def test_j2():
j, m = symbols('j m')
assert Commutator(J2, Jz).doit() == 0
assert qapply(J2 * JzKet(1, 1)) == 2 * hbar**2 * JzKet(1, 1)
assert qapply(J2 * JzKet(
j, m)) == j**2 * hbar**2 * JzKet(j, m) + j * hbar**2 * JzKet(j, m)
assert J2.matrix_element(1, 1, 1, 1) == 2 * hbar**2
def test_fermion_states():
c = FermionOp("c")
# Fock states
assert (FermionFockBra(0) * FermionFockKet(1)).doit() == 0
assert (FermionFockBra(1) * FermionFockKet(1)).doit() == 1
assert qapply(c * FermionFockKet(1)) == FermionFockKet(0)
assert qapply(c * FermionFockKet(0)) == 0
assert qapply(Dagger(c) * FermionFockKet(0)) == FermionFockKet(1)
assert qapply(Dagger(c) * FermionFockKet(1)) == 0
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_superposition_of_states():
assert qapply(CNOT(0,1)*HadamardGate(0)*(1/sqrt(2)*Qubit('01') + 1/sqrt(2)*Qubit('10'))).expand() == (Qubit('01')/2 + Qubit('00')/2 - Qubit('11')/2 +\
Qubit('10')/2)
assert matrix_to_qubit(represent(CNOT(0,1)*HadamardGate(0)\
*(1/sqrt(2)*Qubit('01') + 1/sqrt(2)*Qubit('10')), nqubits=2))\
== (Qubit('01')/2 + Qubit('00')/2 - Qubit('11')/2 + Qubit('10')/2)
def test_x():
assert X.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Commutator(X, Px).doit() == I*hbar
assert qapply(X*XKet(x)) == x*XKet(x)
assert XKet(x).dual_class() == XBra
assert XBra(x).dual_class() == XKet
assert (Dagger(XKet(y))*XKet(x)).doit() == DiracDelta(x - y)
assert (PxBra(px)*XKet(x)).doit() == \
exp(-I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(XKet(x)) == DiracDelta(x - x_1)
assert represent(XBra(x)) == DiracDelta(-x + x_1)
assert XBra(x).position == x
assert represent(XOp()*XKet()) == x*DiracDelta(x - x_2)
assert represent(XOp()*XKet()*XBra('y')) == \
x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)
assert represent(XBra("y")*XKet()) == DiracDelta(x - y)
assert represent(
XKet()*XBra()) == DiracDelta(x - x_2) * DiracDelta(x_1 - x)
rep_p = represent(XOp(), basis=PxOp)
assert rep_p == hbar*I*DiracDelta(px_1 - px_2)*DifferentialOperator(px_1)
assert rep_p == represent(XOp(), basis=PxOp())
assert rep_p == represent(XOp(), basis=PxKet)
assert rep_p == represent(XOp(), basis=PxKet())
assert represent(XOp()*PxKet(), basis=PxKet) == \
hbar*I*DiracDelta(px - px_2)*DifferentialOperator(px)
def test_x():
assert X.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Commutator(X, Px).doit() == I * hbar
assert qapply(X * XKet(x)) == x * XKet(x)
assert XKet(x).dual_class() == XBra
assert XBra(x).dual_class() == XKet
assert (Dagger(XKet(y)) * XKet(x)).doit() == DiracDelta(x - y)
assert (PxBra(px)*XKet(x)).doit() == \
exp(-I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(XKet(x)) == DiracDelta(x - x_1)
assert represent(XBra(x)) == DiracDelta(-x + x_1)
assert XBra(x).position == x
assert represent(XOp() * XKet()) == x * DiracDelta(x - x_2)
assert represent(XOp()*XKet()*XBra('y')) == \
x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)
assert represent(XBra("y") * XKet()) == DiracDelta(x - y)
assert represent(XKet() *
XBra()) == DiracDelta(x - x_2) * DiracDelta(x_1 - x)
rep_p = represent(XOp(), basis=PxOp)
assert rep_p == hbar * I * DiracDelta(px_1 -
px_2) * DifferentialOperator(px_1)
assert rep_p == represent(XOp(), basis=PxOp())
assert rep_p == represent(XOp(), basis=PxKet)
assert rep_p == represent(XOp(), basis=PxKet())
assert represent(XOp()*PxKet(), basis=PxKet) == \
hbar*I*DiracDelta(px - px_2)*DifferentialOperator(px)
def test_H():
assert PIABHamiltonian("H").hilbert_space == L2(
Interval(S.NegativeInfinity, S.Infinity)
)
assert qapply(PIABHamiltonian("H") * PIABKet(n)) == (
n ** 2 * pi ** 2 * hbar ** 2
) / (2 * m * L ** 2) * PIABKet(n)
def test_jx():
assert Commutator(Jx, Jz).doit() == -I * hbar * Jy
assert qapply(Jx * JzKet(1, 1)) == sqrt(2) * hbar * JzKet(1, 0) / 2
assert Jx.rewrite('plusminus') == (Jminus + Jplus) / 2
assert represent(Jx, basis=Jz,
j=1) == (represent(Jplus, basis=Jz, j=1) +
represent(Jminus, basis=Jz, j=1)) / 2
def test_boson_states():
a = BosonOp("a")
# Fock states
n = 3
assert (BosonFockBra(0) * BosonFockKet(1)).doit() == 0
assert (BosonFockBra(1) * BosonFockKet(1)).doit() == 1
assert qapply(BosonFockBra(n) * Dagger(a)**n * BosonFockKet(0)) \
== sqrt(prod(range(1, n+1)))
# Coherent states
alpha1, alpha2 = 1.2, 4.3
assert (BosonCoherentBra(alpha1) * BosonCoherentKet(alpha1)).doit() == 1
assert (BosonCoherentBra(alpha2) * BosonCoherentKet(alpha2)).doit() == 1
assert abs((BosonCoherentBra(alpha1) * BosonCoherentKet(alpha2)).doit() -
exp((alpha1 - alpha2) ** 2 * Rational(-1, 2))) < 1e-12
assert qapply(a * BosonCoherentKet(alpha1)) == \
alpha1 * BosonCoherentKet(alpha1)
def test_p():
assert Px.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert qapply(Px * PxKet(px)) == px * PxKet(px)
assert PxKet(px).dual_class == PxBra
assert PxBra(x).dual_class == PxKet
assert (Dagger(PxKet(py)) * PxKet(px)).doit() == DiracDelta(px - py)
assert (XBra(x)*PxKet(px)).doit() ==\
exp(I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(PxKet(px)) == px
def test_boson_states():
a = BosonOp("a")
# Fock states
n = 3
assert (BosonFockBra(0) * BosonFockKet(1)).doit() == 0
assert (BosonFockBra(1) * BosonFockKet(1)).doit() == 1
assert qapply(BosonFockBra(n) * Dagger(a)**n * BosonFockKet(0)) \
== sqrt(prod(range(1, n+1)))
# Coherent states
alpha1, alpha2 = 1.2, 4.3
assert (BosonCoherentBra(alpha1) * BosonCoherentKet(alpha1)).doit() == 1
assert (BosonCoherentBra(alpha2) * BosonCoherentKet(alpha2)).doit() == 1
assert abs((BosonCoherentBra(alpha1) * BosonCoherentKet(alpha2)).doit() -
exp(-S(1) / 2 * (alpha1 - alpha2) ** 2)) < 1e-12
assert qapply(a * BosonCoherentKet(alpha1)) == \
alpha1 * BosonCoherentKet(alpha1)
def test_p():
assert Px.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert qapply(Px*PxKet(px)) == px*PxKet(px)
assert PxKet(px).dual_class == PxBra
assert PxBra(x).dual_class == PxKet
assert (Dagger(PxKet(py))*PxKet(px)).doit() == DiracDelta(px-py)
assert (XBra(x)*PxKet(px)).doit() ==\
exp(I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(PxKet(px)) == px
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_x():
assert X.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Commutator(X, Px).doit() == I*hbar
assert qapply(X*XKet(x)) == x*XKet(x)
assert XKet(x).dual_class == XBra
assert XBra(x).dual_class == XKet
assert (Dagger(XKet(y))*XKet(x)).doit() == DiracDelta(x-y)
assert (PxBra(px)*XKet(x)).doit() ==\
exp(-I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(XKet(x)) == x
assert XBra(x).position == x
def test_x():
assert X.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Commutator(X, Px).doit() == I * hbar
assert qapply(X * XKet(x)) == x * XKet(x)
assert XKet(x).dual_class == XBra
assert XBra(x).dual_class == XKet
assert (Dagger(XKet(y)) * XKet(x)).doit() == DiracDelta(x - y)
assert (PxBra(px)*XKet(x)).doit() ==\
exp(-I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(XKet(x)) == x
assert XBra(x).position == x
def test_3dpos():
assert Y.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Z.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
test_ket = PositionKet3D(x, y, z)
assert qapply(X*test_ket) == x*test_ket
assert qapply(Y*test_ket) == y*test_ket
assert qapply(Z*test_ket) == z*test_ket
assert qapply(X*Y*test_ket) == x*y*test_ket
assert qapply(X*Y*Z*test_ket) == x*y*z*test_ket
assert qapply(Y*Z*test_ket) == y*z*test_ket
assert PositionKet3D() == test_ket
assert YOp() == Y
assert ZOp() == Z
assert PositionKet3D.dual_class() == PositionBra3D
assert PositionBra3D.dual_class() == PositionKet3D
other_ket = PositionKet3D(x_1, y_1, z_1)
assert (Dagger(other_ket)*test_ket).doit() == \
DiracDelta(x - x_1)*DiracDelta(y - y_1)*DiracDelta(z - z_1)
assert test_ket.position_x == x
assert test_ket.position_y == y
assert test_ket.position_z == z
assert other_ket.position_x == x_1
assert other_ket.position_y == y_1
assert other_ket.position_z == z_1
def test_3dpos():
assert Y.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert Z.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
test_ket = PositionKet3D(x, y, z)
assert qapply(X * test_ket) == x * test_ket
assert qapply(Y * test_ket) == y * test_ket
assert qapply(Z * test_ket) == z * test_ket
assert qapply(X * Y * test_ket) == x * y * test_ket
assert qapply(X * Y * Z * test_ket) == x * y * z * test_ket
assert qapply(Y * Z * test_ket) == y * z * test_ket
assert PositionKet3D() == test_ket
assert YOp() == Y
assert ZOp() == Z
assert PositionKet3D.dual_class() == PositionBra3D
assert PositionBra3D.dual_class() == PositionKet3D
other_ket = PositionKet3D(x_1, y_1, z_1)
assert (Dagger(other_ket)*test_ket).doit() == \
DiracDelta(x - x_1)*DiracDelta(y - y_1)*DiracDelta(z - z_1)
assert test_ket.position_x == x
assert test_ket.position_y == y
assert test_ket.position_z == z
assert other_ket.position_x == x_1
assert other_ket.position_y == y_1
assert other_ket.position_z == z_1
def test_apply_represent_equality():
gates = [HadamardGate(int(3*random.random())),\
XGate(int(3*random.random())), ZGate(int(3*random.random())),\
YGate(int(3*random.random())), ZGate(int(3*random.random())),\
PhaseGate(int(3*random.random()))]
circuit = Qubit(int(random.random()*2),int(random.random()*2),\
int(random.random()*2),int(random.random()*2),int(random.random()*2),\
int(random.random()*2))
for i in range(int(random.random() * 6)):
circuit = gates[int(random.random() * 6)] * circuit
mat = represent(circuit, nqubits=6)
states = qapply(circuit)
state_rep = matrix_to_qubit(mat)
states = states.expand()
state_rep = state_rep.expand()
assert state_rep == states
def test_apply_represent_equality():
gates = [HadamardGate(int(3*random.random())),\
XGate(int(3*random.random())), ZGate(int(3*random.random())),\
YGate(int(3*random.random())), ZGate(int(3*random.random())),\
PhaseGate(int(3*random.random()))]
circuit = Qubit(int(random.random()*2),int(random.random()*2),\
int(random.random()*2),int(random.random()*2),int(random.random()*2),\
int(random.random()*2))
for i in range(int(random.random()*6)):
circuit = gates[int(random.random()*6)]*circuit
mat = represent(circuit, nqubits=6)
states = qapply(circuit)
state_rep = matrix_to_qubit(mat)
states = states.expand()
state_rep = state_rep.expand()
assert state_rep == states
def test_pauli_states():
sx, sz = SigmaX(), SigmaZ()
up = SigmaZKet(0)
down = SigmaZKet(1)
assert qapply(sx * up) == down
assert qapply(sx * down) == up
assert qapply(sz * up) == up
assert qapply(sz * down) == -down
up = SigmaZBra(0)
down = SigmaZBra(1)
assert qapply(up * sx, dagger=True) == down
assert qapply(down * sx, dagger=True) == up
assert qapply(up * sz, dagger=True) == up
assert qapply(down * sz, dagger=True) == -down
assert Dagger(SigmaZKet(0)) == SigmaZBra(0)
assert Dagger(SigmaZBra(1)) == SigmaZKet(1)
def test_pauli_states():
sx, sz = SigmaX(), SigmaZ()
up = SigmaZKet(0)
down = SigmaZKet(1)
assert qapply(sx * up) == down
assert qapply(sx * down) == up
assert qapply(sz * up) == up
assert qapply(sz * down) == - down
up = SigmaZBra(0)
down = SigmaZBra(1)
assert qapply(up * sx, dagger=True) == down
assert qapply(down * sx, dagger=True) == up
assert qapply(up * sz, dagger=True) == up
assert qapply(down * sz, dagger=True) == - down
assert Dagger(SigmaZKet(0)) == SigmaZBra(0)
assert Dagger(SigmaZBra(1)) == SigmaZKet(1)
def test_p():
assert Px.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert qapply(Px*PxKet(px)) == px*PxKet(px)
assert PxKet(px).dual_class() == PxBra
assert PxBra(x).dual_class() == PxKet
assert (Dagger(PxKet(py))*PxKet(px)).doit() == DiracDelta(px-py)
assert (XBra(x)*PxKet(px)).doit() ==\
exp(I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(PxKet(px)) == DiracDelta(px-px_1)
rep_x = represent(PxOp(), basis = XOp)
assert rep_x == -hbar*I*DiracDelta(x_1 - x_2)*DifferentialOperator(x_1)
assert rep_x == represent(PxOp(), basis = XOp())
assert rep_x == represent(PxOp(), basis = XKet)
assert rep_x == represent(PxOp(), basis = XKet())
assert represent(PxOp()*XKet(), basis=XKet) == \
-hbar*I*DiracDelta(x - x_2)*DifferentialOperator(x)
assert represent(XBra("y")*PxOp()*XKet(), basis=XKet) == \
-hbar*I*DiracDelta(x-y)*DifferentialOperator(x)
def test_p():
assert Px.hilbert_space == L2(Interval(S.NegativeInfinity, S.Infinity))
assert qapply(Px*PxKet(px)) == px*PxKet(px)
assert PxKet(px).dual_class() == PxBra
assert PxBra(x).dual_class() == PxKet
assert (Dagger(PxKet(py))*PxKet(px)).doit() == DiracDelta(px - py)
assert (XBra(x)*PxKet(px)).doit() == \
exp(I*x*px/hbar)/sqrt(2*pi*hbar)
assert represent(PxKet(px)) == DiracDelta(px - px_1)
rep_x = represent(PxOp(), basis=XOp)
assert rep_x == -hbar*I*DiracDelta(x_1 - x_2)*DifferentialOperator(x_1)
assert rep_x == represent(PxOp(), basis=XOp())
assert rep_x == represent(PxOp(), basis=XKet)
assert rep_x == represent(PxOp(), basis=XKet())
assert represent(PxOp()*XKet(), basis=XKet) == \
-hbar*I*DiracDelta(x - x_2)*DifferentialOperator(x)
assert represent(XBra("y")*PxOp()*XKet(), basis=XKet) == \
-hbar*I*DiracDelta(x - y)*DifferentialOperator(x)
def test_rewrite():
# Rewrite to same basis
assert JxBra(1, 1).rewrite('Jx') == JxBra(1, 1)
assert JxKet(1, 1).rewrite('Jx') == JxKet(1, 1)
# Rewriting to different basis
assert JxKet(1, 1).rewrite('Jy') == I * JyKet(1, 1)
assert JxKet(1, 0).rewrite('Jy') == JyKet(1, 0)
assert JxKet(1, -1).rewrite('Jy') == -I * JyKet(1, -1)
assert JxKet(1, 1).rewrite(
'Jz') == JzKet(1, 1) / 2 + JzKet(1, 0) / sqrt(2) + JzKet(1, -1) / 2
assert JxKet(1, 0).rewrite(
'Jz') == -sqrt(2) * JzKet(1, 1) / 2 + sqrt(2) * JzKet(1, -1) / 2
assert JxKet(1, -1).rewrite(
'Jz') == JzKet(1, 1) / 2 - JzKet(1, 0) / sqrt(2) + JzKet(1, -1) / 2
assert JyKet(1, 1).rewrite('Jx') == -I * JxKet(1, 1)
assert JyKet(1, 0).rewrite('Jx') == JxKet(1, 0)
assert JyKet(1, -1).rewrite('Jx') == I * JxKet(1, -1)
assert JyKet(1, 1).rewrite('Jz') == JzKet(1, 1) / 2 + sqrt(2) * I * JzKet(
1, 0) / 2 - JzKet(1, -1) / 2
assert JyKet(1, 0).rewrite(
'Jz') == sqrt(2) * I * JzKet(1, 1) / 2 + sqrt(2) * I * JzKet(1, -1) / 2
assert JyKet(1,
-1).rewrite('Jz') == -JzKet(1, 1) / 2 + sqrt(2) * I * JzKet(
1, 0) / 2 + JzKet(1, -1) / 2
assert JzKet(1, 1).rewrite(
'Jx') == JxKet(1, 1) / 2 - sqrt(2) * JxKet(1, 0) / 2 + JxKet(1, -1) / 2
assert JzKet(1, 0).rewrite(
'Jx') == sqrt(2) * JxKet(1, 1) / 2 - sqrt(2) * JxKet(1, -1) / 2
assert JzKet(1, -1).rewrite(
'Jx') == JxKet(1, 1) / 2 + sqrt(2) * JxKet(1, 0) / 2 + JxKet(1, -1) / 2
assert JzKet(1, 1).rewrite('Jy') == JyKet(1, 1) / 2 - sqrt(2) * I * JyKet(
1, 0) / 2 - JyKet(1, -1) / 2
assert JzKet(1, 0).rewrite('Jy') == -sqrt(2) * I * JyKet(
1, 1) / 2 - sqrt(2) * I * JyKet(1, -1) / 2
assert JzKet(1,
-1).rewrite('Jy') == -JyKet(1, 1) / 2 - sqrt(2) * I * JyKet(
1, 0) / 2 + JyKet(1, -1) / 2
# Innerproducts of rewritten states
assert qapply(JxBra(1, 1) * JxKet(1, 1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1, 0) * JxKet(1, 0).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1, -1) * JxKet(1, -1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1, 1) * JxKet(1, 1).rewrite('Jz')).doit() == 1
assert qapply(JxBra(1, 0) * JxKet(1, 0).rewrite('Jz')).doit() == 1
assert qapply(JxBra(1, -1) * JxKet(1, -1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, 1) * JyKet(1, 1).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1, 0) * JyKet(1, 0).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1, -1) * JyKet(1, -1).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1, 1) * JyKet(1, 1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, 0) * JyKet(1, 0).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, -1) * JyKet(1, -1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, 1) * JyKet(1, 1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, 0) * JyKet(1, 0).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1, -1) * JyKet(1, -1).rewrite('Jz')).doit() == 1
assert qapply(JzBra(1, 1) * JzKet(1, 1).rewrite('Jy')).doit() == 1
assert qapply(JzBra(1, 0) * JzKet(1, 0).rewrite('Jy')).doit() == 1
assert qapply(JzBra(1, -1) * JzKet(1, -1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1, 1) * JxKet(1, 0).rewrite('Jy')).doit() == 0
assert qapply(JxBra(1, 1) * JxKet(1, -1).rewrite('Jy')) == 0
assert qapply(JxBra(1, 1) * JxKet(1, 0).rewrite('Jz')).doit() == 0
assert qapply(JxBra(1, 1) * JxKet(1, -1).rewrite('Jz')) == 0
assert qapply(JyBra(1, 1) * JyKet(1, 0).rewrite('Jx')).doit() == 0
assert qapply(JyBra(1, 1) * JyKet(1, -1).rewrite('Jx')) == 0
assert qapply(JyBra(1, 1) * JyKet(1, 0).rewrite('Jz')).doit() == 0
assert qapply(JyBra(1, 1) * JyKet(1, -1).rewrite('Jz')) == 0
assert qapply(JzBra(1, 1) * JzKet(1, 0).rewrite('Jx')).doit() == 0
assert qapply(JzBra(1, 1) * JzKet(1, -1).rewrite('Jx')) == 0
assert qapply(JzBra(1, 1) * JzKet(1, 0).rewrite('Jy')).doit() == 0
assert qapply(JzBra(1, 1) * JzKet(1, -1).rewrite('Jy')) == 0
assert qapply(JxBra(1, 0) * JxKet(1, 1).rewrite('Jy')) == 0
assert qapply(JxBra(1, 0) * JxKet(1, -1).rewrite('Jy')) == 0
assert qapply(JxBra(1, 0) * JxKet(1, 1).rewrite('Jz')) == 0
assert qapply(JxBra(1, 0) * JxKet(1, -1).rewrite('Jz')) == 0
assert qapply(JyBra(1, 0) * JyKet(1, 1).rewrite('Jx')) == 0
assert qapply(JyBra(1, 0) * JyKet(1, -1).rewrite('Jx')) == 0
assert qapply(JyBra(1, 0) * JyKet(1, 1).rewrite('Jz')) == 0
assert qapply(JyBra(1, 0) * JyKet(1, -1).rewrite('Jz')) == 0
assert qapply(JzBra(1, 0) * JzKet(1, 1).rewrite('Jx')) == 0
assert qapply(JzBra(1, 0) * JzKet(1, -1).rewrite('Jx')) == 0
assert qapply(JzBra(1, 0) * JzKet(1, 1).rewrite('Jy')) == 0
assert qapply(JzBra(1, 0) * JzKet(1, -1).rewrite('Jy')) == 0
assert qapply(JxBra(1, -1) * JxKet(1, 1).rewrite('Jy')) == 0
assert qapply(JxBra(1, -1) * JxKet(1, 0).rewrite('Jy')).doit() == 0
assert qapply(JxBra(1, -1) * JxKet(1, 1).rewrite('Jz')) == 0
assert qapply(JxBra(1, -1) * JxKet(1, 0).rewrite('Jz')).doit() == 0
assert qapply(JyBra(1, -1) * JyKet(1, 1).rewrite('Jx')) == 0
assert qapply(JyBra(1, -1) * JyKet(1, 0).rewrite('Jx')).doit() == 0
assert qapply(JyBra(1, -1) * JyKet(1, 1).rewrite('Jz')) == 0
assert qapply(JyBra(1, -1) * JyKet(1, 0).rewrite('Jz')).doit() == 0
assert qapply(JzBra(1, -1) * JzKet(1, 1).rewrite('Jx')) == 0
assert qapply(JzBra(1, -1) * JzKet(1, 0).rewrite('Jx')).doit() == 0
assert qapply(JzBra(1, -1) * JzKet(1, 1).rewrite('Jy')) == 0
assert qapply(JzBra(1, -1) * JzKet(1, 0).rewrite('Jy')).doit() == 0
def test_jz():
assert Commutator(Jz, Jminus).doit() == -hbar * Jminus
assert qapply(Jz * JzKet(2, 1)) == hbar * JzKet(2, 1)
assert Jz.rewrite('plusminus')
def test_jy():
assert Commutator(Jy, Jz).doit() == I * hbar * Jx
assert qapply(Jy * JzKet(1, 1)) == I * sqrt(2) * hbar * JzKet(1, 0) / 2
assert Jy.rewrite('plusminus') == (Jplus - Jminus) / (2 * I)
assert represent(Jy, basis=Jz) == (represent(Jplus, basis=Jz) -
represent(Jminus, basis=Jz)) / (2 * I)
def test_j2():
j, m = symbols('j m')
assert Commutator(J2, Jz).doit() == 0
assert qapply(J2*JzKet(1,1)) == 2*hbar**2*JzKet(1,1)
assert qapply(J2*JzKet(j,m)) == j**2*hbar**2*JzKet(j,m)+j*hbar**2*JzKet(j,m)
assert J2.matrix_element(1,1,1,1) == 2*hbar**2
def test_jx():
assert Commutator(Jx, Jz).doit() == -I*hbar*Jy
assert qapply(Jx*JzKet(1,1)) == sqrt(2)*hbar*JzKet(1,0)/2
assert Jx.rewrite('plusminus') == (Jminus + Jplus)/2
assert represent(Jx, basis=Jz, j=1) == (represent(Jplus, basis=Jz, j=1)+represent(Jminus, basis=Jz, j=1))/2
def test_jminus():
assert qapply(Jminus * JzKet(1, -1)) == 0
assert Jminus.matrix_element(1, 0, 1, 1) == sqrt(2) * hbar
assert Jminus.rewrite('xyz') == Jx - I * Jy
def demo_vgate_app(v):
for i in range(2**v.nqubits):
print 'qapply(v*IntQubit({0}, {1}))'.format(i, v.nqubits)
pprint(qapply(v*IntQubit(i, v.nqubits)))
qapply(v*IntQubit(i, v.nqubits))
def test_jplus():
assert Commutator(Jplus, Jminus).doit() == 2*hbar*Jz
assert qapply(Jplus*JzKet(1,1)) == 0
assert Jplus.matrix_element(1,1,1,1) == 0
assert Jplus.rewrite('xyz') == Jx + I*Jy
def test_jminus():
assert qapply(Jminus*JzKet(1,-1)) == 0
assert Jminus.matrix_element(1,0,1,1) == sqrt(2)*hbar
assert Jminus.rewrite('xyz') == Jx - I*Jy
def demo_vgate_app(v):
for i in range(2**v.nqubits):
print 'qapply(v*IntQubit({0}, {1}))'.format(i, v.nqubits)
pprint(qapply(v * IntQubit(i, v.nqubits)))
qapply(v * IntQubit(i, v.nqubits))
def demo_vgate_app(v):
for i in range(2**v.nqubits):
print('qapply(v*IntQubit(%i, %r))' % (i, v.nqubits))
pprint(qapply(v*IntQubit(i, nqubits=v.nqubits)))
qapply(v*IntQubit(i, nqubits=v.nqubits))
def test_H():
assert PIABHamiltonian('H').hilbert_space ==\
L2(Interval(S.NegativeInfinity,S.Infinity))
assert qapply(PIABHamiltonian('H')*PIABKet(n)) ==\
(n**2*pi**2*hbar**2)/(2*m*L**2)*PIABKet(n)
def demo_vgate_app(v):
for i in range(2**v.nqubits):
print('qapply(v*IntQubit(%i, %r))' % (i, v.nqubits))
pprint(qapply(v * IntQubit(i, nqubits=v.nqubits)))
qapply(v * IntQubit(i, nqubits=v.nqubits))
def test_jy():
assert Commutator(Jy, Jz).doit() == I*hbar*Jx
assert qapply(Jy*JzKet(1,1)) == I*sqrt(2)*hbar*JzKet(1,0)/2
assert Jy.rewrite('plusminus') == (Jplus - Jminus)/(2*I)
assert represent(Jy, basis=Jz) == (represent(Jplus, basis=Jz) - represent(Jminus, basis=Jz))/(2*I)
def eval_state(self, state):
return qapply(Dagger(state) * self.args[0] * state, dagger=True).doit()
def test_jplus():
assert Commutator(Jplus, Jminus).doit() == 2 * hbar * Jz
assert qapply(Jplus * JzKet(1, 1)) == 0
assert Jplus.matrix_element(1, 1, 1, 1) == 0
assert Jplus.rewrite('xyz') == Jx + I * Jy
def test_jz():
assert Commutator(Jz, Jminus).doit() == -hbar*Jminus
assert qapply(Jz*JzKet(2,1)) == hbar*JzKet(2,1)
assert Jz.rewrite('plusminus')
def test_rewrite():
# Rewrite to same basis
assert JxBra(1,1).rewrite('Jx') == JxBra(1,1)
assert JxKet(1,1).rewrite('Jx') == JxKet(1,1)
# Rewriting to different basis
assert JxKet(1,1).rewrite('Jy') == I*JyKet(1,1)
assert JxKet(1,0).rewrite('Jy') == JyKet(1,0)
assert JxKet(1,-1).rewrite('Jy') == -I*JyKet(1,-1)
assert JxKet(1,1).rewrite('Jz') == JzKet(1,1)/2+JzKet(1,0)/sqrt(2)+JzKet(1,-1)/2
assert JxKet(1,0).rewrite('Jz') == -sqrt(2)*JzKet(1,1)/2+sqrt(2)*JzKet(1,-1)/2
assert JxKet(1,-1).rewrite('Jz') == JzKet(1,1)/2-JzKet(1,0)/sqrt(2)+JzKet(1,-1)/2
assert JyKet(1,1).rewrite('Jx') == -I*JxKet(1,1)
assert JyKet(1,0).rewrite('Jx') == JxKet(1,0)
assert JyKet(1,-1).rewrite('Jx') == I*JxKet(1,-1)
assert JyKet(1,1).rewrite('Jz') == JzKet(1,1)/2+sqrt(2)*I*JzKet(1,0)/2-JzKet(1,-1)/2
assert JyKet(1,0).rewrite('Jz') == sqrt(2)*I*JzKet(1,1)/2+sqrt(2)*I*JzKet(1,-1)/2
assert JyKet(1,-1).rewrite('Jz') == -JzKet(1,1)/2+sqrt(2)*I*JzKet(1,0)/2+JzKet(1,-1)/2
assert JzKet(1,1).rewrite('Jx') == JxKet(1,1)/2-sqrt(2)*JxKet(1,0)/2+JxKet(1,-1)/2
assert JzKet(1,0).rewrite('Jx') == sqrt(2)*JxKet(1,1)/2-sqrt(2)*JxKet(1,-1)/2
assert JzKet(1,-1).rewrite('Jx') == JxKet(1,1)/2+sqrt(2)*JxKet(1,0)/2+JxKet(1,-1)/2
assert JzKet(1,1).rewrite('Jy') == JyKet(1,1)/2-sqrt(2)*I*JyKet(1,0)/2-JyKet(1,-1)/2
assert JzKet(1,0).rewrite('Jy') == -sqrt(2)*I*JyKet(1,1)/2-sqrt(2)*I*JyKet(1,-1)/2
assert JzKet(1,-1).rewrite('Jy') == -JyKet(1,1)/2-sqrt(2)*I*JyKet(1,0)/2+JyKet(1,-1)/2
# Innerproducts of rewritten states
assert qapply(JxBra(1,1)*JxKet(1,1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1,0)*JxKet(1,0).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1,-1)*JxKet(1,-1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1,1)*JxKet(1,1).rewrite('Jz')).doit() == 1
assert qapply(JxBra(1,0)*JxKet(1,0).rewrite('Jz')).doit() == 1
assert qapply(JxBra(1,-1)*JxKet(1,-1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,1)*JyKet(1,1).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1,0)*JyKet(1,0).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1,-1)*JyKet(1,-1).rewrite('Jx')).doit() == 1
assert qapply(JyBra(1,1)*JyKet(1,1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,0)*JyKet(1,0).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,-1)*JyKet(1,-1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,1)*JyKet(1,1).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,0)*JyKet(1,0).rewrite('Jz')).doit() == 1
assert qapply(JyBra(1,-1)*JyKet(1,-1).rewrite('Jz')).doit() == 1
assert qapply(JzBra(1,1)*JzKet(1,1).rewrite('Jy')).doit() == 1
assert qapply(JzBra(1,0)*JzKet(1,0).rewrite('Jy')).doit() == 1
assert qapply(JzBra(1,-1)*JzKet(1,-1).rewrite('Jy')).doit() == 1
assert qapply(JxBra(1,1)*JxKet(1,0).rewrite('Jy')).doit() == 0
assert qapply(JxBra(1,1)*JxKet(1,-1).rewrite('Jy')) == 0
assert qapply(JxBra(1,1)*JxKet(1,0).rewrite('Jz')).doit() == 0
assert qapply(JxBra(1,1)*JxKet(1,-1).rewrite('Jz')) == 0
assert qapply(JyBra(1,1)*JyKet(1,0).rewrite('Jx')).doit() == 0
assert qapply(JyBra(1,1)*JyKet(1,-1).rewrite('Jx')) == 0
assert qapply(JyBra(1,1)*JyKet(1,0).rewrite('Jz')).doit() == 0
assert qapply(JyBra(1,1)*JyKet(1,-1).rewrite('Jz')) == 0
assert qapply(JzBra(1,1)*JzKet(1,0).rewrite('Jx')).doit() == 0
assert qapply(JzBra(1,1)*JzKet(1,-1).rewrite('Jx')) == 0
assert qapply(JzBra(1,1)*JzKet(1,0).rewrite('Jy')).doit() == 0
assert qapply(JzBra(1,1)*JzKet(1,-1).rewrite('Jy')) == 0
assert qapply(JxBra(1,0)*JxKet(1,1).rewrite('Jy')) == 0
assert qapply(JxBra(1,0)*JxKet(1,-1).rewrite('Jy')) == 0
assert qapply(JxBra(1,0)*JxKet(1,1).rewrite('Jz')) == 0
assert qapply(JxBra(1,0)*JxKet(1,-1).rewrite('Jz')) == 0
assert qapply(JyBra(1,0)*JyKet(1,1).rewrite('Jx')) == 0
assert qapply(JyBra(1,0)*JyKet(1,-1).rewrite('Jx')) == 0
assert qapply(JyBra(1,0)*JyKet(1,1).rewrite('Jz')) == 0
assert qapply(JyBra(1,0)*JyKet(1,-1).rewrite('Jz')) == 0
assert qapply(JzBra(1,0)*JzKet(1,1).rewrite('Jx')) == 0
assert qapply(JzBra(1,0)*JzKet(1,-1).rewrite('Jx')) == 0
assert qapply(JzBra(1,0)*JzKet(1,1).rewrite('Jy')) == 0
assert qapply(JzBra(1,0)*JzKet(1,-1).rewrite('Jy')) == 0
assert qapply(JxBra(1,-1)*JxKet(1,1).rewrite('Jy')) == 0
assert qapply(JxBra(1,-1)*JxKet(1,0).rewrite('Jy')).doit() == 0
assert qapply(JxBra(1,-1)*JxKet(1,1).rewrite('Jz')) == 0
assert qapply(JxBra(1,-1)*JxKet(1,0).rewrite('Jz')).doit() == 0
assert qapply(JyBra(1,-1)*JyKet(1,1).rewrite('Jx')) == 0
assert qapply(JyBra(1,-1)*JyKet(1,0).rewrite('Jx')).doit() == 0
assert qapply(JyBra(1,-1)*JyKet(1,1).rewrite('Jz')) == 0
assert qapply(JyBra(1,-1)*JyKet(1,0).rewrite('Jz')).doit() == 0
assert qapply(JzBra(1,-1)*JzKet(1,1).rewrite('Jx')) == 0
assert qapply(JzBra(1,-1)*JzKet(1,0).rewrite('Jx')).doit() == 0
assert qapply(JzBra(1,-1)*JzKet(1,1).rewrite('Jy')) == 0
assert qapply(JzBra(1,-1)*JzKet(1,0).rewrite('Jy')).doit() == 0
