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# Source code for qiskit.quantum_info.states.densitymatrix

# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2019.
#
# obtain a copy of this license in the LICENSE.txt file in the root directory
#
# 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.

"""
DensityMatrix quantum state class.
"""

import warnings
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.states.quantum_state import QuantumState
from qiskit.quantum_info.operators.operator import Operator
from qiskit.quantum_info.operators.scalar_op import ScalarOp
from qiskit.quantum_info.operators.predicates import is_hermitian_matrix
from qiskit.quantum_info.operators.predicates import is_positive_semidefinite_matrix
from qiskit.quantum_info.operators.channel.quantum_channel import QuantumChannel
from qiskit.quantum_info.operators.channel.superop import SuperOp
from qiskit.quantum_info.states.statevector import Statevector

[docs]class DensityMatrix(QuantumState):
"""DensityMatrix class"""

[docs]    def __init__(self, data, dims=None):
"""Initialize a density matrix object.

Args:
data (matrix_like or vector_like): a density matrix or
statevector. If a vector the density matrix is constructed
as the projector of that vector.
dims (int or tuple or list): Optional. The subsystem dimension
of the state (See additional information).

Raises:
QiskitError: if input data is not valid.

The dims kwarg can be None, an integer, or an iterable of
integers.

* Iterable -- the subsystem dimensions are the values in the list
with the total number of subsystems given by the length of the list.

* Int or None -- the leading dimension of the input matrix
specifies the total dimension of the density matrix. If it is a
power of two the state will be initialized as an N-qubit state.
If it is not a power of two the state will have a single
d-dimensional subsystem.
"""
if isinstance(data, (list, np.ndarray)):
# Finally we check if the input is a raw matrix in either a
# python list or numpy array format.
self._data = np.asarray(data, dtype=complex)
elif hasattr(data, 'to_operator'):
# If the data object has a 'to_operator' attribute this is given
# higher preference than the 'to_matrix' method for initializing
# an Operator object.
op = data.to_operator()
self._data = op.data
if dims is None:
dims = op._output_dims
elif hasattr(data, 'to_matrix'):
# If no 'to_operator' attribute exists we next look for a
# 'to_matrix' attribute to a matrix that will be cast into
# a complex numpy matrix.
self._data = np.asarray(data.to_matrix(), dtype=complex)
else:
raise QiskitError("Invalid input data format for DensityMatrix")
# Convert statevector into a density matrix
ndim = self._data.ndim
shape = self._data.shape
if ndim == 2 and shape == shape:
pass  # We good
elif ndim == 1:
self._data = np.outer(self._data, np.conj(self._data))
elif ndim == 2 and shape == 1:
self._data = np.reshape(self._data, shape)
shape = self._data.shape
else:
raise QiskitError(
"Invalid DensityMatrix input: not a square matrix.")
super().__init__(self._automatic_dims(dims, shape))

def __eq__(self, other):
return super().__eq__(other) and np.allclose(
self._data, other._data, rtol=self.rtol, atol=self.atol)

def __repr__(self):
prefix = 'DensityMatrix('
pad = len(prefix) * ' '
return '{}{},\n{}dims={})'.format(
prefix, np.array2string(
self._data, separator=', ', prefix=prefix),

@property
def data(self):
"""Return data."""
return self._data

[docs]    def is_valid(self, atol=None, rtol=None):
"""Return True if trace 1 and positive semidefinite."""
if atol is None:
atol = self.atol
if rtol is None:
rtol = self.rtol
# Check trace == 1
if not np.allclose(self.trace(), 1, rtol=rtol, atol=atol):
return False
# Check Hermitian
if not is_hermitian_matrix(self.data, rtol=rtol, atol=atol):
return False
# Check positive semidefinite
return is_positive_semidefinite_matrix(self.data, rtol=rtol, atol=atol)

[docs]    def to_operator(self):
"""Convert to Operator"""
dims = self.dims()
return Operator(self.data, input_dims=dims, output_dims=dims)

[docs]    def conjugate(self):
"""Return the conjugate of the density matrix."""
return DensityMatrix(np.conj(self.data), dims=self.dims())

[docs]    def trace(self):
"""Return the trace of the density matrix."""
return np.trace(self.data)

[docs]    def purity(self):
"""Return the purity of the quantum state."""
# For a valid statevector the purity is always 1, however if we simply
# have an arbitrary vector (not correctly normalized) then the
# purity is equivalent to the trace squared:
# P(|psi>) = Tr[|psi><psi|psi><psi|] = |<psi|psi>|^2
return np.trace(np.dot(self.data, self.data))

[docs]    def tensor(self, other):
"""Return the tensor product state self ⊗ other.

Args:
other (DensityMatrix): a quantum state object.

Returns:
DensityMatrix: the tensor product operator self ⊗ other.

Raises:
QiskitError: if other is not a quantum state.
"""
if not isinstance(other, DensityMatrix):
other = DensityMatrix(other)
dims = other.dims() + self.dims()
data = np.kron(self._data, other._data)
return DensityMatrix(data, dims)

[docs]    def expand(self, other):
"""Return the tensor product state other ⊗ self.

Args:
other (DensityMatrix): a quantum state object.

Returns:
DensityMatrix: the tensor product state other ⊗ self.

Raises:
QiskitError: if other is not a quantum state.
"""
if not isinstance(other, DensityMatrix):
other = DensityMatrix(other)
dims = self.dims() + other.dims()
data = np.kron(other._data, self._data)
return DensityMatrix(data, dims)

"""Return the linear combination self + other.

Args:
other (DensityMatrix): a quantum state object.

Returns:
DensityMatrix: the linear combination self + other.

Raises:
QiskitError: if other is not a quantum state, or has
incompatible dimensions.
"""
if not isinstance(other, DensityMatrix):
other = DensityMatrix(other)
if self.dim != other.dim:
raise QiskitError("other DensityMatrix has different dimensions.")
return DensityMatrix(self.data + other.data, self.dims())

def _multiply(self, other):
"""Return the scalar multiplied state other * self.

Args:
other (complex): a complex number.

Returns:
DensityMatrix: the scalar multiplied state other * self.

Raises:
QiskitError: if other is not a valid complex number.
"""
if not isinstance(other, Number):
raise QiskitError("other is not a number")
return DensityMatrix(other * self.data, self.dims())

[docs]    def evolve(self, other, qargs=None):
"""Evolve a quantum state by an operator.

Args:
other (Operator or QuantumChannel
or Instruction or Circuit): The operator to evolve by.
qargs (list): a list of QuantumState subsystem positions to apply
the operator on.

Returns:
QuantumState: the output quantum state.

Raises:
QiskitError: if the operator dimension does not match the
specified QuantumState subsystem dimensions.
"""
if qargs is None:
qargs = getattr(other, 'qargs', None)

# Evolution by a circuit or instruction
if isinstance(other, (QuantumCircuit, Instruction)):
return self._evolve_instruction(other, qargs=qargs)

# Evolution by a QuantumChannel
if hasattr(other, 'to_quantumchannel'):
return other.to_quantumchannel()._evolve(self, qargs=qargs)
if isinstance(other, QuantumChannel):
return other._evolve(self, qargs=qargs)

# Unitary evolution by an Operator
if not isinstance(other, Operator):
other = Operator(other)
return self._evolve_operator(other, qargs=qargs)

[docs]    def expectation_value(self, oper, qargs=None):
"""Compute the expectation value of an operator.

Args:
oper (Operator): an operator to evaluate expval.
qargs (None or list): subsystems to apply the operator on.

Returns:
complex: the expectation value.
"""
if not isinstance(oper, Operator):
oper = Operator(oper)

[docs]    def probabilities(self, qargs=None, decimals=None):
"""Return the subsystem measurement probability vector.

Measurement probabilities are with respect to measurement in the
computation (diagonal) basis.

Args:
qargs (None or list): subsystems to return probabilities for,
if None return for all subsystems (Default: None).
decimals (None or int): the number of decimal places to round
values. If None no rounding is done (Default: None).

Returns:
np.array: The Numpy vector array of probabilities.

Examples:

Consider a 2-qubit product state :math:\\rho=\\rho_1\\otimes\\rho_0
with :math:\\rho_1=|+\\rangle\\!\\langle+|,
:math:\\rho_0=|0\\rangle\\!\\langle0|.

.. jupyter-execute::

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('+0')

# Probabilities for measuring both qubits
probs = rho.probabilities()
print('probs: {}'.format(probs))

# Probabilities for measuring only qubit-0
probs_qubit_0 = rho.probabilities()
print('Qubit-0 probs: {}'.format(probs_qubit_0))

# Probabilities for measuring only qubit-1
probs_qubit_1 = rho.probabilities()
print('Qubit-1 probs: {}'.format(probs_qubit_1))

We can also permute the order of qubits in the qargs list
to change the qubit position in the probabilities output

.. jupyter-execute::

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('+0')

# Probabilities for measuring both qubits
probs = rho.probabilities([0, 1])
print('probs: {}'.format(probs))

# Probabilities for measuring both qubits
# but swapping qubits 0 and 1 in output
probs_swapped = rho.probabilities([1, 0])
print('Swapped probs: {}'.format(probs_swapped))
"""
probs = self._subsystem_probabilities(
np.abs(self.data.diagonal()), self._dims, qargs=qargs)
if decimals is not None:
probs = probs.round(decimals=decimals)
return probs

[docs]    def reset(self, qargs=None):
"""Reset state or subsystems to the 0-state.

Args:
qargs (list or None): subsystems to reset, if None all
subsystems will be reset to their 0-state
(Default: None).

Returns:
DensityMatrix: the reset state.

If all subsystems are reset this will return the ground state
on all subsystems. If only a some subsystems are reset this
function will perform evolution by the reset
:class:~qiskit.quantum_info.SuperOp of the reset subsystems.
"""
if qargs is None:
# Resetting all qubits does not require sampling or RNG
state = np.zeros(2 * (self._dim, ), dtype=complex)
state[0, 0] = 1
return DensityMatrix(state, dims=self._dims)

# Reset by evolving by reset SuperOp
dims = self.dims(qargs)
reset_superop = SuperOp(ScalarOp(dims, coeff=0))
reset_superop.data = Operator(ScalarOp(dims)).data.ravel()
return self.evolve(reset_superop, qargs=qargs)

[docs]    @classmethod
def from_label(cls, label):
r"""Return a tensor product of Pauli X,Y,Z eigenstates.

.. list-table:: Single-qubit state labels

* - Label
- Statevector
* - "0"
- :math:\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
* - "1"
- :math:\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
* - "+"
- :math:\frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}
* - "-"
- :math:\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}
* - "r"
- :math:\frac{1}{2}\begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}
* - "l"
- :math:\frac{1}{2}\begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}

Args:
label (string): a eigenstate string ket label (see table for
allowed values).

Returns:
Statevector: The N-qubit basis state density matrix.

Raises:
QiskitError: if the label contains invalid characters, or the length
of the label is larger than an explicitly specified num_qubits.
"""
return DensityMatrix(Statevector.from_label(label))

[docs]    @staticmethod
def from_int(i, dims):
"""Return a computational basis state density matrix.

Args:
i (int): the basis state element.
dims (int or tuple or list): The subsystem dimensions of the statevector

Returns:
DensityMatrix: The computational basis state :math:|i\\rangle\\!\\langle i|.

The dims kwarg can be an integer or an iterable of integers.

* Iterable -- the subsystem dimensions are the values in the list
with the total number of subsystems given by the length of the list.

* Int -- the integer specifies the total dimension of the
state. If it is a power of two the state will be initialized
as an N-qubit state. If it is not a power of  two the state
will have a single d-dimensional subsystem.
"""
size = np.product(dims)
state = np.zeros((size, size), dtype=complex)
state[i, i] = 1.0
return DensityMatrix(state, dims=dims)

[docs]    @classmethod
def from_instruction(cls, instruction):
"""Return the output density matrix of an instruction.

The statevector is initialized in the state :math:|{0,\\ldots,0}\\rangle of
the same number of qubits as the input instruction or circuit, evolved
by the input instruction, and the output statevector returned.

Args:
instruction (qiskit.circuit.Instruction or QuantumCircuit): instruction or circuit

Returns:
DensityMatrix: the final density matrix.

Raises:
QiskitError: if the instruction contains invalid instructions for
density matrix simulation.
"""
# Convert circuit to an instruction
if isinstance(instruction, QuantumCircuit):
instruction = instruction.to_instruction()
# Initialize an the statevector in the all |0> state
num_qubits = instruction.num_qubits
init = np.zeros((2**num_qubits, 2**num_qubits), dtype=complex)
init[0, 0] = 1
vec = DensityMatrix(init, dims=num_qubits * (2, ))
vec._append_instruction(instruction)
return vec

[docs]    def to_dict(self, decimals=None):
r"""Convert the density matrix to dictionary form.

This dictionary representation uses a Ket-like notation where the
dictionary keys are qudit strings for the subsystem basis vectors.
If any subsystem has a dimension greater than 10 comma delimiters are
inserted between integers so that subsystems can be distinguished.

Args:
decimals (None or int): the number of decimal places to round
values. If None no rounding is done
(Default: None).

Returns:
dict: the dictionary form of the DensityMatrix.

Examples:

The ket-form of a 2-qubit density matrix
:math:rho = |-\rangle\!\langle -|\otimes |0\rangle\!\langle 0|

.. jupyter-execute::

from qiskit.quantum_info import DensityMatrix

rho = DensityMatrix.from_label('-0')
print(rho.to_dict())

For non-qubit subsystems the integer range can go from 0 to 9. For
example in a qutrit system

.. jupyter-execute::

import numpy as np
from qiskit.quantum_info import DensityMatrix

mat = np.zeros((9, 9))
mat[0, 0] = 0.25
mat[3, 3] = 0.25
mat[6, 6] = 0.25
mat[-1, -1] = 0.25
rho = DensityMatrix(mat, dims=(3, 3))
print(rho.to_dict())

For large subsystem dimensions delimeters are required. The
following example is for a 20-dimensional system consisting of
a qubit and 10-dimensional qudit.

.. jupyter-execute::

import numpy as np
from qiskit.quantum_info import DensityMatrix

mat = np.zeros((2 * 10, 2 * 10))
mat[0, 0] = 0.5
mat[-1, -1] = 0.5
rho = DensityMatrix(mat, dims=(2, 10))
print(rho.to_dict())
"""
return self._matrix_to_dict(self.data,
self._dims,
decimals=decimals,
string_labels=True)

@property
def _shape(self):
"""Return the tensor shape of the matrix operator"""
return 2 * tuple(reversed(self.dims()))

def _evolve_operator(self, other, qargs=None):
"""Evolve density matrix by an operator"""
if qargs is None:
# Evolution on full matrix
if self._dim != other._input_dim:
raise QiskitError(
"Operator input dimension is not equal to density matrix dimension."
)
op_mat = other.data
mat = np.dot(op_mat, self.data).dot(op_mat.T.conj())
return DensityMatrix(mat, dims=other._output_dims)
# Otherwise we are applying an operator only to subsystems
# Check dimensions of subsystems match the operator
if self.dims(qargs) != other.input_dims():
raise QiskitError(
"Operator input dimensions are not equal to statevector subsystem dimensions."
)
# Reshape statevector and operator
tensor = np.reshape(self.data, self._shape)
# Construct list of tensor indices of statevector to be contracted
num_indices = len(self.dims())
indices = [num_indices - 1 - qubit for qubit in qargs]
# Left multiple by mat
mat = np.reshape(other.data, other._shape)
tensor = Operator._einsum_matmul(tensor, mat, indices)
# Right multiply by mat ** dagger
tensor = Operator._einsum_matmul(tensor, mat_adj, indices, num_indices,
True)
# Replace evolved dimensions
new_dims = list(self.dims())
for i, qubit in enumerate(qargs):
new_dims[qubit] = other._output_dims[i]
new_dim = np.product(new_dims)
return DensityMatrix(np.reshape(tensor, (new_dim, new_dim)),
dims=new_dims)

def _append_instruction(self, other, qargs=None):
"""Update the current Statevector by applying an instruction."""
from qiskit.circuit.reset import Reset
from qiskit.circuit.barrier import Barrier

# Try evolving by a matrix operator (unitary-like evolution)
mat = Operator._instruction_to_matrix(other)
if mat is not None:
self._data = self._evolve_operator(Operator(mat), qargs=qargs).data
return

# Special instruction types
if isinstance(other, Reset):
self._data = self.reset(qargs)._data
return
if isinstance(other, Barrier):
return

# Otherwise try evolving by a Superoperator
chan = SuperOp._instruction_to_superop(other)
if chan is not None:
# Evolve current state by the superoperator
self._data = chan._evolve(self, qargs=qargs).data
return
# 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 other.definition is None:
raise QiskitError('Cannot apply Instruction: {}'.format(
other.name))
if not isinstance(other.definition, QuantumCircuit):
raise QiskitError('{} instruction definition is {}; expected QuantumCircuit'.format(
other.name, type(other.definition)))
for instr, qregs, cregs in other.definition:
if cregs:
raise QiskitError(
'Cannot apply instruction with classical registers: {}'.
format(instr.name))
# 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)

def _evolve_instruction(self, obj, qargs=None):
"""Return a new statevector by applying an instruction."""
if isinstance(obj, QuantumCircuit):
obj = obj.to_instruction()
vec = DensityMatrix(self.data, dims=self._dims)
vec._append_instruction(obj, qargs=qargs)
return vec

[docs]    def to_statevector(self, atol=None, rtol=None):
"""Return a statevector from a pure density matrix.

Args:
atol (float): Absolute tolerance for checking operation validity.
rtol (float): Relative tolerance for checking operation validity.

Returns:
Statevector: The pure density matrix's corresponding statevector.
Corresponds to the eigenvector of the only non-zero eigenvalue.

Raises:
QiskitError: if the state is not pure.
"""
if atol is None:
atol = self.atol
if rtol is None:
rtol = self.rtol

if not is_hermitian_matrix(self._data, atol=atol, rtol=rtol):
raise QiskitError("Not a valid density matrix (non-hermitian).")

evals, evecs = np.linalg.eig(self._data)

nonzero_evals = evals[abs(evals) > atol]
if len(nonzero_evals) != 1 or not np.isclose(nonzero_evals, 1,
atol=atol, rtol=rtol):
raise QiskitError("Density matrix is not a pure state")

psi = evecs[:, np.argmax(evals)]  # eigenvectors returned in columns.
return Statevector(psi)

[docs]    def to_counts(self):
"""Returns the density matrix as a counts dict of probabilities.

DEPRECATED: use :meth:probabilities_dict instead.

Returns:
dict: Counts of probabilities.
"""
warnings.warn(
'The Statevector.to_counts method is deprecated as of 0.13.0,'
' and will be removed no earlier than 3 months after that '
'release date. You should use the Statevector.probabilities_dict'