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# 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
# 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.

# pylint: disable=unpacking-non-sequence

Superoperator representation of a Quantum Channel."""

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.operator import Operator
from import QuantumChannel
from import _to_superop
from import _bipartite_tensor

[문서]class SuperOp(QuantumChannel): r"""Superoperator representation of a quantum channel. The Superoperator representation of a quantum channel :math:`\mathcal{E}` is a matrix :math:`S` such that the evolution of a :class:`~qiskit.quantum_info.DensityMatrix` :math:`\rho` is given by .. math:: |\mathcal{E}(\rho)\rangle\!\rangle = S |\rho\rangle\!\rangle where the double-ket notation :math:`|A\rangle\!\rangle` denotes a vector formed by stacking the columns of the matrix :math:`A` *(column-vectorization)*. 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] <>`_ """
[문서] def __init__(self, data, input_dims=None, output_dims=None): """Initialize a quantum channel Superoperator 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 superoperator. Additional Information: If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a Numpy array of shape (4**N, 4**N) qubit systems will be used. If the input operator is not an N-qubit operator, it will assign a single subsystem with dimension specified by the shape of the input. """ # If the input is a raw list or matrix we assume that it is # already a superoperator. if isinstance(data, (list, np.ndarray)): # We initialize directly from superoperator matrix super_mat = np.asarray(data, dtype=complex) # Determine total input and output dimensions dout, din = super_mat.shape input_dim = int(np.sqrt(din)) output_dim = int(np.sqrt(dout)) if output_dim**2 != dout or input_dim**2 != din: raise QiskitError("Invalid shape for SuperOp matrix.") 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 # perform a simulation to construct the circuit superoperator. # This will only work if the circuit or instruction can be # defined in terms of instructions which have no classical # register components. The instructions can be gates, reset, # or Kraus instructions. Any conditional gates or measure # will cause an exception to be raised. data = self._init_instruction(data) else: # We use the QuantumChannel init transform to initialize # other objects into a QuantumChannel or Operator object. data = self._init_transformer(data) # Now that the input is an operator we convert it to a # SuperOp object input_dim, output_dim = data.dim rep = getattr(data, '_channel_rep', 'Operator') super_mat = _to_superop(rep, data._data, input_dim, output_dim) if input_dims is None: input_dims = data.input_dims() if output_dims is None: output_dims = data.output_dims() # Finally we format and validate the channel input and # output dimensions input_dims = self._automatic_dims(input_dims, input_dim) output_dims = self._automatic_dims(output_dims, output_dim) super().__init__(super_mat, input_dims, output_dims, 'SuperOp')
@property def _shape(self): """Return the tensor shape of the superoperator matrix""" return 2 * tuple(reversed(self.output_dims())) + 2 * tuple( reversed(self.input_dims())) @property def _bipartite_shape(self): """Return the shape for bipartite matrix""" return (self._output_dim, self._output_dim, self._input_dim, self._input_dim)
[문서] def conjugate(self): """Return the conjugate of the QuantumChannel.""" return SuperOp(np.conj(self._data), self.input_dims(), self.output_dims())
[문서] def transpose(self): """Return the transpose of the QuantumChannel.""" return SuperOp(np.transpose(self._data), input_dims=self.output_dims(), output_dims=self.input_dims())
[문서] def compose(self, other, qargs=None, front=False): """Return the composed quantum channel self @ other. Args: other (QuantumChannel): a quantum channel. qargs (list or None): a list of subsystem positions to apply other on. If None apply on all subsystems [default: None]. front (bool): If True compose using right operator multiplication, instead of left multiplication [default: False]. Returns: SuperOp: The quantum channel self @ other. Raises: QiskitError: if other has incompatible dimensions. Additional Information: Composition (``@``) is defined as `left` matrix multiplication for :class:`SuperOp` matrices. That is that ``A @ B`` is equal to ``B * A``. Setting ``front=True`` returns `right` matrix multiplication ``A * B`` and is equivalent to the :meth:`dot` method. """ if qargs is None: qargs = getattr(other, 'qargs', None) # Convert other to SuperOp if not isinstance(other, SuperOp): other = SuperOp(other) # Validate dimensions are compatible and return the composed # operator dimensions input_dims, output_dims = self._get_compose_dims( other, qargs, front) # Full composition of superoperators if qargs is None: if front: data =, else: data =, self._data) return SuperOp(data, input_dims, output_dims) # Compute tensor contraction indices from qargs if front: num_indices = len(self._input_dims) shift = 2 * len(self._output_dims) right_mul = True else: num_indices = len(self._output_dims) shift = 0 right_mul = False # Reshape current matrix # Note that we must reverse the subsystem dimension order as # qubit 0 corresponds to the right-most position in the tensor # product, which is the last tensor wire index. tensor = np.reshape(, self._shape) mat = np.reshape(, other._shape) # Add first set of indices indices = [2 * num_indices - 1 - qubit for qubit in qargs ] + [num_indices - 1 - qubit for qubit in qargs] final_shape = [np.product(output_dims)**2, np.product(input_dims)**2] data = np.reshape( Operator._einsum_matmul(tensor, mat, indices, shift, right_mul), final_shape) return SuperOp(data, input_dims, output_dims)
[문서] def power(self, n): """Return the compose of a QuantumChannel with itself n times. Args: n (int): compute the matrix power of the superoperator matrix. Returns: SuperOp: the n-times composition channel as a SuperOp object. Raises: QiskitError: if the input and output dimensions of the QuantumChannel are not equal, or the power is not an integer. """ if not isinstance(n, (int, np.integer)): raise QiskitError("Can only power with integer powers.") if self._input_dim != self._output_dim: raise QiskitError("Can only power with input_dim = output_dim.") # Override base class power so we can implement more efficiently # using Numpy.matrix_power return SuperOp(np.linalg.matrix_power(self._data, n), self.input_dims(), self.output_dims())
[문서] def tensor(self, other): """Return the tensor product channel self ⊗ other. Args: other (QuantumChannel): a quantum channel. Returns: SuperOp: the tensor product channel self ⊗ other as a SuperOp object. Raises: QiskitError: if other cannot be converted to a channel. """ # Convert other to SuperOp if not isinstance(other, SuperOp): other = SuperOp(other) input_dims = other.input_dims() + self.input_dims() output_dims = other.output_dims() + self.output_dims() data = _bipartite_tensor(self._data,, shape1=self._bipartite_shape, shape2=other._bipartite_shape) return SuperOp(data, input_dims, output_dims)
[문서] def expand(self, other): """Return the tensor product channel other ⊗ self. Args: other (QuantumChannel): a quantum channel. Returns: SuperOp: the tensor product channel other ⊗ self as a SuperOp object. Raises: QiskitError: if other cannot be converted to a channel. """ # Convert other to SuperOp if not isinstance(other, SuperOp): other = SuperOp(other) input_dims = self.input_dims() + other.input_dims() output_dims = self.output_dims() + other.output_dims() data = _bipartite_tensor(, self._data, shape1=other._bipartite_shape, shape2=self._bipartite_shape) return SuperOp(data, input_dims, output_dims)
def _evolve(self, state, qargs=None): """Evolve a quantum state by the quantum channel. Args: state (DensityMatrix or Statevector): The input state. qargs (list): a list of quantum state subsystem positions to apply the quantum channel on. Returns: DensityMatrix: the output quantum state as a density matrix. Raises: QiskitError: if the quantum channel dimension does not match the specified quantum state subsystem dimensions. """ # Prevent cyclic imports by importing DensityMatrix here # pylint: disable=cyclic-import from qiskit.quantum_info.states.densitymatrix import DensityMatrix if not isinstance(state, DensityMatrix): state = DensityMatrix(state) if qargs is None: # Evolution on full matrix if state._dim != self._input_dim: raise QiskitError( "Operator input dimension is not equal to density matrix dimension." ) # We reshape in column-major vectorization (Fortran order in Numpy) # since that is how the SuperOp is defined vec = np.ravel(, order='F') mat = np.reshape(, vec), (self._output_dim, self._output_dim), order='F') return DensityMatrix(mat, dims=self.output_dims()) # Otherwise we are applying an operator only to subsystems # Check dimensions of subsystems match the operator if state.dims(qargs) != self.input_dims(): raise QiskitError( "Operator input dimensions are not equal to statevector subsystem dimensions." ) # Reshape statevector and operator tensor = np.reshape(, state._shape) mat = np.reshape(, self._shape) # Construct list of tensor indices of statevector to be contracted num_indices = len(state.dims()) indices = [num_indices - 1 - qubit for qubit in qargs ] + [2 * num_indices - 1 - qubit for qubit in qargs] tensor = Operator._einsum_matmul(tensor, mat, indices) # Replace evolved dimensions new_dims = list(state.dims()) for i, qubit in enumerate(qargs): new_dims[qubit] = self._output_dims[i] new_dim = np.product(new_dims) # reshape tensor to density matrix tensor = np.reshape(tensor, (new_dim, new_dim)) return DensityMatrix(tensor, dims=new_dims) @classmethod def _init_instruction(cls, instruction): """Convert a QuantumCircuit or Instruction to a SuperOp.""" # Convert circuit to an instruction if isinstance(instruction, QuantumCircuit): instruction = instruction.to_instruction() # Initialize an identity superoperator of the correct size # of the circuit op = SuperOp(np.eye(4**instruction.num_qubits)) op._append_instruction(instruction) return op @classmethod def _instruction_to_superop(cls, obj): """Return superop for instruction if defined or None otherwise.""" if not isinstance(obj, Instruction): raise QiskitError('Input is not an instruction.') chan = None if == 'reset': # For superoperator evolution we can simulate a reset as # a non-unitary superoperator matrix chan = SuperOp( np.array([[1, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])) if == 'kraus': kraus = obj.params dim = len(kraus[0]) chan = SuperOp(_to_superop('Kraus', (kraus, None), dim, dim)) elif hasattr(obj, 'to_matrix'): # If instruction is a gate first we see if it has a # `to_matrix` definition and if so use that. try: kraus = [obj.to_matrix()] dim = len(kraus[0]) chan = SuperOp(_to_superop('Kraus', (kraus, None), dim, dim)) except QiskitError: pass return chan def _append_instruction(self, obj, qargs=None): """Update the current Operator by apply an instruction.""" from qiskit.circuit.barrier import Barrier chan = self._instruction_to_superop(obj) if chan is not None: # Perform the composition and inplace update the current state # of the operator op = self.compose(chan, qargs=qargs) self._data = elif isinstance(obj, Barrier): return else: # If the instruction doesn't have a matrix defined we use its # circuit decomposition definition if it exists, otherwise we # cannot compose this gate and raise an error. if obj.definition is None: raise QiskitError('Cannot apply Instruction: {}'.format( if not isinstance(obj.definition, QuantumCircuit): raise QiskitError('{} instruction definition is {}; ' 'expected QuantumCircuit'.format(, type(obj.definition))) for instr, qregs, cregs in if cregs: raise QiskitError( 'Cannot apply instruction with classical registers: {}' .format( # Get the integer position of the flat register if qargs is None: new_qargs = [tup.index for tup in qregs] else: new_qargs = [qargs[tup.index] for tup in qregs] self._append_instruction(instr, qargs=new_qargs)

© Copyright 2020, Qiskit Development Team. 최종 업데이트: 2021/01/17

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