def entangling_power(U):
"""
Calculate the entangling power of a two-qubit gate U, which
is zero of nonentangling gates and 1 and 2/9 for maximally
entangling gates.
Parameters
----------
U : qobj
Qobj instance representing a two-qubit gate.
Returns
-------
ep : float
The entanglement power of U (real number between 0 and 1)
References:
Explorations in Quantum Computing, Colin P. Williams (Springer, 2011)
"""
if not U.isoper:
raise Exception("U must be an operator.")
if U.dims != [[2, 2], [2, 2]]:
raise Exception("U must be a two-qubit gate.")
a = (tensor(U, U).dag() * swap(N=4, targets=[1, 3]) * tensor(U, U) *
swap(N=4, targets=[1, 3]))
b = (tensor(swap() * U,
swap() * U).dag() * swap(N=4, targets=[1, 3]) *
tensor(swap() * U,
swap() * U) * swap(N=4, targets=[1, 3]))
return 5.0 / 9 - 1.0 / 36 * (a.tr() + b.tr()).real
def entangling_power(U):
"""
Calculate the entangling power of a two-qubit gate U, which
is zero of nonentangling gates and 1 and 2/9 for maximally
entangling gates.
Parameters
----------
U : qobj
Qobj instance representing a two-qubit gate.
Returns
-------
ep : float
The entanglement power of U (real number between 0 and 1)
References:
Explorations in Quantum Computing, Colin P. Williams (Springer, 2011)
"""
if not U.isoper:
raise Exception("U must be an operator.")
if U.dims != [[2, 2], [2, 2]]:
raise Exception("U must be a two-qubit gate.")
a = (tensor(U, U).dag() * swap(N=4, targets=[1, 3]) *
tensor(U, U) * swap(N=4, targets=[1, 3]))
b = (tensor(swap() * U, swap() * U).dag() * swap(N=4, targets=[1, 3]) *
tensor(swap() * U, swap() * U) * swap(N=4, targets=[1, 3]))
return 5.0/9 - 1.0/36 * (a.tr() + b.tr()).real
def test_known_iscptp(self):
"""
Superoperator: iscp, istp and iscptp known cases.
"""
# Check that unitaries are CPTP.
assert_(identity(2).iscptp)
assert_(sigmax().iscptp)
# The partial transpose map, whose Choi matrix is SWAP, is TP but not
# CP.
W = Qobj(swap(), type='super', superrep='choi')
assert_(W.istp)
assert_(not W.iscp)
assert_(not W.iscptp)
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
assert_(subnorm_map.iscp)
assert_(not subnorm_map.istp)
assert_(not subnorm_map.iscptp)
# Check that things which aren't even operators aren't identified as
# CPTP.
assert_(not (basis(2).iscptp))
def qft_steps(N=1, swapping=True):
"""
Quantum Fourier Transform operator on N qubits returning the individual
steps as unitary matrices operating from left to right.
Parameters
----------
N: int
Number of qubits.
swap: boolean
Flag indicating sequence of swap gates to be applied at the end or not.
Returns
-------
U_step_list: list of qobj
List of Hadamard and controlled rotation gates implementing QFT.
"""
if N < 1:
raise ValueError("Minimum value of N can be 1")
U_step_list = []
if N == 1:
U_step_list.append(snot())
else:
for i in range(N):
for j in range(i):
U_step_list.append(
cphase(np.pi / (2**(i - j)), N, control=i, target=j))
U_step_list.append(snot(N, i))
if swapping is True:
for i in range(N // 2):
U_step_list.append(swap(N, [N - i - 1, i]))
return U_step_list
def test_known_iscptp(self):
"""
Superoperator: iscp, istp and iscptp known cases.
"""
# Check that unitaries are CPTP.
assert_(identity(2).iscptp)
assert_(sigmax().iscptp)
# The partial transpose map, whose Choi matrix is SWAP, is TP but not
# CP.
W = Qobj(swap(), type='super', superrep='choi')
assert_(W.istp)
assert_(not W.iscp)
assert_(not W.iscptp)
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
assert_(subnorm_map.iscp)
assert_(not subnorm_map.istp)
assert_(not subnorm_map.iscptp)
# Check that things which aren't even operators aren't identified as
# CPTP.
assert_(not (basis(2).iscptp))
def qft_steps(N=1, swapping=True):
"""
Quantum Fourier Transform operator on N qubits returning the individual
steps as unitary matrices operating from left to right.
Parameters
----------
N: int
Number of qubits.
swap: boolean
Flag indicating sequence of swap gates to be applied at the end or not.
Returns
-------
U_step_list: list of qobj
List of Hadamard and controlled rotation gates implementing QFT.
"""
if N < 1:
raise ValueError("Minimum value of N can be 1")
U_step_list = []
if N == 1:
U_step_list.append(snot())
else:
for i in range(N):
for j in range(i):
U_step_list.append(cphase(np.pi/(2**(i-j)), N,
control=i, target=j))
U_step_list.append(snot(N, i))
if swapping == True:
for i in range(N//2):
U_step_list.append(swap(N, [N-i-1, i]))
return U_step_list
def test_known_iscptp(self):
"""
Superoperator: ishp, iscp, istp and iscptp known cases.
"""
def case(qobj, shouldhp, shouldcp, shouldtp):
hp = qobj.ishp
cp = qobj.iscp
tp = qobj.istp
cptp = qobj.iscptp
shouldcptp = shouldcp and shouldtp
if (
hp == shouldhp and
cp == shouldcp and
tp == shouldtp and
cptp == shouldcptp
):
return
fails = []
if hp != shouldhp:
fails.append(("ishp", shouldhp, hp))
if tp != shouldtp:
fails.append(("istp", shouldtp, tp))
if cp != shouldcp:
fails.append(("iscp", shouldcp, cp))
if cptp != shouldcptp:
fails.append(("iscptp", shouldcptp, cptp))
raise AssertionError("Expected {}.".format(" and ".join([
"{} == {} (got {})".format(fail, expected, got)
for fail, expected, got in fails
])))
# Conjugation by a creation operator should
# have be CP (and hence HP), but not TP.
a = create(2).dag()
S = sprepost(a, a.dag())
yield case, S, True, True, False
# A single off-diagonal element should not be CP,
# nor even HP.
S = sprepost(a, a)
yield case, S, False, False, False
# Check that unitaries are CPTP and HP.
yield case, identity(2), True, True, True
yield case, sigmax(), True, True, True
# The partial transpose map, whose Choi matrix is SWAP, is TP
# and HP but not CP (one negative eigenvalue).
W = Qobj(swap(), type='super', superrep='choi')
yield case, W, True, False, True
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
yield case, subnorm_map, True, True, False
# Check that things which aren't even operators aren't identified as
# CPTP.
yield case, basis(2), False, False, False
def test_known_iscptp(self):
"""
Superoperator: ishp, iscp, istp and iscptp known cases.
"""
def case(qobj, shouldhp, shouldcp, shouldtp):
hp = qobj.ishp
cp = qobj.iscp
tp = qobj.istp
cptp = qobj.iscptp
shouldcptp = shouldcp and shouldtp
if (
hp == shouldhp and
cp == shouldcp and
tp == shouldtp and
cptp == shouldcptp
):
return
fails = []
if hp != shouldhp:
fails.append(("ishp", shouldhp, hp))
if tp != shouldtp:
fails.append(("istp", shouldtp, tp))
if cp != shouldcp:
fails.append(("iscp", shouldcp, cp))
if cptp != shouldcptp:
fails.append(("iscptp", shouldcptp, cptp))
raise AssertionError("Expected {}.".format(" and ".join([
"{} == {} (got {})".format(fail, expected, got)
for fail, expected, got in fails
])))
# Conjugation by a creation operator should
# have be CP (and hence HP), but not TP.
a = create(2).dag()
S = sprepost(a, a.dag())
case(S, True, True, False)
# A single off-diagonal element should not be CP,
# nor even HP.
S = sprepost(a, a)
case(S, False, False, False)
# Check that unitaries are CPTP and HP.
case(identity(2), True, True, True)
case(sigmax(), True, True, True)
# The partial transpose map, whose Choi matrix is SWAP, is TP
# and HP but not CP (one negative eigenvalue).
W = Qobj(swap(), type='super', superrep='choi')
case(W, True, False, True)
# Subnormalized maps (representing erasure channels, for instance)
# can be CP but not TP.
subnorm_map = Qobj(identity(4) * 0.9, type='super', superrep='super')
case(subnorm_map, True, True, False)
# Check that things which aren't even operators aren't identified as
# CPTP.
case(basis(2), False, False, False)
def test_dnorm_qubit_known_cases():
"""
Metrics: check agreement for known qubit channels.
"""
def case(chan1, chan2, expected, significant=4):
# We again take a generous tolerance so that we don't kill off
# SCS solvers.
assert_approx_equal(dnorm(chan1, chan2),
expected,
significant=significant)
id_chan = to_choi(qeye(2))
S_eye = to_super(id_chan)
X_chan = to_choi(sigmax())
depol = to_choi(
Qobj(diag(ones((4, ))), dims=[[[2], [2]], [[2], [2]]], superrep='chi'))
S_H = to_super(hadamard_transform())
W = swap()
# We need to restrict the number of iterations for things on the boundary,
# such as perfectly distinguishable channels.
yield case, id_chan, X_chan, 2
yield case, id_chan, depol, 1.5
# Next, we'll generate some test cases based on comparisons to pre-existing
# dnorm() implementations. In particular, the targets for the following
# test cases were generated using QuantumUtils for MATLAB (https://goo.gl/oWXhO9).
def overrotation(x):
return to_super((1j * np.pi * x * sigmax() / 2).expm())
for x, target in {
1.000000e-03: 3.141591e-03,
3.100000e-03: 9.738899e-03,
1.000000e-02: 3.141463e-02,
3.100000e-02: 9.735089e-02,
1.000000e-01: 3.128689e-01,
3.100000e-01: 9.358596e-01
}.items():
yield case, overrotation(x), id_chan, target
def had_mixture(x):
return (1 - x) * S_eye + x * S_H
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.200000e-03,
1.000000e-02: 2.000000e-02,
3.100000e-02: 6.200000e-02,
1.000000e-01: 2.000000e-01,
3.100000e-01: 6.200000e-01
}.items():
yield case, had_mixture(x), id_chan, target
def swap_map(x):
S = (1j * x * W).expm()
S._type = None
S.dims = [[[2], [2]], [[2], [2]]]
S.superrep = 'super'
return S
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.199997e-03,
1.000000e-02: 1.999992e-02,
3.100000e-02: 6.199752e-02,
1.000000e-01: 1.999162e-01,
3.100000e-01: 6.173918e-01
}.items():
yield case, swap_map(x), id_chan, target
# Finally, we add a known case from Johnston's QETLAB documentation,
# || Phi - I ||,_♢ where Phi(X) = UXU⁺ and U = [[1, 1], [-1, 1]] / sqrt(2).
yield case, Qobj([[1, 1], [-1, 1]]) / np.sqrt(2), qeye(2), np.sqrt(2)
def test_dnorm_qubit_known_cases():
"""
Metrics: check agreement for known qubit channels.
"""
def case(chan1, chan2, expected, significant=4):
# We again take a generous tolerance so that we don't kill off
# SCS solvers.
assert_approx_equal(
dnorm(chan1, chan2), expected,
significant=significant
)
id_chan = to_choi(qeye(2))
S_eye = to_super(id_chan)
X_chan = to_choi(sigmax())
depol = to_choi(Qobj(
diag(ones((4,))),
dims=[[[2], [2]], [[2], [2]]], superrep='chi'
))
S_H = to_super(hadamard_transform())
W = swap()
# We need to restrict the number of iterations for things on the boundary,
# such as perfectly distinguishable channels.
yield case, id_chan, X_chan, 2
yield case, id_chan, depol, 1.5
# Next, we'll generate some test cases based on comparisons to pre-existing
# dnorm() implementations. In particular, the targets for the following
# test cases were generated using QuantumUtils for MATLAB (https://goo.gl/oWXhO9).
def overrotation(x):
return to_super((1j * np.pi * x * sigmax() / 2).expm())
for x, target in {
1.000000e-03: 3.141591e-03,
3.100000e-03: 9.738899e-03,
1.000000e-02: 3.141463e-02,
3.100000e-02: 9.735089e-02,
1.000000e-01: 3.128689e-01,
3.100000e-01: 9.358596e-01
}.items():
yield case, overrotation(x), id_chan, target
def had_mixture(x):
return (1 - x) * S_eye + x * S_H
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.200000e-03,
1.000000e-02: 2.000000e-02,
3.100000e-02: 6.200000e-02,
1.000000e-01: 2.000000e-01,
3.100000e-01: 6.200000e-01
}.items():
yield case, had_mixture(x), id_chan, target
def swap_map(x):
S = (1j * x * W).expm()
S._type = None
S.dims = [[[2], [2]], [[2], [2]]]
S.superrep = 'super'
return S
for x, target in {
1.000000e-03: 2.000000e-03,
3.100000e-03: 6.199997e-03,
1.000000e-02: 1.999992e-02,
3.100000e-02: 6.199752e-02,
1.000000e-01: 1.999162e-01,
3.100000e-01: 6.173918e-01
}.items():
yield case, swap_map(x), id_chan, target
# Finally, we add a known case from Johnston's QETLAB documentation,
# || Phi - I ||,_♢ where Phi(X) = UXU⁺ and U = [[1, 1], [-1, 1]] / sqrt(2).
yield case, Qobj([[1, 1], [-1, 1]]) / np.sqrt(2), qeye(2), np.sqrt(2)
ket_zero = qubit_states(N=1, states=[0])
print(ket_zero)
print("\n")
print(x * ket_zero)
print("\n")
print(tensor([identity(N=2), hadamard_transform(N=1)]))
print("\n")
print(snot(N=2, target=1))
print("\n")
print(swap(N=2, targets=[0, 1]))
print("\n")
print(cnot(N=2, control=0, target=1))
print("\n")
print(cnot(N=4, control=0, target=3))
print("\n")
print(hadamard_transform(N=2) * qubit_states(N=2, states=[0, 0]))
print("\n")
qc3 = QubitCircuit(N=3)
qc3.add_gate("CNOT", 1, 0)
qc3.add_gate("RX", 0, None, pi / 2, r"\pi/2")
qc3.add_gate("RY", 1, None, pi / 2, r"\pi/2")
# taking the trace
f = uu.tr()
# absolute value of the trace
fi = abs(f)
return fi
success_amt = []
n_trials = 100
opt_repeat = 5 #50
fid_check = []
g_check = []
for p in range(n_trials):
# g_target = random_sample()
# U_target = u_W_deltaa_T_g(W, deltaa, T, g_target)
U_target = swap()
fid_this_repeat = []
g_this_repeat = []
for q in range(opt_repeat):
g0 = random(1) # random_sample()
def one_minus_fid_Utarg_Uofg(nelder_g):
nelder_g_num = nelder_g[0]
U_nelder = u_W_deltaa_T_g(W, deltaa, T, nelder_g_num)
fi = 1 - fid_U_targ_U(U_target, U_nelder)
return fi
optim = minimize(one_minus_fid_Utarg_Uofg,
g0,