# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2019.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""
Stinespring representation of a Quantum Channel.
"""
from __future__ import annotations
import copy
from numbers import Number
import numpy as np
from qiskit.circuit.quantumcircuit import QuantumCircuit
from qiskit.circuit.instruction import Instruction
from qiskit.exceptions import QiskitError
from qiskit.quantum_info.operators.predicates import is_identity_matrix
from qiskit.quantum_info.operators.channel.quantum_channel import QuantumChannel
from qiskit.quantum_info.operators.op_shape import OpShape
from qiskit.quantum_info.operators.channel.kraus import Kraus
from qiskit.quantum_info.operators.channel.choi import Choi
from qiskit.quantum_info.operators.channel.superop import SuperOp
from qiskit.quantum_info.operators.channel.transformations import _to_stinespring
from qiskit.quantum_info.operators.mixins import generate_apidocs
from qiskit.quantum_info.operators.base_operator import BaseOperator
[documentos]class Stinespring(QuantumChannel):
r"""Stinespring representation of a quantum channel.
The Stinespring representation of a quantum channel :math:`\mathcal{E}`
is a rectangular matrix :math:`A` such that the evolution of a
:class:`~qiskit.quantum_info.DensityMatrix` :math:`\rho` is given by
.. math::
\mathcal{E}(蟻) = \mbox{Tr}_2\left[A 蟻 A^\dagger\right]
where :math:`\mbox{Tr}_2` is the :func:`partial_trace` over subsystem 2.
A general operator map :math:`\mathcal{G}` can also be written using the
generalized Stinespring representation which is given by two matrices
:math:`A`, :math:`B` such that
.. math::
\mathcal{G}(蟻) = \mbox{Tr}_2\left[A 蟻 B^\dagger\right]
See reference [1] for further details.
References:
1. C.J. Wood, J.D. Biamonte, D.G. Cory, *Tensor networks and graphical calculus
for open quantum systems*, Quant. Inf. Comp. 15, 0579-0811 (2015).
`arXiv:1111.6950 [quant-ph] <https://arxiv.org/abs/1111.6950>`_
"""
def __init__(
self,
data: QuantumCircuit | Instruction | BaseOperator | np.ndarray,
input_dims: int | tuple | None = None,
output_dims: int | tuple | None = None,
):
"""Initialize a quantum channel Stinespring operator.
Args:
data (QuantumCircuit or
Instruction or
BaseOperator or
matrix): data to initialize superoperator.
input_dims (tuple): the input subsystem dimensions.
[Default: None]
output_dims (tuple): the output subsystem dimensions.
[Default: None]
Raises:
QiskitError: if input data cannot be initialized as a
a list of Kraus matrices.
Additional Information:
If the input or output dimensions are None, they will be
automatically determined from the input data. This can fail for the
Stinespring operator if the output dimension cannot be automatically
determined.
"""
# If the input is a list or tuple we assume it is a pair of general
# Stinespring matrices. If it is a numpy array we assume that it is
# a single Stinespring matrix.
if isinstance(data, (list, tuple, np.ndarray)):
if not isinstance(data, tuple):
# Convert single Stinespring set to length 1 tuple
stine = (np.asarray(data, dtype=complex), None)
if isinstance(data, tuple) and len(data) == 2:
if data[1] is None:
stine = (np.asarray(data[0], dtype=complex), None)
else:
stine = (np.asarray(data[0], dtype=complex), np.asarray(data[1], dtype=complex))
dim_left, dim_right = stine[0].shape
# If two Stinespring matrices check they are same shape
if stine[1] is not None:
if stine[1].shape != (dim_left, dim_right):
raise QiskitError("Invalid Stinespring input.")
input_dim = dim_right
if output_dims:
output_dim = np.prod(output_dims)
else:
output_dim = input_dim
if dim_left % output_dim != 0:
raise QiskitError("Invalid output_dim")
op_shape = OpShape.auto(
dims_l=output_dims, dims_r=input_dims, shape=(output_dim, input_dim)
)
else:
# Otherwise we initialize by conversion from another Qiskit
# object into the QuantumChannel.
if isinstance(data, (QuantumCircuit, Instruction)):
# If the input is a Terra QuantumCircuit or Instruction we
# convert it to a SuperOp
data = SuperOp._init_instruction(data)
else:
# We use the QuantumChannel init transform to intialize
# other objects into a QuantumChannel or Operator object.
data = self._init_transformer(data)
op_shape = data._op_shape
output_dim, input_dim = op_shape.shape
# Now that the input is an operator we convert it to a
# Stinespring operator
rep = getattr(data, "_channel_rep", "Operator")
stine = _to_stinespring(rep, data._data, input_dim, output_dim)
# Initialize either single or general Stinespring
if stine[1] is None or (stine[1] == stine[0]).all():
# Standard Stinespring map
data = (stine[0], None)
else:
# General (non-CPTP) Stinespring map
data = stine
super().__init__(data, op_shape=op_shape)
@property
def data(self):
# Override to deal with data being either tuple or not
if self._data[1] is None:
return self._data[0]
else:
return self._data
[documentos] def is_cptp(self, atol=None, rtol=None):
"""Return True if completely-positive trace-preserving."""
if atol is None:
atol = self.atol
if rtol is None:
rtol = self.rtol
if self._data[1] is not None:
return False
check = np.dot(np.transpose(np.conj(self._data[0])), self._data[0])
return is_identity_matrix(check, rtol=self.rtol, atol=self.atol)
def _evolve(self, state, qargs=None):
return SuperOp(self)._evolve(state, qargs)
# ---------------------------------------------------------------------
# BaseOperator methods
# ---------------------------------------------------------------------
[documentos] def conjugate(self):
ret = copy.copy(self)
stine_l = np.conjugate(self._data[0])
stine_r = None
if self._data[1] is not None:
stine_r = np.conjugate(self._data[1])
ret._data = (stine_l, stine_r)
return ret
[documentos] def transpose(self):
ret = copy.copy(self)
ret._op_shape = self._op_shape.transpose()
din, dout = self.dim
dtr = self._data[0].shape[0] // dout
stine = [None, None]
for i, mat in enumerate(self._data):
if mat is not None:
stine[i] = np.reshape(
np.transpose(np.reshape(mat, (dout, dtr, din)), (2, 1, 0)), (din * dtr, dout)
)
ret._data = (stine[0], stine[1])
return ret
[documentos] def compose(
self, other: Stinespring, qargs: list | None = None, front: bool = False
) -> Stinespring:
if qargs is None:
qargs = getattr(other, "qargs", None)
if qargs is not None:
return Stinespring(SuperOp(self).compose(other, qargs=qargs, front=front))
# Otherwise we convert via Kraus representation rather than
# superoperator to avoid unnecessary representation conversions
return Stinespring(Kraus(self).compose(other, front=front))
[documentos] def tensor(self, other: Stinespring) -> Stinespring:
if not isinstance(other, Stinespring):
other = Stinespring(other)
return self._tensor(self, other)
[documentos] def expand(self, other: Stinespring) -> Stinespring:
if not isinstance(other, Stinespring):
other = Stinespring(other)
return self._tensor(other, self)
@classmethod
def _tensor(cls, a, b):
# Tensor Stinespring ops
sa_l, sa_r = a._data
sb_l, sb_r = b._data
# Reshuffle tensor dimensions
din_a, dout_a = a.dim
din_b, dout_b = b.dim
dtr_a = sa_l.shape[0] // dout_a
dtr_b = sb_l.shape[0] // dout_b
shape_in = (dout_a, dtr_a, dout_b, dtr_b, din_a * din_b)
shape_out = (dout_a * dtr_a * dout_b * dtr_b, din_a * din_b)
sab_l = np.kron(sa_l, sb_l)
# Reravel indices
sab_l = np.reshape(np.transpose(np.reshape(sab_l, shape_in), (0, 2, 1, 3, 4)), shape_out)
# Compute right Stinespring op
if sa_r is None and sb_r is None:
sab_r = None
else:
if sa_r is None:
sa_r = sa_l
elif sb_r is None:
sb_r = sb_l
sab_r = np.kron(sa_r, sb_r)
# Reravel indices
sab_r = np.reshape(
np.transpose(np.reshape(sab_r, shape_in), (0, 2, 1, 3, 4)), shape_out
)
ret = copy.copy(a)
ret._op_shape = a._op_shape.tensor(b._op_shape)
ret._data = (sab_l, sab_r)
return ret
def __add__(self, other):
qargs = getattr(other, "qargs", None)
if not isinstance(other, QuantumChannel):
other = Choi(other)
return self._add(other, qargs=qargs)
def __sub__(self, other):
qargs = getattr(other, "qargs", None)
if not isinstance(other, QuantumChannel):
other = Choi(other)
return self._add(-other, qargs=qargs)
def _add(self, other, qargs=None):
# Since we cannot directly add two channels in the Stinespring
# representation we convert to the Choi representation
return Stinespring(Choi(self)._add(other, qargs=qargs))
def _multiply(self, other):
if not isinstance(other, Number):
raise QiskitError("other is not a number")
ret = copy.copy(self)
# If the number is complex or negative we need to convert to
# general Stinespring representation so we first convert to
# the Choi representation
if isinstance(other, complex) or other < 1:
# Convert to Choi-matrix
ret._data = Stinespring(Choi(self)._multiply(other))._data
return ret
# If the number is real we can update the Kraus operators
# directly
num = np.sqrt(other)
stine_l, stine_r = self._data
stine_l = num * self._data[0]
stine_r = None
if self._data[1] is not None:
stine_r = num * self._data[1]
ret._data = (stine_l, stine_r)
return ret
# Update docstrings for API docs
generate_apidocs(Stinespring)