"""Abstract tensor product."""
from __future__ import print_function, division
from sympy import Expr, Add, Mul, Matrix, Pow, sympify
from sympy.core.compatibility import range
from sympy.core.trace import Tr
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.state import Ket, Bra
from sympy.physics.quantum.matrixutils import (
numpy_ndarray,
scipy_sparse_matrix,
matrix_tensor_product
)
__all__ = [
'TensorProduct',
'tensor_product_simp'
]
#-----------------------------------------------------------------------------
# Tensor product
#-----------------------------------------------------------------------------
_combined_printing = False
def combined_tensor_printing(combined):
"""Set flag controlling whether tensor products of states should be
printed as a combined bra/ket or as an explicit tensor product of different
bra/kets. This is a global setting for all TensorProduct class instances.
Parameters
----------
combine : bool
When true, tensor product states are combined into one ket/bra, and
when false explicit tensor product notation is used between each
ket/bra.
"""
global _combined_printing
_combined_printing = combined
[docs]class TensorProduct(Expr):
"""The tensor product of two or more arguments.
For matrices, this uses ``matrix_tensor_product`` to compute the Kronecker
or tensor product matrix. For other objects a symbolic ``TensorProduct``
instance is returned. The tensor product is a non-commutative
multiplication that is used primarily with operators and states in quantum
mechanics.
Currently, the tensor product distinguishes between commutative and
non-commutative arguments. Commutative arguments are assumed to be scalars
and are pulled out in front of the ``TensorProduct``. Non-commutative
arguments remain in the resulting ``TensorProduct``.
Parameters
==========
args : tuple
A sequence of the objects to take the tensor product of.
Examples
========
Start with a simple tensor product of sympy matrices::
>>> from sympy import I, Matrix, symbols
>>> from sympy.physics.quantum import TensorProduct
>>> m1 = Matrix([[1,2],[3,4]])
>>> m2 = Matrix([[1,0],[0,1]])
>>> TensorProduct(m1, m2)
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2],
[3, 0, 4, 0],
[0, 3, 0, 4]])
>>> TensorProduct(m2, m1)
Matrix([
[1, 2, 0, 0],
[3, 4, 0, 0],
[0, 0, 1, 2],
[0, 0, 3, 4]])
We can also construct tensor products of non-commutative symbols:
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> tp = TensorProduct(A, B)
>>> tp
AxB
We can take the dagger of a tensor product (note the order does NOT reverse
like the dagger of a normal product):
>>> from sympy.physics.quantum import Dagger
>>> Dagger(tp)
Dagger(A)xDagger(B)
Expand can be used to distribute a tensor product across addition:
>>> C = Symbol('C',commutative=False)
>>> tp = TensorProduct(A+B,C)
>>> tp
(A + B)xC
>>> tp.expand(tensorproduct=True)
AxC + BxC
"""
is_commutative = False
def __new__(cls, *args):
if isinstance(args[0], (Matrix, numpy_ndarray, scipy_sparse_matrix)):
return matrix_tensor_product(*args)
c_part, new_args = cls.flatten(sympify(args))
c_part = Mul(*c_part)
if len(new_args) == 0:
return c_part
elif len(new_args) == 1:
return c_part * new_args[0]
else:
tp = Expr.__new__(cls, *new_args)
return c_part * tp
@classmethod
def flatten(cls, args):
# TODO: disallow nested TensorProducts.
c_part = []
nc_parts = []
for arg in args:
cp, ncp = arg.args_cnc()
c_part.extend(list(cp))
nc_parts.append(Mul._from_args(ncp))
return c_part, nc_parts
def _eval_adjoint(self):
return TensorProduct(*[Dagger(i) for i in self.args])
def _eval_rewrite(self, pattern, rule, **hints):
sargs = self.args
terms = [t._eval_rewrite(pattern, rule, **hints) for t in sargs]
return TensorProduct(*terms).expand(tensorproduct=True)
def _sympystr(self, printer, *args):
from sympy.printing.str import sstr
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + '('
s = s + sstr(self.args[i])
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + ')'
if i != length - 1:
s = s + 'x'
return s
def _pretty(self, printer, *args):
if (_combined_printing and
(all([isinstance(arg, Ket) for arg in self.args]) or
all([isinstance(arg, Bra) for arg in self.args]))):
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print('', *args)
length_i = len(self.args[i].args)
for j in range(length_i):
part_pform = printer._print(self.args[i].args[j], *args)
next_pform = prettyForm(*next_pform.right(part_pform))
if j != length_i - 1:
next_pform = prettyForm(*next_pform.right(', '))
if len(self.args[i].args) > 1:
next_pform = prettyForm(
*next_pform.parens(left='{', right='}'))
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
pform = prettyForm(*pform.right(',' + ' '))
pform = prettyForm(*pform.left(self.args[0].lbracket))
pform = prettyForm(*pform.right(self.args[0].rbracket))
return pform
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print(self.args[i], *args)
if isinstance(self.args[i], (Add, Mul)):
next_pform = prettyForm(
*next_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
if printer._use_unicode:
pform = prettyForm(*pform.right(u'\N{N-ARY CIRCLED TIMES OPERATOR}' + u' '))
else:
pform = prettyForm(*pform.right('x' + ' '))
return pform
def _latex(self, printer, *args):
if (_combined_printing and
(all([isinstance(arg, Ket) for arg in self.args]) or
all([isinstance(arg, Bra) for arg in self.args]))):
def _label_wrap(label, nlabels):
return label if nlabels == 1 else r"\left\{%s\right\}" % label
s = r", ".join([_label_wrap(arg._print_label_latex(printer, *args),
len(arg.args)) for arg in self.args])
return r"{%s%s%s}" % (self.args[0].lbracket_latex, s,
self.args[0].rbracket_latex)
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\left('
# The extra {} brackets are needed to get matplotlib's latex
# rendered to render this properly.
s = s + '{' + printer._print(self.args[i], *args) + '}'
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\right)'
if i != length - 1:
s = s + '\\otimes '
return s
def doit(self, **hints):
return TensorProduct(*[item.doit(**hints) for item in self.args])
def _eval_expand_tensorproduct(self, **hints):
"""Distribute TensorProducts across addition."""
args = self.args
add_args = []
stop = False
for i in range(len(args)):
if isinstance(args[i], Add):
for aa in args[i].args:
tp = TensorProduct(*args[:i] + (aa,) + args[i + 1:])
if isinstance(tp, TensorProduct):
tp = tp._eval_expand_tensorproduct()
add_args.append(tp)
break
if add_args:
return Add(*add_args)
else:
return self
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', None)
exp = tensor_product_simp(self)
if indices is None or len(indices) == 0:
return Mul(*[Tr(arg).doit() for arg in exp.args])
else:
return Mul(*[Tr(value).doit() if idx in indices else value
for idx, value in enumerate(exp.args)])
def tensor_product_simp_Mul(e):
"""Simplify a Mul with TensorProducts.
Current the main use of this is to simplify a ``Mul`` of ``TensorProduct``s
to a ``TensorProduct`` of ``Muls``. It currently only works for relatively
simple cases where the initial ``Mul`` only has scalars and raw
``TensorProduct``s, not ``Add``, ``Pow``, ``Commutator``s of
``TensorProduct``s.
Parameters
==========
e : Expr
A ``Mul`` of ``TensorProduct``s to be simplified.
Returns
=======
e : Expr
A ``TensorProduct`` of ``Mul``s.
Examples
========
This is an example of the type of simplification that this function
performs::
>>> from sympy.physics.quantum.tensorproduct import \
tensor_product_simp_Mul, 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)
>>> e = TensorProduct(A,B)*TensorProduct(C,D)
>>> e
AxB*CxD
>>> tensor_product_simp_Mul(e)
(A*C)x(B*D)
"""
# TODO: This won't work with Muls that have other composites of
# TensorProducts, like an Add, Commutator, etc.
# TODO: This only works for the equivalent of single Qbit gates.
if not isinstance(e, Mul):
return e
c_part, nc_part = e.args_cnc()
n_nc = len(nc_part)
if n_nc == 0:
return e
elif n_nc == 1:
if isinstance(nc_part[0], Pow):
return Mul(*c_part) * tensor_product_simp_Pow(nc_part[0])
return e
elif e.has(TensorProduct):
current = nc_part[0]
if not isinstance(current, TensorProduct):
if isinstance(current, Pow):
if isinstance(current.base, TensorProduct):
current = tensor_product_simp_Pow(current)
else:
raise TypeError('TensorProduct expected, got: %r' % current)
n_terms = len(current.args)
new_args = list(current.args)
for next in nc_part[1:]:
# TODO: check the hilbert spaces of next and current here.
if isinstance(next, TensorProduct):
if n_terms != len(next.args):
raise QuantumError(
'TensorProducts of different lengths: %r and %r' %
(current, next)
)
for i in range(len(new_args)):
new_args[i] = new_args[i] * next.args[i]
else:
if isinstance(next, Pow):
if isinstance(next.base, TensorProduct):
new_tp = tensor_product_simp_Pow(next)
for i in range(len(new_args)):
new_args[i] = new_args[i] * new_tp.args[i]
else:
raise TypeError('TensorProduct expected, got: %r' % next)
else:
raise TypeError('TensorProduct expected, got: %r' % next)
current = next
return Mul(*c_part) * TensorProduct(*new_args)
elif e.has(Pow):
new_args = [ tensor_product_simp_Pow(nc) for nc in nc_part ]
return tensor_product_simp_Mul(Mul(*c_part) * TensorProduct(*new_args))
else:
return e
def tensor_product_simp_Pow(e):
"""Evaluates ``Pow`` expressions whose base is ``TensorProduct``"""
if not isinstance(e, Pow):
return e
if isinstance(e.base, TensorProduct):
return TensorProduct(*[ b**e.exp for b in e.base.args])
else:
return e
[docs]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
