def test_anticommutator():
A = Operator('A')
B = Operator('B')
ac = AntiCommutator(A, B)
ac_tall = AntiCommutator(A**2, B)
assert str(ac) == '{A,B}'
assert pretty(ac) == '{A,B}'
assert upretty(ac) == u'{A,B}'
assert latex(ac) == r'\left\{A,B\right\}'
sT(ac, "AntiCommutator(Operator(Symbol('A')),Operator(Symbol('B')))")
assert str(ac_tall) == '{A**2,B}'
ascii_str = \
"""\
/ 2 \\\n\
<A ,B>\n\
\\ /\
"""
ucode_str = \
u"""\
⎧ 2 ⎫\n\
⎨A ,B⎬\n\
⎩ ⎭\
"""
assert pretty(ac_tall) == ascii_str
assert upretty(ac_tall) == ucode_str
assert latex(ac_tall) == r'\left\{\left(A\right)^{2},B\right\}'
sT(
ac_tall,
"AntiCommutator(Pow(Operator(Symbol('A')), Integer(2)),Operator(Symbol('B')))"
)
def test_anticommutator():
A = Operator('A')
B = Operator('B')
ac = AntiCommutator(A, B)
ac_tall = AntiCommutator(A**2, B)
assert str(ac) == '{A,B}'
assert pretty(ac) == '{A,B}'
assert upretty(ac) == '{A,B}'
assert latex(ac) == r'\left\{A,B\right\}'
sT(ac, "AntiCommutator(Operator(Symbol('A')),Operator(Symbol('B')))")
assert str(ac_tall) == '{A**2,B}'
ascii_str = \
"""\
/ 2 \\\n\
<A ,B>\n\
\\ /\
"""
ucode_str = \
"""\
⎧ 2 ⎫\n\
⎨A ,B⎬\n\
⎩ ⎭\
"""
assert pretty(ac_tall) == ascii_str
assert upretty(ac_tall) == ucode_str
#FIXME ajgpitch 2019-09-22:
# It's not clear to me why this would be expected
# assert latex(ac_tall) == r'\left\{\left(A\right)^{2},B\right\}'
# This renders fine
assert latex(ac_tall) == r'\left\{A^{2},B\right\}'
sT(
ac_tall,
"AntiCommutator(Pow(Operator(Symbol('A')), Integer(2)),Operator(Symbol('B')))"
)
def test_anticommutator():
A = Operator("A")
B = Operator("B")
ac = AntiCommutator(A, B)
ac_tall = AntiCommutator(A**2, B)
assert str(ac) == "{A,B}"
assert pretty(ac) == "{A,B}"
assert upretty(ac) == u"{A,B}"
assert latex(ac) == r"\left\{A,B\right\}"
sT(ac, "AntiCommutator(Operator(Symbol('A')),Operator(Symbol('B')))")
assert str(ac_tall) == "{A**2,B}"
ascii_str = """\
/ 2 \\\n\
<A ,B>\n\
\\ /\
"""
ucode_str = u("""\
⎧ 2 ⎫\n\
⎨A ,B⎬\n\
⎩ ⎭\
""")
assert pretty(ac_tall) == ascii_str
assert upretty(ac_tall) == ucode_str
assert latex(ac_tall) == r"\left\{A^{2},B\right\}"
sT(
ac_tall,
"AntiCommutator(Pow(Operator(Symbol('A')), Integer(2)),Operator(Symbol('B')))",
)
def test_one_qubit_anticommutators():
"""Test single qubit gate anticommutation relations."""
for g1 in (IdentityGate, X, Y, Z, H):
for g2 in (IdentityGate, X, Y, Z, H):
e = AntiCommutator(g1(0), g2(0))
a = matrix_to_zero(represent(e, nqubits=1, format='sympy'))
b = matrix_to_zero(represent(e.doit(), nqubits=1, format='sympy'))
assert a == b
e = AntiCommutator(g1(0), g2(1))
a = matrix_to_zero(represent(e, nqubits=2, format='sympy'))
b = matrix_to_zero(represent(e.doit(), nqubits=2, format='sympy'))
assert a == b
def gate_sort(circuit):
"""Sorts the gates while keeping track of commutation relations
This function uses a bubble sort to rearrange the order of gate
application. Keeps track of Quantum computations special commutation
relations (e.g. things that apply to the same Qubit do not commute with
each other)
circuit is the Mul of gates that are to be sorted.
"""
# Make sure we have an Add or Mul.
if isinstance(circuit, Add):
return sum(gate_sort(t) for t in circuit.args)
if isinstance(circuit, Pow):
return gate_sort(circuit.base) ** circuit.exp
elif isinstance(circuit, Gate):
return circuit
if not isinstance(circuit, Mul):
return circuit
changes = True
while changes:
changes = False
circ_array = circuit.args
for i in range(len(circ_array) - 1):
# Go through each element and switch ones that are in wrong order
if isinstance(circ_array[i], (Gate, Pow)) and isinstance(
circ_array[i + 1], (Gate, Pow)
):
# If we have a Pow object, look at only the base
first_base, first_exp = circ_array[i].as_base_exp()
second_base, second_exp = circ_array[i + 1].as_base_exp()
# Use sympy's hash based sorting. This is not mathematical
# sorting, but is rather based on comparing hashes of objects.
# See Basic.compare for details.
if first_base.compare(second_base) > 0:
if Commutator(first_base, second_base).doit() == 0:
new_args = (
circuit.args[:i]
+ (circuit.args[i + 1],)
+ (circuit.args[i],)
+ circuit.args[i + 2 :]
)
circuit = Mul(*new_args)
changes = True
break
if AntiCommutator(first_base, second_base).doit() == 0:
new_args = (
circuit.args[:i]
+ (circuit.args[i + 1],)
+ (circuit.args[i],)
+ circuit.args[i + 2 :]
)
sign = Integer(-1) ** (first_exp * second_exp)
circuit = sign * Mul(*new_args)
changes = True
break
return circuit
def tensor_product_simp(e, **hints):
"""Try to simplify and combine TensorProducts.
In general this will try to pull expressions inside of ``TensorProducts``.
It currently only works for relatively simple cases where the products have
only scalars, raw ``TensorProducts``, not ``Add``, ``Pow``, ``Commutators``
of ``TensorProducts``. It is best to see what it does by showing examples.
Examples
========
>>> from sympy.physics.quantum import tensor_product_simp
>>> from sympy.physics.quantum import TensorProduct
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> C = Symbol('C',commutative=False)
>>> D = Symbol('D',commutative=False)
First see what happens to products of tensor products:
>>> e = TensorProduct(A,B)*TensorProduct(C,D)
>>> e
AxB*CxD
>>> tensor_product_simp(e)
(A*C)x(B*D)
This is the core logic of this function, and it works inside, powers, sums,
commutators and anticommutators as well:
>>> tensor_product_simp(e**2)
(A*C)x(B*D)**2
"""
if isinstance(e, Add):
return Add(*[tensor_product_simp(arg) for arg in e.args])
elif isinstance(e, Pow):
if isinstance(e.base, TensorProduct):
return tensor_product_simp_Pow(e)
else:
return tensor_product_simp(e.base) ** e.exp
elif isinstance(e, Mul):
return tensor_product_simp_Mul(e)
elif isinstance(e, Commutator):
return Commutator(*[tensor_product_simp(arg) for arg in e.args])
elif isinstance(e, AntiCommutator):
return AntiCommutator(*[tensor_product_simp(arg) for arg in e.args])
else:
return e
def test_one_qubit_anticommutators():
"""Test single qubit gate anticommutation relations."""
for g1 in (IdentityGate, X, Y, Z, H):
for g2 in (IdentityGate, X, Y, Z, H):
e = AntiCommutator(g1(0), g2(0))
a = matrix_to_zero(represent(e, nqubits=1, format='sympy'))
b = matrix_to_zero(represent(e.doit(), nqubits=1, format='sympy'))
assert a == b
e = AntiCommutator(g1(0), g2(1))
a = matrix_to_zero(represent(e, nqubits=2, format='sympy'))
b = matrix_to_zero(represent(e.doit(), nqubits=2, format='sympy'))
assert a == b
def test_anticommutator():
assert qapply(AntiCommutator(Jz, Foo('F')) * po) == 2 * hbar * po
assert qapply(AntiCommutator(Foo('F'), Jz) * po) == 2 * hbar * po
def test_sympy__physics__quantum__anticommutator__AntiCommutator():
from sympy.physics.quantum.anticommutator import AntiCommutator
assert _test_args(AntiCommutator(x, y))
def test_anticommutator():
ac = AComm(A, B)
assert isinstance(ac, AComm)
assert ac.is_commutative == False
assert ac.subs(A, C) == AComm(C, B)
def test_big_expr():
f = Function('f')
x = symbols('x')
e1 = Dagger(
AntiCommutator(
Operator('A') + Operator('B'),
Pow(DifferentialOperator(Derivative(f(x), x), f(x)), 3)) *
TensorProduct(Jz**2,
Operator('A') + Operator('B'))) * (JzBra(1, 0) + JzBra(
1, 1)) * (JzKet(0, 0) + JzKet(1, -1))
e2 = Commutator(Jz**2,
Operator('A') + Operator('B')) * AntiCommutator(
Dagger(Operator('C') * Operator('D')),
Operator('E').inv()**2) * Dagger(Commutator(Jz, J2))
e3 = Wigner3j(1, 2, 3, 4, 5, 6) * TensorProduct(
Commutator(
Operator('A') + Dagger(Operator('B')),
Operator('C') + Operator('D')), Jz - J2) * Dagger(
OuterProduct(Dagger(JzBra(1, 1)), JzBra(
1, 0))) * TensorProduct(
JzKetCoupled(1, 1,
(1, 1)) + JzKetCoupled(1, 0, (1, 1)),
JzKetCoupled(1, -1, (1, 1)))
e4 = (ComplexSpace(1) * ComplexSpace(2) +
FockSpace()**2) * (L2(Interval(0, oo)) + HilbertSpace())
assert str(
e1
) == '(Jz**2)x(Dagger(A) + Dagger(B))*{Dagger(DifferentialOperator(Derivative(f(x), x),f(x)))**3,Dagger(A) + Dagger(B)}*(<1,0| + <1,1|)*(|0,0> + |1,-1>)'
ascii_str = \
"""\
/ 3 \\ \n\
|/ +\\ | \n\
2 / + +\\ <| /d \\ | + +> \n\
/J \\ x \\A + B /*||DifferentialOperator|--(f(x)),f(x)| | ,A + B |*(<1,0| + <1,1|)*(|0,0> + |1,-1>)\n\
\\ z/ \\\\ \dx / / / \
"""
ucode_str = \
u"""\
⎧ 3 ⎫ \n\
⎪⎛ †⎞ ⎪ \n\
2 ⎛ † †⎞ ⎨⎜ ⎛d ⎞ ⎟ † †⎬ \n\
⎛J ⎞ ⨂ ⎝A + B ⎠⋅⎪⎜DifferentialOperator⎜──(f(x)),f(x)⎟ ⎟ ,A + B ⎪⋅(⟨1,0❘ + ⟨1,1❘)⋅(❘0,0⟩ + ❘1,-1⟩)\n\
⎝ z⎠ ⎩⎝ ⎝dx ⎠ ⎠ ⎭ \
"""
assert pretty(e1) == ascii_str
assert upretty(e1) == ucode_str
assert latex(e1) == \
r'{\left(J_z\right)^{2}}\otimes \left({A^{\dag} + B^{\dag}}\right) \left\{\left(DifferentialOperator\left(\frac{\partial}{\partial x} \operatorname{f}{\left (x \right )},\operatorname{f}{\left (x \right )}\right)^{\dag}\right)^{3},A^{\dag} + B^{\dag}\right\} \left({\left\langle 1,0\right|} + {\left\langle 1,1\right|}\right) \left({\left|0,0\right\rangle } + {\left|1,-1\right\rangle }\right)'
sT(
e1,
"Mul(TensorProduct(Pow(JzOp(Symbol('J')), Integer(2)), Add(Dagger(Operator(Symbol('A'))), Dagger(Operator(Symbol('B'))))), AntiCommutator(Pow(Dagger(DifferentialOperator(Derivative(Function('f')(Symbol('x')), Symbol('x')),Function('f')(Symbol('x')))), Integer(3)),Add(Dagger(Operator(Symbol('A'))), Dagger(Operator(Symbol('B'))))), Add(JzBra(Integer(1),Integer(0)), JzBra(Integer(1),Integer(1))), Add(JzKet(Integer(0),Integer(0)), JzKet(Integer(1),Integer(-1))))"
)
assert str(e2) == '[Jz**2,A + B]*{E**(-2),Dagger(D)*Dagger(C)}*[J2,Jz]'
ascii_str = \
"""\
[ 2 ] / -2 + +\\ [ 2 ]\n\
[/J \\ ,A + B]*<E ,D *C >*[J ,J ]\n\
[\\ z/ ] \\ / [ z]\
"""
ucode_str = \
u"""\
⎡ 2 ⎤ ⎧ -2 † †⎫ ⎡ 2 ⎤\n\
⎢⎛J ⎞ ,A + B⎥⋅⎨E ,D ⋅C ⎬⋅⎢J ,J ⎥\n\
⎣⎝ z⎠ ⎦ ⎩ ⎭ ⎣ z⎦\
"""
assert pretty(e2) == ascii_str
assert upretty(e2) == ucode_str
assert latex(e2) == \
r'\left[\left(J_z\right)^{2},A + B\right] \left\{\left(E\right)^{-2},D^{\dag} C^{\dag}\right\} \left[J^2,J_z\right]'
sT(
e2,
"Mul(Commutator(Pow(JzOp(Symbol('J')), Integer(2)),Add(Operator(Symbol('A')), Operator(Symbol('B')))), AntiCommutator(Pow(Operator(Symbol('E')), Integer(-2)),Mul(Dagger(Operator(Symbol('D'))), Dagger(Operator(Symbol('C'))))), Commutator(J2Op(Symbol('J')),JzOp(Symbol('J'))))"
)
assert str(e3) == \
"Wigner3j(1, 2, 3, 4, 5, 6)*[Dagger(B) + A,C + D]x(-J2 + Jz)*|1,0><1,1|*(|1,0,j1=1,j2=1> + |1,1,j1=1,j2=1>)x|1,-1,j1=1,j2=1>"
ascii_str = \
"""\
[ + ] / 2 \\ \n\
/1 3 5\\*[B + A,C + D]x |- J + J |*|1,0><1,1|*(|1,0,j1=1,j2=1> + |1,1,j1=1,j2=1>)x |1,-1,j1=1,j2=1>\n\
| | \\ z/ \n\
\\2 4 6/ \
"""
ucode_str = \
u"""\
⎡ † ⎤ ⎛ 2 ⎞ \n\
⎛1 3 5⎞⋅⎣B + A,C + D⎦⨂ ⎜- J + J ⎟⋅❘1,0⟩⟨1,1❘⋅(❘1,0,j₁=1,j₂=1⟩ + ❘1,1,j₁=1,j₂=1⟩)⨂ ❘1,-1,j₁=1,j₂=1⟩\n\
⎜ ⎟ ⎝ z⎠ \n\
⎝2 4 6⎠ \
"""
assert pretty(e3) == ascii_str
assert upretty(e3) == ucode_str
assert latex(e3) == \
r'\left(\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right) {\left[B^{\dag} + A,C + D\right]}\otimes \left({- J^2 + J_z}\right) {\left|1,0\right\rangle }{\left\langle 1,1\right|} \left({{\left|1,0,j_{1}=1,j_{2}=1\right\rangle } + {\left|1,1,j_{1}=1,j_{2}=1\right\rangle }}\right)\otimes {{\left|1,-1,j_{1}=1,j_{2}=1\right\rangle }}'
sT(
e3,
"Mul(Wigner3j(Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6)), TensorProduct(Commutator(Add(Dagger(Operator(Symbol('B'))), Operator(Symbol('A'))),Add(Operator(Symbol('C')), Operator(Symbol('D')))), Add(Mul(Integer(-1), J2Op(Symbol('J'))), JzOp(Symbol('J')))), OuterProduct(JzKet(Integer(1),Integer(0)),JzBra(Integer(1),Integer(1))), TensorProduct(Add(JzKetCoupled(Integer(1),Integer(0),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1)))), JzKetCoupled(Integer(1),Integer(1),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1))))), JzKetCoupled(Integer(1),Integer(-1),Tuple(Integer(1), Integer(1)),Tuple(Tuple(Integer(1), Integer(2), Integer(1))))))"
)
assert str(e4) == '(C(1)*C(2)+F**2)*(L2([0, oo))+H)'
ascii_str = \
"""\
// 1 2\\ x2\\ / 2 \\\n\
\\\\C x C / + F / x \L + H/\
"""
ucode_str = \
u"""\
⎛⎛ 1 2⎞ ⨂2⎞ ⎛ 2 ⎞\n\
⎝⎝C ⨂ C ⎠ ⊕ F ⎠ ⨂ ⎝L ⊕ H⎠\
"""
assert pretty(e4) == ascii_str
assert upretty(e4) == ucode_str
assert latex(e4) == \
r'\left(\left(\mathcal{C}^{1}\otimes \mathcal{C}^{2}\right)\oplus {\mathcal{F}}^{\otimes 2}\right)\otimes \left({\mathcal{L}^2}\left( \left[0, \infty\right) \right)\oplus \mathcal{H}\right)'
sT(
e4,
"TensorProductHilbertSpace((DirectSumHilbertSpace(TensorProductHilbertSpace(ComplexSpace(Integer(1)),ComplexSpace(Integer(2))),TensorPowerHilbertSpace(FockSpace(),Integer(2)))),(DirectSumHilbertSpace(L2(Interval(Integer(0), oo, False, True)),HilbertSpace())))"
)
(Dagger(k), Dagger(Avec)),
# Operator
(A, Amat),
(Dagger(A), Dagger(Amat)),
# OuterProduct
(OuterProduct(k, b), Avec * Avec.H),
# TensorProduct
(TensorProduct(A, B), matrix_tensor_product(Amat, Bmat)),
# Pow
(A**2, Amat**2),
# Add/Mul
(A * B + 2 * A, Amat * Bmat + 2 * Amat),
# Commutator
(Commutator(A, B), Amat * Bmat - Bmat * Amat),
# AntiCommutator
(AntiCommutator(A, B), Amat * Bmat + Bmat * Amat),
# InnerProduct
(InnerProduct(b, k), (Avec.H * Avec)[0])
]
def test_format_sympy():
for test in _tests:
lhs = represent(test[0], basis=A, format='sympy')
rhs = to_sympy(test[1])
assert lhs == rhs
def test_scalar_sympy():
assert represent(Integer(1)) == Integer(1)
assert represent(Float(1.0)) == Float(1.0)
def test_anticommutator():
ac = AComm(A,B)
assert isinstance(ac, AComm)
assert ac.is_commutative == False
assert ac.subs(A,C) == AComm(C,B)
