mapsto {2}, 2 mapsto {3,4}
# Corrections operator:
# I ot I_c1 ot (X^c1) ot I_c2 ot (X^c2 Z^c2) ot (X^c2 Z^c1+c2)
corr_g1 = np.kron(np.kron(np.kron(np.kron(id_4x4,
X_c1),
id_2x2),
np.matmul(X_c2, Z_c2)),
np.matmul(X_c2, Z_c1c2))
corr_g1_trans = np.transpose(corr_g1)
# g2: 1 mapsto {3}, 2 mapsto {3,4}:
# Corrections op:
# I ot I_c1 ot I ot I_c2 ot (X^c1+c2 Z^c2) ot (X^c2 Z^c1+c2)
corr_g2 = np.kron(np.kron(id_16x16,
np.matmul(X_c1c2, Z_c2)),
np.matmul(X_c2, Z_c1c2))
corr_g2_trans = np.transpose(corr_g2)
# g3: 1 mapsto {3}, 2 mapsto {4} - NOTE: 2 < 1 in this case, which means
# that the correction operator is relative to the swapped computational
# basis state (i.e. the ordering of the tensor factors is 2 ot 1 ot 3 ot
# 4) and the labels c1 and c2 denote tensor factor position rather than
# label (i.e. c1 is for the result of measuring qubit 2 which is
# position 1).
# Corrections op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (Z^c1 X^c2) ot (X^c1 Z^c2)
corr_g3 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
np.matmul(Z_c1, X_c2)),
np.matmul(X_c1, Z_c2))
corr_g3_trans = np.transpose(corr_g3)
# g4: 1 mapsto {4}, 2 mapsto {3,4}
# Corrections op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (X^c2 Z^c1+c2) ot (X^c1+c2 Z^c2)
corr_g4 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
np.matmul(X_c2, Z_c1c2)),
np.matmul(X_c1c2, Z_c2))
corr_g4_trans = np.transpose(corr_g4)
# g5: 1 mapsto {2,3,4}, 2 mapsto {3,4}
# Corrections op:
# I ot I_c1 ot (X^c1 Z^c1) ot I_c2 ot (X^c1+c2 Z^c1+c2) ot(X^c1+c2 Z^c2)
corr_g5 = np.kron(np.kron(np.kron(np.kron(id_4x4,
np.matmul(X_c1, Z_c1)),
id_2x2),
np.matmul(X_c1c2, Z_c1c2)),
np.matmul(X_c1c2, Z_c2))
corr_g5_trans = np.transpose(corr_g5)
# g6: 1 mapsto {2}, 2 mapsto {2,4}
# Corrections op:
# I ot I_c1 ot (X^c1) ot I_c2 ot (Z^c2) ot (X^c2, Z^c1+c2)
corr_g6 = np.kron(np.kron(np.kron(np.kron(id_4x4,
X_c1),
id_2x2),
Z_c2),
np.matmul(X_c2, Z_c1c2))
corr_g6_trans = np.transpose(corr_g6)
# g7: 1 mapsto {3}, 2 mapsto {2,4}
# Corrections op:
# I ot I_c1 ot I ot I_c2 ot (X^c1 Z^c2) ot (X^c2 Z^c1+c2)
corr_g7 = np.kron(np.kron(id_16x16,
np.matmul(X_c1, Z_c2)),
np.matmul(X_c2, Z_c1c2))
corr_g7_trans = np.transpose(corr_g7)
# g8: 1 mapsto {3}, 2 mapsto {2,3,4} NOTE: 2 < 1 in this case, so the
# same convention regarding the swapped state and notation applies as in
# the case for g3 (see above).
# Corrections op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (X^c1+c2 Z^c1) ot (X^c1 Z^c2)
corr_g8 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
np.matmul(X_c1c2, Z_c1)),
np.matmul(X_c1, Z_c2))
corr_g8_trans = np.transpose(corr_g8)
# g9: 1 mapsto {4}, 2 mapsto {2,4}
# Corrections op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (Z^c1+c2) ot (X^c1+c2 Z^c2)
corr_g9 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
Z_c1c2),
np.matmul(X_c1c2, Z_c2))
corr_g9_trans = np.transpose(corr_g9)
# g10: 1 mapsto {2,3,4}, 2 mapsto {2,4}
# Corrections op:
# I ot I_c1 ot (X^c1 Z^c1) ot I_c2 ot (X^c1 Z^c1+c2) ot (X^c1+c2 Z^c2)
corr_g10 = np.kron(np.kron(np.kron(np.kron(id_4x4,
np.matmul(X_c1, Z_c1)),
id_2x2),
np.matmul(X_c1, Z_c1c2)),
np.matmul(X_c1c2, Z_c2))
corr_g10_trans = np.transpose(corr_g10)
# g11: 1 mapsto {2}, 2 mapsto {2,3}
# Corrections op:
# I ot I_c1 ot (X^c1) ot I_c2 ot (X^c2) ot (Z^c1)
corr_g11 = np.kron(np.kron(np.kron(np.kron(id_4x4,
X_c1),
id_2x2),
X_c2),
Z_c1)
corr_g11_trans = np.transpose(corr_g11)
# g12: 1 mapsto {3}, 2 mapsto {2} NOTE: 2 < 1 in this case, so the same
# convention regarding the swapped state and notation applies as in the
# case for g3 (see above).
# Corrections op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (X^c2) ot (Z^c1+c2)
corr_g12 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
X_c2),
Z_c1c2)
corr_g12_trans = np.transpose(corr_g12)
# g13: 1 mapsto {3}, 2 mapsto {2,3}
# Correction op:
# I ot I_c1 ot I ot I_c2 ot (X^c1+c2) ot (Z^c1)
corr_g13 = np.kron(np.kron(id_16x16,
X_c1c2),
Z_c1)
corr_g13_trans = np.transpose(corr_g13)
# g14: 1 mapsto {4}, 2 mapsto {2,3}
# Correction op:
# I ot I_c1 ot (Z^c1) ot I_c2 ot (X^c2 Z^c1) ot (X^c1)
corr_g14 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
np.matmul(X_c2, Z_c1)),
X_c1)
corr_g14_trans = np.transpose(corr_g14)
# g15: 1 mapsto {2,3,4}, 2 mapsto {2,3}
# Correction op:
# I ot I_c1 ot (X^c1 Z^c1) ot I_c2 ot (X^c1+c2 Z^c1) ot (X^c1)
corr_g15 = np.kron(np.kron(np.kron(np.kron(id_4x4,
np.matmul(X_c1, Z_c1)),
id_2x2),
np.matmul(X_c1c2, Z_c1)),
X_c1)
corr_g15_trans = np.transpose(corr_g15)
# The following applies the correction operators to the current comp
# basis state multiplied by the graph state phase factor for each of the
# sigmas.
sigma_g1 += graph_state_factor*np.matmul(np.matmul(corr_g1,
pre_corr_state),
corr_g1_trans)
sigma_g2 += graph_state_factor*np.matmul(np.matmul(corr_g2,
pre_corr_state),
corr_g2_trans)
sigma_g3 += graph_state_factor*np.matmul(np.matmul(corr_g3,
swapped_pre_corr_state),
corr_g3_trans)
sigma_g4 += graph_state_factor*np.matmul(np.matmul(corr_g4,
pre_corr_state),
corr_g4_trans)
sigma_g5 += graph_state_factor*np.matmul(np.matmul(corr_g5,
pre_corr_state),
corr_g5_trans)
sigma_g6 += graph_state_factor*np.matmul(np.matmul(corr_g6,
pre_corr_state),
corr_g6_trans)
sigma_g7 += graph_state_factor*np.matmul(np.matmul(corr_g7,
pre_corr_state),
corr_g7_trans)
sigma_g8 += graph_state_factor*np.matmul(np.matmul(corr_g8,
swapped_pre_corr_state),
corr_g8_trans)
sigma_g9 += graph_state_factor*np.matmul(np.matmul(corr_g9,
pre_corr_state),
corr_g9_trans)
sigma_g10 += graph_state_factor*np.matmul(np.matmul(corr_g10,
pre_corr_state),
corr_g10_trans)
sigma_g11 += graph_state_factor*np.matmul(np.matmul(corr_g11,
pre_corr_state),
corr_g11_trans)
sigma_g12 += graph_state_factor*np.matmul(np.matmul(corr_g12,
swapped_pre_corr_state),
corr_g12_trans)
sigma_g13 += graph_state_factor*np.matmul(np.matmul(corr_g13,
pre_corr_state),
corr_g13_trans)
sigma_g14 += graph_state_factor*np.matmul(np.matmul(corr_g14,
pre_corr_state),
corr_g14_trans)
sigma_g15 += graph_state_factor*np.matmul(np.matmul(corr_g15,
pre_corr_state),
corr_g15_trans)
return [sigma_g1,sigma_g2,sigma_g3,sigma_g4,sigma_g5,
sigma_g6,sigma_g7,sigma_g8,sigma_g9,sigma_g10,
sigma_g11,sigma_g12,sigma_g13,sigma_g14,sigma_g15]
# # # # # # # # # # # # # # Problem Specific Combs # # # # # # # # # # # # # #
def grey_box_MBQC_meas_planes(rounds=1, dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{mp} (sum_{g~mp} (1/5) bigot_{j=1}^{m} sigma_{MBQC}^{g})
ot P(mp)ket{mp}bra{mp}
where mp stands for measurement plane, m is the number of rounds and
g~mp denotes the gflows associated to the given measurement plane.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself.
Returns:
np.array which is square of size 3*((1*2*2*2*2*4*1)**rounds)
"""
sigmas=generate_sigmas(dtype)
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 3*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=dtype)
P_mp = 1/3 # We assume uniform P(mp)
for mp in range(3):
intermediate_sigma = P_mp*elementary_matrix(3, [mp])
mp_comb = np.zeros((rounds_dim, rounds_dim), dtype=dtype)
for g in range(5):
g_round_comb = (1/5)*np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(sigmas[mp*5 + g], g_round_comb)
mp_comb += g_round_comb
sigma_MBQC += np.kron(mp_comb, intermediate_sigma)
return sigma_MBQC
def grey_box_MBQC_two_meas_planes(rounds=1, dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{mp=Xy,XZ}(sum_{g~mp}(1/5 bigot_{j=1}^{m} sigma_{MBQC}^{g})
ot P(mp)ket{mp}bra{mp}
where mp stands for measurement plane (restricted to just two choices),
m is the number of rounds and g~mp denotes the gflows associated to the
given measurement plane.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself.
Returns:
np.array which is square of size 2*((1*2*2*2*2*4*1)**rounds)
"""
sigmas=generate_sigmas(dtype)
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 2*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=dtype)
P_mp = 1/2 # We assume uniform P(mp)
for mp in range(2):
intermediate_sigma = P_mp*elementary_matrix(2, [mp])
mp_comb = np.zeros((rounds_dim, rounds_dim), dtype=dtype)
for g in range(5):
g_round_comb = (1/5)*np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(sigmas[mp*5 + g], g_round_comb)
mp_comb += g_round_comb
sigma_MBQC += np.kron(mp_comb, intermediate_sigma)
return sigma_MBQC
def grey_box_MBQC_gflows(rounds=1, dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{g} bigot_{j=1}^{m} sigma_{MBQC}^{g}) ot P(g)ket{g}bra{g}
where m is the number of rounds and the sum is over all gflows.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself.
Returns:
np.array which is square of size 15*((1*2*2*2*2*4*1)**rounds)
"""
sigmas=generate_sigmas(dtype)
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 15*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=dtype)
P_g = 1/15 # We assume uniform P(g)
for g in range(15):
intermediate_sigma = P_g*elementary_matrix(15, [g])
g_round_comb = np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(g_round_comb, sigmas[g])
sigma_MBQC += np.kron(g_round_comb, intermediate_sigma)
return sigma_MBQC
def grey_box_MBQC_XY_gflows(rounds=1, dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{g~XY} bigot_{j=1}^{m} sigma_{MBQC}^{g})
ot P(g)ket{g}bra{g}
where m is the number of rounds and the sum is over all XY-plane gflows.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself.
Returns:
np.array which is square of size 5*((1*2*2*2*2*4*1)**rounds)
"""
sigmas = generate_sigmas(dtype)[0:5]
#print("Len of sigmas is {}".format(len(sigmas)))
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 5*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=dtype)
P_g = 1/5 # We assume uniform P(g)
for g in range(5):
intermediate_sigma = P_g*elementary_matrix(5, [g])
g_round_comb = np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(g_round_comb, sigmas[g])
sigma_MBQC += np.kron(g_round_comb, intermediate_sigma)
return sigma_MBQC
def grey_box_MBQC_XY_1_2_partial_order(rounds=1, dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{g = 1,2,4,5} bigot_{j=1}^{m} sigma_{MBQC}^{g})
ot P(g)ket{g}bra{g}
where m is the number of rounds.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself.
Returns:
np.array which is square of size 4*((1*2*2*2*2*4*1)**rounds)
"""
sigmas = generate_sigmas(dtype)[0:5]
sigmas.pop(2)
#print("length of sigmas is {}".format(len(sigmas)))
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 4*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=dtype)
P_g = 1/4 # We assume uniform P(g)
for g in range(4):
intermediate_sigma = P_g*elementary_matrix(4, [g])
g_round_comb = np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(g_round_comb, sigmas[g])
sigma_MBQC += np.kron(g_round_comb, intermediate_sigma)
return sigma_MBQC
def grey_box_MBQC_gflows_XY_1_2_noisy(rounds=1,theta=[0,0],dtype=DTYPE):
""" Computes the operator
D_MBQC = sum_{g = 1,2,4,5} bigot_{j=1}^{m} sigma_{MBQC}^{g})
ot P(g)ket{g}bra{g}
where m is the number of rounds. The sigma_{MBQC}^{g} contains a noisy state
preparation of a graph state acted upon by amplitude damping channels on
qubits 2 and 3 for specified damping parameters.
Inputs:
rounds Int indicating the number of times sigma^{g} is tensored
with itself;
theta List of floats specifying the damping parameter for each
qubit.
Returns:
np.array which is square of size 4*((1*2*2*2*2*4*1)**rounds)
"""
# The following code is copied from the relevant parts of generate_sigmas()
# above:
sigma_g1_rot = np.zeros((2**6, 2**6), dtype=DTYPE)
sigma_g2_rot = np.zeros((2**6, 2**6), dtype=DTYPE)
sigma_g4_rot = np.zeros((2**6, 2**6), dtype=DTYPE)
sigma_g5_rot = np.zeros((2**6, 2**6), dtype=DTYPE)
for i in range(2**10):
binary = format(i, 'b')
if len(binary) < 10:
binary = (10 - len(binary))*'0' + binary
a1 = int(binary[0])
a2 = int(binary[2])
a3 = int(binary[4])
a4 = int(binary[6])
b1 = int(binary[1])
b2 = int(binary[3])
b3 = int(binary[5])
b4 = int(binary[7])
c1 = int(binary[8])
c2 = int(binary[9])
comp_ket = [a1, c1, a2, c2, a3, a4]
comp_bra = [b1, c1, b2, c2, b3, b4]
pre_corr_state = elementary_matrix(2, comp_ket, comp_bra)
graph_state_factor = (1/16)*(-1)**(a1*a2+a1*a3+a1*a4+a2*a4+a3*a4+
b1*b2+b1*b3+b1*b4+b2*b4+b3*b4)
X_c1 = np.linalg.matrix_power(X, c1)
X_c2 = np.linalg.matrix_power(X, c2)
X_c1c2 = np.linalg.matrix_power(X, (c1 + c2) % 2)
Z_c1 = np.linalg.matrix_power(Z, c1)
Z_c2 = np.linalg.matrix_power(Z, c2)
Z_c1c2 = np.linalg.matrix_power(Z, (c1 + c2) % 2)
# The following correction operators for g_1,g_2,g_4,g_5 are the same as
# in generate_sigmas() above:
corr_g1 = np.kron(np.kron(np.kron(np.kron(id_4x4,
X_c1),
id_2x2),
np.matmul(X_c2, Z_c2)),
np.matmul(X_c2, Z_c1c2))
corr_g1_trans = np.transpose(corr_g1)
corr_g2 = np.kron(np.kron(id_16x16,
np.matmul(X_c1c2, Z_c2)),
np.matmul(X_c2, Z_c1c2))
corr_g2_trans = np.transpose(corr_g2)
corr_g4 = np.kron(np.kron(np.kron(np.kron(id_4x4,
Z_c1),
id_2x2),
np.matmul(X_c2, Z_c1c2)),
np.matmul(X_c1c2, Z_c2))
corr_g4_trans = np.transpose(corr_g4)
corr_g5 = np.kron(np.kron(np.kron(np.kron(id_4x4,
np.matmul(X_c1, Z_c1)),
id_2x2),
np.matmul(X_c1c2, Z_c1c2)),
np.matmul(X_c1c2, Z_c2))
corr_g5_trans = np.transpose(corr_g5)
# The following operators are the Kraus operators for indpendent
# amplitude damping channels on qubits 2 and 3 of the graph state:
E_0_0 = np.kron(np.kron(np.kron(id_4x4,amplitude_damping(theta[0],0)),
id_2x2),
np.kron(amplitude_damping(theta[1],0),
id_2x2))
E_0_1 = np.kron(np.kron(np.kron(id_4x4,amplitude_damping(theta[0],0)),
id_2x2),
np.kron(amplitude_damping(theta[1],1),
id_2x2))
E_1_0 = np.kron(np.kron(np.kron(id_4x4,amplitude_damping(theta[0],1)),
id_2x2),
np.kron(amplitude_damping(theta[1],0),
id_2x2))
E_1_1 = np.kron(np.kron(np.kron(id_4x4,amplitude_damping(theta[0],1)),
id_2x2),
np.kron(amplitude_damping(theta[1],1),
id_2x2))
E_0_0_trans = np.transpose(E_0_0)
E_0_1_trans = np.transpose(E_0_1)
E_1_0_trans = np.transpose(E_1_0)
E_1_1_trans = np.transpose(E_1_1)
# Apply the Kraus operators to the current computational basis state:
E_0_0_pre_corr_state = np.matmul(np.matmul(E_0_0,pre_corr_state),
E_0_0_trans)
E_0_1_pre_corr_state = np.matmul(np.matmul(E_0_1,pre_corr_state),
E_0_1_trans)
E_1_0_pre_corr_state = np.matmul(np.matmul(E_1_0,pre_corr_state),
E_1_0_trans)
E_1_1_pre_corr_state = np.matmul(np.matmul(E_1_1,pre_corr_state),
E_1_1_trans)
rotated_pre_corr_state = (E_0_0_pre_corr_state + E_0_1_pre_corr_state
+ E_1_0_pre_corr_state + E_1_1_pre_corr_state)
# The following applies the correction operators to the current comp
# basis state multiplied by the graph state phase factor for each of
# the sigmas:
sigma_g1_rot += graph_state_factor*np.matmul(np.matmul(corr_g1,
rotated_pre_corr_state),
corr_g1_trans)
sigma_g2_rot += graph_state_factor*np.matmul(np.matmul(corr_g2,
rotated_pre_corr_state),
corr_g2_trans)
sigma_g4_rot += graph_state_factor*np.matmul(np.matmul(corr_g4,
rotated_pre_corr_state),
corr_g4_trans)
sigma_g5_rot += graph_state_factor*np.matmul(np.matmul(corr_g5,
rotated_pre_corr_state),
corr_g5_trans)
sigmas_XY_1_2_noisy = [sigma_g1_rot,
sigma_g2_rot,
sigma_g4_rot,
sigma_g5_rot]
rounds_dim = (1*2*2*2*2*4*1)**rounds
sigma_MBQC_dim = 4*rounds_dim
sigma_MBQC = np.zeros((sigma_MBQC_dim, sigma_MBQC_dim),dtype=DTYPE)
P_g = 1/4 # We assume uniform P(g)
for g in range(4):
intermediate_sigma = P_g*elementary_matrix(4, [g])
g_round_comb = np.identity(1)
for _ in range(rounds):
g_round_comb = np.kron(g_round_comb, sigmas_XY_1_2_noisy[g])
sigma_MBQC += np.kron(g_round_comb, intermediate_sigma)
return sigma_MBQC
# # # # # # # # # # # # # # # Auxilliary Functions # # # # # # # # # # # # # # #
def observational_meas_small(theta1, phi1, theta2a, phi2a, theta2b = 0,
phi2b = 0, adaptive = False):
""" Generates a dual comb to the combs D above, representing projective
measurements on the first two qubits. The classical outcomes are ONLY sent
back into the gflow-comb.
If adaptive == True:
A 0-outcome on the first qubit will lead to measurement settings
theta2a, phi2a, on qubit 2;
A 1-outcome will lead to measurement settings theta2b, phi2b.
Else:
The second qubit is always measured with theta2a, phi2a.
Pure states are parametrized as:
cos(theta/2) |0> + e^(i phi) sin(theta/2) | 1 >
For the system order, we make the following convention:
qubit1, outcome1, qubit2, outcome2
Returns the associated (probabilistic) comb for each outcome combination.
"""
# Create the pure states for the 0 outcomes:
ket1 = np.array([[np.cos(theta1/2.0)],
[np.exp(1.0j * phi1 )*np.sin(theta1/2.0)]],
dtype = np.cfloat)
ket2a = np.array([[np.cos(theta2a/2.0)],
[np.exp(1.0j * phi2a )*np.sin(theta2a/2.0)]],
dtype = np.cfloat)
ket2b = np.array([[np.cos(theta2b/2.0)],
[np.exp(1.0j * phi2b )*np.sin(theta2b/2.0)]],
dtype = np.cfloat)
# Write the states as density matrices:
pure1 = ket1 @ (ket1.copy().conj().transpose())
pure2a = ket2a @ (ket2a.copy().conj().transpose())
pure2b = ket2b @ (ket2b.copy().conj().transpose())
# The projectors for outcomes 1:
opposite1 = id_2x2 - pure1
opposite2a = id_2x2 - pure2a
opposite2b = id_2x2 - pure2b
# Apply the partial transpositions:
pure1T = pure1.transpose()
pure2aT = pure2a.transpose()
pure2bT = pure2b.transpose()
opposite1T = opposite1.transpose()
opposite2aT = opposite2a.transpose()
opposite2bT = opposite2b.transpose()
# This will make it easier to access the operators in loops:
qubit1Meas = [pure1T, opposite1T]
qubit2Meas = [pure2aT, opposite2aT , pure2bT, opposite2bT]
# Represent the classical outcomes as operators:
zeroOutcome = np.array([[1,0],[0,0]], dtype = np.cfloat)
oneOutcome = np.array([[0,0],[0,1]], dtype = np.cfloat)
outcomes = [zeroOutcome, oneOutcome]
# Construct the comb, respecting the order convention:
# qubit1, outcome1, qubit2, outcome2
result = np.zeros(shape = (2,2,2**4, 2**4), dtype = np.cfloat)
for out1 in range(2):
for out2 in range(2):
change = np.kron(qubit1Meas[out1] , outcomes[out1])
if adaptive:
change = np.kron(change, qubit2Meas[2*out1+out2])
else:
change = np.kron(change, qubit2Meas[out2])
change = np.kron(change, outcomes[out2])
result[out1,out2 , : , :] = change.copy()
return result.copy()