def test_superposition_basis():
nbits = 2
first_half_state = IntQubit(0, nbits)/2 + IntQubit(1, nbits)/2
second_half_state = IntQubit(2, nbits)/2 + IntQubit(3, nbits)/2
assert first_half_state + second_half_state == superposition_basis(nbits)
nbits = 3
firstq = (1/sqrt(8))*IntQubit(0, nbits) + (1/sqrt(8))*IntQubit(1, nbits)
secondq = (1/sqrt(8))*IntQubit(2, nbits) + (1/sqrt(8))*IntQubit(3, nbits)
thirdq = (1/sqrt(8))*IntQubit(4, nbits) + (1/sqrt(8))*IntQubit(5, nbits)
fourthq = (1/sqrt(8))*IntQubit(6, nbits) + (1/sqrt(8))*IntQubit(7, nbits)
assert firstq + secondq + thirdq + fourthq == superposition_basis(nbits)
def test_superposition_basis():
nbits = 2
first_half_state = IntQubit(0, nqubits=nbits)/2 + IntQubit(1, nqubits=nbits)/2
second_half_state = IntQubit(2, nbits)/2 + IntQubit(3, nbits)/2
assert first_half_state + second_half_state == superposition_basis(nbits)
nbits = 3
firstq = (1/sqrt(8))*IntQubit(0, nqubits=nbits) + (1/sqrt(8))*IntQubit(1, nqubits=nbits)
secondq = (1/sqrt(8))*IntQubit(2, nbits) + (1/sqrt(8))*IntQubit(3, nbits)
thirdq = (1/sqrt(8))*IntQubit(4, nbits) + (1/sqrt(8))*IntQubit(5, nbits)
fourthq = (1/sqrt(8))*IntQubit(6, nbits) + (1/sqrt(8))*IntQubit(7, nbits)
assert firstq + secondq + thirdq + fourthq == superposition_basis(nbits)
def main():
print()
print('Demonstration of Grover\'s Algorithm')
print('The OracleGate or V Gate carries the unknown function f(x)')
print(
'> V|x> = ((-1)^f(x))|x> where f(x) = 1 when x = a (True in our case)')
print('> and 0 (False in our case) otherwise')
print()
nqubits = 2
print('nqubits = ', nqubits)
v = OracleGate(nqubits, black_box)
print('Oracle or v = OracleGate(%r, black_box)' % nqubits)
print()
psi = superposition_basis(nqubits)
print('psi:')
pprint(psi)
demo_vgate_app(v)
print('qapply(v*psi)')
pprint(qapply(v * psi))
print()
w = WGate(nqubits)
print('WGate or w = WGate(%r)' % nqubits)
print('On a 2 Qubit system like psi, 1 iteration is enough to yield |1>')
print('qapply(w*v*psi)')
pprint(qapply(w * v * psi))
print()
nqubits = 3
print('On a 3 Qubit system, it requires 2 iterations to achieve')
print('|1> with high enough probability')
psi = superposition_basis(nqubits)
print('psi:')
pprint(psi)
v = OracleGate(nqubits, black_box)
print('Oracle or v = OracleGate(%r, black_box)' % nqubits)
print()
print('iter1 = grover.grover_iteration(psi, v)')
iter1 = qapply(grover_iteration(psi, v))
pprint(iter1)
print()
print('iter2 = grover.grover_iteration(iter1, v)')
iter2 = qapply(grover_iteration(iter1, v))
pprint(iter2)
print()
def main():
print()
print("Demonstration of Grover's Algorithm")
print("The OracleGate or V Gate carries the unknown function f(x)")
print(
"> V|x> = ((-1)^f(x))|x> where f(x) = 1 when x = a (True in our case)")
print("> and 0 (False in our case) otherwise")
print()
nqubits = 2
print("nqubits = ", nqubits)
v = OracleGate(nqubits, black_box)
print("Oracle or v = OracleGate(%r, black_box)" % nqubits)
print()
psi = superposition_basis(nqubits)
print("psi:")
pprint(psi)
demo_vgate_app(v)
print("qapply(v*psi)")
pprint(qapply(v * psi))
print()
w = WGate(nqubits)
print("WGate or w = WGate(%r)" % nqubits)
print("On a 2 Qubit system like psi, 1 iteration is enough to yield |1>")
print("qapply(w*v*psi)")
pprint(qapply(w * v * psi))
print()
nqubits = 3
print("On a 3 Qubit system, it requires 2 iterations to achieve")
print("|1> with high enough probability")
psi = superposition_basis(nqubits)
print("psi:")
pprint(psi)
v = OracleGate(nqubits, black_box)
print("Oracle or v = OracleGate(%r, black_box)" % nqubits)
print()
print("iter1 = grover.grover_iteration(psi, v)")
iter1 = qapply(grover_iteration(psi, v))
pprint(iter1)
print()
print("iter2 = grover.grover_iteration(iter1, v)")
iter2 = qapply(grover_iteration(iter1, v))
pprint(iter2)
print()
def main():
print
print "Demonstration of Grover's Algorithm"
print "The OracleGate or V Gate carries the unknown function f(x)"
print "> V|x> = ((-1)^f(x))|x> where f(x) = 1 when x = a (True in our case)"
print "> and 0 (False in our case) otherwise"
print
nqubits = 2
print "nqubits = ", nqubits
v = OracleGate(nqubits, black_box)
print "Oracle or v = OracleGate(%r, black_box)" % nqubits
print
psi = superposition_basis(nqubits)
print "psi:"
pprint(psi)
demo_vgate_app(v)
print "qapply(v*psi)"
pprint(qapply(v * psi))
print
w = WGate(nqubits)
print "WGate or w = WGate(%r)" % nqubits
print "On a 2 Qubit system like psi, 1 iteration is enough to yield |1>"
print "qapply(w*v*psi)"
pprint(qapply(w * v * psi))
print
nqubits = 3
print "On a 3 Qubit system, it requires 2 iterations to achieve"
print "|1> with high enough probability"
psi = superposition_basis(nqubits)
print "psi:"
pprint(psi)
v = OracleGate(nqubits, black_box)
print "Oracle or v = OracleGate(%r, black_box)" % nqubits
print
print "iter1 = grover.grover_iteration(psi, v)"
iter1 = qapply(grover_iteration(psi, v))
pprint(iter1)
print
print "iter2 = grover.grover_iteration(iter1, v)"
iter2 = qapply(grover_iteration(iter1, v))
pprint(iter2)
print
def main():
print()
print('Demonstration of Grover\'s Algorithm')
print('The OracleGate or V Gate carries the unknown function f(x)')
print('> V|x> = ((-1)^f(x))|x> where f(x) = 1 when x = a (True in our case)')
print('> and 0 (False in our case) otherwise')
print()
nqubits = 2
print('nqubits = ', nqubits)
v = OracleGate(nqubits, black_box)
print('Oracle or v = OracleGate(%r, black_box)' % nqubits)
print()
psi = superposition_basis(nqubits)
print('psi:')
pprint(psi)
demo_vgate_app(v)
print('qapply(v*psi)')
pprint(qapply(v*psi))
print()
w = WGate(nqubits)
print('WGate or w = WGate(%r)' % nqubits)
print('On a 2 Qubit system like psi, 1 iteration is enough to yield |1>')
print('qapply(w*v*psi)')
pprint(qapply(w*v*psi))
print()
nqubits = 3
print('On a 3 Qubit system, it requires 2 iterations to achieve')
print('|1> with high enough probability')
psi = superposition_basis(nqubits)
print('psi:')
pprint(psi)
v = OracleGate(nqubits, black_box)
print('Oracle or v = OracleGate(%r, black_box)' % nqubits)
print()
print('iter1 = grover.grover_iteration(psi, v)')
iter1 = qapply(grover_iteration(psi, v))
pprint(iter1)
print()
print('iter2 = grover.grover_iteration(iter1, v)')
iter2 = qapply(grover_iteration(iter1, v))
pprint(iter2)
print()
def test_WGate():
nqubits = 2
basis_states = superposition_basis(nqubits)
assert qapply(WGate(nqubits)*basis_states) == basis_states
expected = ((2/sqrt(pow(2, nqubits)))*basis_states) - IntQubit(1, nqubits=nqubits)
assert qapply(WGate(nqubits)*IntQubit(1, nqubits=nqubits)) == expected
def test_WGate():
nqubits = 2
basis_states = superposition_basis(nqubits)
assert qapply(WGate(nqubits)*basis_states) == basis_states
expected = ((2/sqrt(pow(2, nqubits)))*basis_states) - IntQubit(1, nqubits)
assert qapply(WGate(nqubits)*IntQubit(1, nqubits)) == expected
def test_grover():
nqubits = 2
assert apply_grover(return_one_on_one, nqubits) == IntQubit(1, nqubits=nqubits)
nqubits = 4
basis_states = superposition_basis(nqubits)
expected = (-13*basis_states)/64 + 264*IntQubit(2, nqubits)/256
assert apply_grover(return_one_on_two, 4) == qapply(expected)
def test_grover():
nqubits = 2
assert apply_grover(return_one_on_one, nqubits) == IntQubit(1, nqubits)
nqubits = 4
basis_states = superposition_basis(nqubits)
expected = (-13*basis_states)/64 + 264*IntQubit(2, nqubits)/256
assert apply_grover(return_one_on_two, 4) == qapply(expected)
def test_grover_iteration_2():
numqubits = 4
basis_states = superposition_basis(numqubits)
v = OracleGate(numqubits, return_one_on_two)
# After (pi/4)sqrt(pow(2, n)), IntQubit(2) should have highest prob
# In this case, after around pi times (3 or 4)
iterated = grover_iteration(basis_states, v)
iterated = qapply(iterated)
iterated = grover_iteration(iterated, v)
iterated = qapply(iterated)
iterated = grover_iteration(iterated, v)
iterated = qapply(iterated)
# In this case, probability was highest after 3 iterations
# Probability of Qubit('0010') was 251/256 (3) vs 781/1024 (4)
# Ask about measurement
expected = (-13*basis_states)/64 + 264*IntQubit(2, numqubits)/256
assert qapply(expected) == iterated
def test_grover_iteration_2():
numqubits = 4
basis_states = superposition_basis(numqubits)
v = OracleGate(numqubits, return_one_on_two)
# After (pi/4)sqrt(pow(2, n)), IntQubit(2) should have highest prob
# In this case, after around pi times (3 or 4)
iterated = grover_iteration(basis_states, v)
iterated = qapply(iterated)
iterated = grover_iteration(iterated, v)
iterated = qapply(iterated)
iterated = grover_iteration(iterated, v)
iterated = qapply(iterated)
# In this case, probability was highest after 3 iterations
# Probability of Qubit('0010') was 251/256 (3) vs 781/1024 (4)
# Ask about measurement
expected = (-13*basis_states)/64 + 264*IntQubit(2, numqubits)/256
assert qapply(expected) == iterated
def main():
psi = superposition_basis(2)
psi
# Dense coding demo:
# Assume Alice has the left QBit in psi
print(
"An even superposition of 2 qubits. Assume Alice has the left QBit.")
pprint(psi)
# The corresponding gates applied to Alice's QBit are:
# Identity Gate (1), Not Gate (X), Z Gate (Z), Z Gate and Not Gate (ZX)
# Then there's the controlled not gate (with Alice's as control):CNOT(1, 0)
# And the Hadamard gate applied to Alice's Qbit: H(1)
# To Send Bob the message |0>|0>
print("To Send Bob the message |00>.")
circuit = H(1) * CNOT(1, 0)
result = qapply(circuit * psi)
result
pprint(result)
# To send Bob the message |0>|1>
print("To Send Bob the message |01>.")
circuit = H(1) * CNOT(1, 0) * X(1)
result = qapply(circuit * psi)
result
pprint(result)
# To send Bob the message |1>|0>
print("To Send Bob the message |10>.")
circuit = H(1) * CNOT(1, 0) * Z(1)
result = qapply(circuit * psi)
result
pprint(result)
# To send Bob the message |1>|1>
print("To Send Bob the message |11>.")
circuit = H(1) * CNOT(1, 0) * Z(1) * X(1)
result = qapply(circuit * psi)
result
pprint(result)
def main():
psi = superposition_basis(2)
psi
# Dense coding demo:
# Assume Alice has the left QBit in psi
print "An even superposition of 2 qubits. Assume Alice has the left QBit."
pprint(psi)
# The corresponding gates applied to Alice's QBit are:
# Identity Gate (1), Not Gate (X), Z Gate (Z), Z Gate and Not Gate (ZX)
# Then there's the controlled not gate (with Alice's as control):CNOT(1, 0)
# And the Hadamard gate applied to Alice's Qbit: H(1)
# To Send Bob the message |0>|0>
print "To Send Bob the message |00>."
circuit = H(1)*CNOT(1, 0)
result = qapply(circuit*psi)
result
pprint(result)
# To send Bob the message |0>|1>
print "To Send Bob the message |01>."
circuit = H(1)*CNOT(1, 0)*X(1)
result = qapply(circuit*psi)
result
pprint(result)
# To send Bob the message |1>|0>
print "To Send Bob the message |10>."
circuit = H(1)*CNOT(1, 0)*Z(1)
result = qapply(circuit*psi)
result
pprint(result)
# To send Bob the message |1>|1>
print "To Send Bob the message |11>."
circuit = H(1)*CNOT(1, 0)*Z(1)*X(1)
result = qapply(circuit*psi)
result
pprint(result)
def test_grover_iteration_1():
numqubits = 2
basis_states = superposition_basis(numqubits)
v = OracleGate(numqubits, return_one_on_one)
expected = IntQubit(1, nqubits=numqubits)
assert qapply(grover_iteration(basis_states, v)) == expected
def test_g_bbht_search(nqubits=5):
basis_states = superposition_basis(nqubits)
oracle = random_oracle(nqubits, min_img=5, max_img=15, q_type="bin")
x = g_bbht_search(basis_states, oracle)[0]
assert oracle.search_function(x) == 1
def superposition(self):
return superposition_basis(self.qubit.nqubits)
def test_grover_iteration_1():
numqubits = 2
basis_states = superposition_basis(numqubits)
v = OracleGate(numqubits, return_one_on_one)
expected = IntQubit(1, numqubits)
assert qapply(grover_iteration(basis_states, v)) == expected