# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2018.
#
# 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.
# pylint: disable=invalid-name
# pylint: disable=missing-param-doc,missing-type-doc,unused-argument
"""
Visualization functions for quantum states.
"""
from typing import Optional, List, Union
from functools import reduce
import colorsys
import numpy as np
from qiskit import user_config
from qiskit.quantum_info.states.statevector import Statevector
from qiskit.quantum_info.operators.symplectic import PauliList, SparsePauliOp
from qiskit.quantum_info.states.densitymatrix import DensityMatrix
from qiskit.utils.deprecation import deprecate_arg, deprecate_func
from qiskit.utils import optionals as _optionals
from qiskit.circuit.tools.pi_check import pi_check
from .array import _num_to_latex, array_to_latex
from .utils import matplotlib_close_if_inline
from .exceptions import VisualizationError
[documentos]@deprecate_arg("rho", new_alias="state", since="0.15.1")
@_optionals.HAS_MATPLOTLIB.require_in_call
def plot_state_hinton(
state, title="", figsize=None, ax_real=None, ax_imag=None, *, rho=None, filename=None
):
"""Plot a hinton diagram for the density matrix of a quantum state.
The hinton diagram represents the values of a matrix using
squares, whose size indicate the magnitude of their corresponding value
and their color, its sign. A white square means the value is positive and
a black one means negative.
Args:
state (Statevector or DensityMatrix or ndarray): An N-qubit quantum state.
title (str): a string that represents the plot title
figsize (tuple): Figure size in inches.
filename (str): file path to save image to.
ax_real (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. If this is specified without an
ax_imag only the real component plot will be generated.
Additionally, if specified there will be no returned Figure since
it is redundant.
ax_imag (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. If this is specified without an
ax_imag only the real component plot will be generated.
Additionally, if specified there will be no returned Figure since
it is redundant.
Returns:
:class:`matplotlib:matplotlib.figure.Figure` :
The matplotlib.Figure of the visualization if
neither ax_real or ax_imag is set.
Raises:
MissingOptionalLibraryError: Requires matplotlib.
VisualizationError: if input is not a valid N-qubit state.
Examples:
.. plot::
:include-source:
import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import DensityMatrix
from qiskit.visualization import plot_state_hinton
qc = QuantumCircuit(2)
qc.h([0, 1])
qc.cz(0,1)
qc.ry(np.pi/3 , 0)
qc.rx(np.pi/5, 1)
state = DensityMatrix(qc)
plot_state_hinton(state, title="New Hinton Plot")
"""
from matplotlib import pyplot as plt
# Figure data
rho = DensityMatrix(state)
num = rho.num_qubits
if num is None:
raise VisualizationError("Input is not a multi-qubit quantum state.")
max_weight = 2 ** np.ceil(np.log(np.abs(rho.data).max()) / np.log(2))
datareal = np.real(rho.data)
dataimag = np.imag(rho.data)
if figsize is None:
figsize = (8, 5)
if not ax_real and not ax_imag:
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=figsize)
else:
if ax_real:
fig = ax_real.get_figure()
else:
fig = ax_imag.get_figure()
ax1 = ax_real
ax2 = ax_imag
# Reversal is to account for Qiskit's endianness.
column_names = [bin(i)[2:].zfill(num) for i in range(2**num)]
row_names = [bin(i)[2:].zfill(num) for i in range(2**num)][::-1]
ly, lx = datareal.shape
# Real
if ax1:
ax1.patch.set_facecolor("gray")
ax1.set_aspect("equal", "box")
ax1.xaxis.set_major_locator(plt.NullLocator())
ax1.yaxis.set_major_locator(plt.NullLocator())
for (x, y), w in np.ndenumerate(datareal):
# Convert from matrix co-ordinates to plot co-ordinates.
plot_x, plot_y = y, lx - x - 1
color = "white" if w > 0 else "black"
size = np.sqrt(np.abs(w) / max_weight)
rect = plt.Rectangle(
[0.5 + plot_x - size / 2, 0.5 + plot_y - size / 2],
size,
size,
facecolor=color,
edgecolor=color,
)
ax1.add_patch(rect)
ax1.set_xticks(0.5 + np.arange(lx))
ax1.set_yticks(0.5 + np.arange(ly))
ax1.set_xlim([0, lx])
ax1.set_ylim([0, ly])
ax1.set_yticklabels(row_names, fontsize=14)
ax1.set_xticklabels(column_names, fontsize=14, rotation=90)
ax1.set_title("Re[$\\rho$]", fontsize=14)
# Imaginary
if ax2:
ax2.patch.set_facecolor("gray")
ax2.set_aspect("equal", "box")
ax2.xaxis.set_major_locator(plt.NullLocator())
ax2.yaxis.set_major_locator(plt.NullLocator())
for (x, y), w in np.ndenumerate(dataimag):
# Convert from matrix co-ordinates to plot co-ordinates.
plot_x, plot_y = y, lx - x - 1
color = "white" if w > 0 else "black"
size = np.sqrt(np.abs(w) / max_weight)
rect = plt.Rectangle(
[0.5 + plot_x - size / 2, 0.5 + plot_y - size / 2],
size,
size,
facecolor=color,
edgecolor=color,
)
ax2.add_patch(rect)
ax2.set_xticks(0.5 + np.arange(lx))
ax2.set_yticks(0.5 + np.arange(ly))
ax2.set_xlim([0, lx])
ax2.set_ylim([0, ly])
ax2.set_yticklabels(row_names, fontsize=14)
ax2.set_xticklabels(column_names, fontsize=14, rotation=90)
ax2.set_title("Im[$\\rho$]", fontsize=14)
fig.tight_layout()
if title:
fig.suptitle(title, fontsize=16)
if ax_real is None and ax_imag is None:
matplotlib_close_if_inline(fig)
if filename is None:
return fig
else:
return fig.savefig(filename)
[documentos]@_optionals.HAS_MATPLOTLIB.require_in_call
def plot_bloch_vector(
bloch, title="", ax=None, figsize=None, coord_type="cartesian", font_size=None
):
"""Plot the Bloch sphere.
Plot a Bloch sphere with the specified coordinates, that can be given in both
cartesian and spherical systems.
Args:
bloch (list[double]): array of three elements where [<x>, <y>, <z>] (Cartesian)
or [<r>, <theta>, <phi>] (spherical in radians)
<theta> is inclination angle from +z direction
<phi> is azimuth from +x direction
title (str): a string that represents the plot title
ax (matplotlib.axes.Axes): An Axes to use for rendering the bloch
sphere
figsize (tuple): Figure size in inches. Has no effect is passing ``ax``.
coord_type (str): a string that specifies coordinate type for bloch
(Cartesian or spherical), default is Cartesian
font_size (float): Font size.
Returns:
:class:`matplotlib:matplotlib.figure.Figure` : A matplotlib figure instance if ``ax = None``.
Raises:
MissingOptionalLibraryError: Requires matplotlib.
Examples:
.. plot::
:include-source:
from qiskit.visualization import plot_bloch_vector
plot_bloch_vector([0,1,0], title="New Bloch Sphere")
.. plot::
:include-source:
import numpy as np
from qiskit.visualization import plot_bloch_vector
# You can use spherical coordinates instead of cartesian.
plot_bloch_vector([1, np.pi/2, np.pi/3], coord_type='spherical')
"""
from .bloch import Bloch
if figsize is None:
figsize = (5, 5)
B = Bloch(axes=ax, font_size=font_size)
if coord_type == "spherical":
r, theta, phi = bloch[0], bloch[1], bloch[2]
bloch[0] = r * np.sin(theta) * np.cos(phi)
bloch[1] = r * np.sin(theta) * np.sin(phi)
bloch[2] = r * np.cos(theta)
B.add_vectors(bloch)
B.render(title=title)
if ax is None:
fig = B.fig
fig.set_size_inches(figsize[0], figsize[1])
matplotlib_close_if_inline(fig)
return fig
return None
[documentos]@deprecate_arg("rho", new_alias="state", since="0.15.1")
@_optionals.HAS_MATPLOTLIB.require_in_call
def plot_bloch_multivector(
state,
title="",
figsize=None,
*,
rho=None,
reverse_bits=False,
filename=None,
font_size=None,
title_font_size=None,
title_pad=1,
):
r"""Plot a Bloch sphere for each qubit.
Each component :math:`(x,y,z)` of the Bloch sphere labeled as 'qubit i' represents the expected
value of the corresponding Pauli operator acting only on that qubit, that is, the expected value
of :math:`I_{N-1} \otimes\dotsb\otimes I_{i+1}\otimes P_i \otimes I_{i-1}\otimes\dotsb\otimes
I_0`, where :math:`N` is the number of qubits, :math:`P\in \{X,Y,Z\}` and :math:`I` is the
identity operator.
Args:
state (Statevector or DensityMatrix or ndarray): an N-qubit quantum state.
title (str): a string that represents the plot title
figsize (tuple): size of each individual Bloch sphere figure, in inches.
reverse_bits (bool): If True, plots qubits following Qiskit's convention [Default:False].
font_size (float): Font size for the Bloch ball figures.
title_font_size (float): Font size for the title.
title_pad (float): Padding for the title (suptitle `y` position is `y=1+title_pad/100`).
Returns:
:class:`matplotlib:matplotlib.figure.Figure` :
A matplotlib figure instance.
Raises:
MissingOptionalLibraryError: Requires matplotlib.
VisualizationError: if input is not a valid N-qubit state.
Examples:
.. plot::
:include-source:
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_bloch_multivector
qc = QuantumCircuit(2)
qc.h(0)
qc.x(1)
state = Statevector(qc)
plot_bloch_multivector(state)
.. plot::
:include-source:
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_bloch_multivector
qc = QuantumCircuit(2)
qc.h(0)
qc.x(1)
# You can reverse the order of the qubits.
from qiskit.quantum_info import DensityMatrix
qc = QuantumCircuit(2)
qc.h([0, 1])
qc.t(1)
qc.s(0)
qc.cx(0,1)
matrix = DensityMatrix(qc)
plot_bloch_multivector(matrix, title='My Bloch Spheres', reverse_bits=True)
"""
from matplotlib import pyplot as plt
# Data
bloch_data = (
_bloch_multivector_data(state)[::-1] if reverse_bits else _bloch_multivector_data(state)
)
num = len(bloch_data)
if figsize is not None:
width, height = figsize
width *= num
else:
width, height = plt.figaspect(1 / num)
default_title_font_size = font_size if font_size is not None else 16
title_font_size = title_font_size if title_font_size is not None else default_title_font_size
fig = plt.figure(figsize=(width, height))
for i in range(num):
pos = num - 1 - i if reverse_bits else i
ax = fig.add_subplot(1, num, i + 1, projection="3d")
plot_bloch_vector(
bloch_data[i], "qubit " + str(pos), ax=ax, figsize=figsize, font_size=font_size
)
fig.suptitle(title, fontsize=title_font_size, y=1.0 + title_pad / 100)
matplotlib_close_if_inline(fig)
if filename is None:
return fig
else:
return fig.savefig(filename)
[documentos]@deprecate_arg("rho", new_alias="state", since="0.15.1")
@_optionals.HAS_MATPLOTLIB.require_in_call
def plot_state_city(
state,
title="",
figsize=None,
color=None,
alpha=1,
ax_real=None,
ax_imag=None,
*,
rho=None,
filename=None,
):
"""Plot the cityscape of quantum state.
Plot two 3d bar graphs (two dimensional) of the real and imaginary
part of the density matrix rho.
Args:
state (Statevector or DensityMatrix or ndarray): an N-qubit quantum state.
title (str): a string that represents the plot title
figsize (tuple): Figure size in inches.
color (list): A list of len=2 giving colors for real and
imaginary components of matrix elements.
alpha (float): Transparency value for bars
ax_real (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. If this is specified without an
ax_imag only the real component plot will be generated.
Additionally, if specified there will be no returned Figure since
it is redundant.
ax_imag (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. If this is specified without an
ax_real only the imaginary component plot will be generated.
Additionally, if specified there will be no returned Figure since
it is redundant.
Returns:
:class:`matplotlib:matplotlib.figure.Figure` :
The matplotlib.Figure of the visualization if the
``ax_real`` and ``ax_imag`` kwargs are not set
Raises:
MissingOptionalLibraryError: Requires matplotlib.
ValueError: When 'color' is not a list of len=2.
VisualizationError: if input is not a valid N-qubit state.
Examples:
.. plot::
:include-source:
# You can choose different colors for the real and imaginary parts of the density matrix.
from qiskit import QuantumCircuit
from qiskit.quantum_info import DensityMatrix
from qiskit.visualization import plot_state_city
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
state = DensityMatrix(qc)
plot_state_city(state, color=['midnightblue', 'crimson'], title="New State City")
.. plot::
:include-source:
# You can make the bars more transparent to better see the ones that are behind
# if they overlap.
import numpy as np
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_state_city
from qiskit import QuantumCircuit
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
qc = QuantumCircuit(2)
qc.h([0, 1])
qc.cz(0,1)
qc.ry(np.pi/3, 0)
qc.rx(np.pi/5, 1)
state = Statevector(qc)
plot_state_city(state, alpha=0.6)
"""
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
rho = DensityMatrix(state)
num = rho.num_qubits
if num is None:
raise VisualizationError("Input is not a multi-qubit quantum state.")
# get the real and imag parts of rho
datareal = np.real(rho.data)
dataimag = np.imag(rho.data)
# get the labels
column_names = [bin(i)[2:].zfill(num) for i in range(2**num)]
row_names = [bin(i)[2:].zfill(num) for i in range(2**num)]
lx = len(datareal[0]) # Work out matrix dimensions
ly = len(datareal[:, 0])
xpos = np.arange(0, lx, 1) # Set up a mesh of positions
ypos = np.arange(0, ly, 1)
xpos, ypos = np.meshgrid(xpos + 0.25, ypos + 0.25)
xpos = xpos.flatten()
ypos = ypos.flatten()
zpos = np.zeros(lx * ly)
dx = 0.5 * np.ones_like(zpos) # width of bars
dy = dx.copy()
dzr = datareal.flatten()
dzi = dataimag.flatten()
if color is None:
color = ["#648fff", "#648fff"]
else:
if len(color) != 2:
raise ValueError("'color' must be a list of len=2.")
if color[0] is None:
color[0] = "#648fff"
if color[1] is None:
color[1] = "#648fff"
if ax_real is None and ax_imag is None:
# set default figure size
if figsize is None:
figsize = (15, 5)
fig = plt.figure(figsize=figsize)
ax1 = fig.add_subplot(1, 2, 1, projection="3d")
ax2 = fig.add_subplot(1, 2, 2, projection="3d")
elif ax_real is not None:
fig = ax_real.get_figure()
ax1 = ax_real
ax2 = ax_imag
else:
fig = ax_imag.get_figure()
ax1 = None
ax2 = ax_imag
max_dzr = max(dzr)
min_dzr = min(dzr)
min_dzi = np.min(dzi)
max_dzi = np.max(dzi)
# There seems to be a rounding error in which some zero bars are negative
dzr = np.clip(dzr, 0, None)
if ax1 is not None:
fc1 = generate_facecolors(xpos, ypos, zpos, dx, dy, dzr, color[0])
for idx, cur_zpos in enumerate(zpos):
if dzr[idx] > 0:
zorder = 2
else:
zorder = 0
b1 = ax1.bar3d(
xpos[idx],
ypos[idx],
cur_zpos,
dx[idx],
dy[idx],
dzr[idx],
alpha=alpha,
zorder=zorder,
)
b1.set_facecolors(fc1[6 * idx : 6 * idx + 6])
xlim, ylim = ax1.get_xlim(), ax1.get_ylim()
x = [xlim[0], xlim[1], xlim[1], xlim[0]]
y = [ylim[0], ylim[0], ylim[1], ylim[1]]
z = [0, 0, 0, 0]
verts = [list(zip(x, y, z))]
pc1 = Poly3DCollection(verts, alpha=0.15, facecolor="k", linewidths=1, zorder=1)
if min(dzr) < 0 < max(dzr):
ax1.add_collection3d(pc1)
ax1.set_xticks(np.arange(0.5, lx + 0.5, 1))
ax1.set_yticks(np.arange(0.5, ly + 0.5, 1))
if max_dzr != min_dzr:
ax1.axes.set_zlim3d(np.min(dzr), max(np.max(dzr) + 1e-9, max_dzi))
else:
if min_dzr == 0:
ax1.axes.set_zlim3d(np.min(dzr), max(np.max(dzr) + 1e-9, np.max(dzi)))
else:
ax1.axes.set_zlim3d(auto=True)
ax1.get_autoscalez_on()
ax1.xaxis.set_ticklabels(row_names, fontsize=14, rotation=45, ha="right", va="top")
ax1.yaxis.set_ticklabels(column_names, fontsize=14, rotation=-22.5, ha="left", va="center")
ax1.set_zlabel("Re[$\\rho$]", fontsize=14)
for tick in ax1.zaxis.get_major_ticks():
tick.label1.set_fontsize(14)
if ax2 is not None:
fc2 = generate_facecolors(xpos, ypos, zpos, dx, dy, dzi, color[1])
for idx, cur_zpos in enumerate(zpos):
if dzi[idx] > 0:
zorder = 2
else:
zorder = 0
b2 = ax2.bar3d(
xpos[idx],
ypos[idx],
cur_zpos,
dx[idx],
dy[idx],
dzi[idx],
alpha=alpha,
zorder=zorder,
)
b2.set_facecolors(fc2[6 * idx : 6 * idx + 6])
xlim, ylim = ax2.get_xlim(), ax2.get_ylim()
x = [xlim[0], xlim[1], xlim[1], xlim[0]]
y = [ylim[0], ylim[0], ylim[1], ylim[1]]
z = [0, 0, 0, 0]
verts = [list(zip(x, y, z))]
pc2 = Poly3DCollection(verts, alpha=0.2, facecolor="k", linewidths=1, zorder=1)
if min(dzi) < 0 < max(dzi):
ax2.add_collection3d(pc2)
ax2.set_xticks(np.arange(0.5, lx + 0.5, 1))
ax2.set_yticks(np.arange(0.5, ly + 0.5, 1))
if min_dzi != max_dzi:
eps = 0
ax2.axes.set_zlim3d(np.min(dzi), max(np.max(dzr) + 1e-9, np.max(dzi) + eps))
else:
if min_dzi == 0:
ax2.set_zticks([0])
eps = 1e-9
ax2.axes.set_zlim3d(np.min(dzi), max(np.max(dzr) + 1e-9, np.max(dzi) + eps))
else:
ax2.axes.set_zlim3d(auto=True)
ax2.xaxis.set_ticklabels(row_names, fontsize=14, rotation=45, ha="right", va="top")
ax2.yaxis.set_ticklabels(column_names, fontsize=14, rotation=-22.5, ha="left", va="center")
ax2.set_zlabel("Im[$\\rho$]", fontsize=14)
for tick in ax2.zaxis.get_major_ticks():
tick.label1.set_fontsize(14)
ax2.get_autoscalez_on()
fig.suptitle(title, fontsize=16)
if ax_real is None and ax_imag is None:
matplotlib_close_if_inline(fig)
if filename is None:
return fig
else:
return fig.savefig(filename)
[documentos]@deprecate_arg("rho", new_alias="state", since="0.15.1")
@_optionals.HAS_MATPLOTLIB.require_in_call
def plot_state_paulivec(
state, title="", figsize=None, color=None, ax=None, *, rho=None, filename=None
):
r"""Plot the Pauli-vector representation of a quantum state as bar graph.
The Pauli-vector of a density matrix :math:`\rho` is defined by the expectation of each
possible tensor product of single-qubit Pauli operators (including the identity), that is
.. math ::
\rho = \frac{1}{2^n} \sum_{\sigma \in \{I, X, Y, Z\}^{\otimes n}}
\mathrm{Tr}(\sigma \rho) \sigma.
This function plots the coefficients :math:`\mathrm{Tr}(\sigma\rho)` as bar graph.
Args:
state (Statevector or DensityMatrix or ndarray): an N-qubit quantum state.
title (str): a string that represents the plot title
figsize (tuple): Figure size in inches.
color (list or str): Color of the coefficient value bars.
ax (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. Additionally, if specified there
will be no returned Figure since it is redundant.
Returns:
:class:`matplotlib:matplotlib.figure.Figure` :
The matplotlib.Figure of the visualization if the
``ax`` kwarg is not set
Raises:
MissingOptionalLibraryError: Requires matplotlib.
VisualizationError: if input is not a valid N-qubit state.
Examples:
.. plot::
:include-source:
# You can set a color for all the bars.
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_state_paulivec
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
state = Statevector(qc)
plot_state_paulivec(state, color='midnightblue', title="New PauliVec plot")
.. plot::
:include-source:
# If you introduce a list with less colors than bars, the color of the bars will
# alternate following the sequence from the list.
import numpy as np
from qiskit.quantum_info import DensityMatrix
from qiskit import QuantumCircuit
from qiskit.visualization import plot_state_paulivec
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
qc = QuantumCircuit(2)
qc.h([0, 1])
qc.cz(0, 1)
qc.ry(np.pi/3, 0)
qc.rx(np.pi/5, 1)
matrix = DensityMatrix(qc)
plot_state_paulivec(matrix, color=['crimson', 'midnightblue', 'seagreen'])
"""
from matplotlib import pyplot as plt
labels, values = _paulivec_data(state)
numelem = len(values)
if figsize is None:
figsize = (7, 5)
if color is None:
color = "#648fff"
ind = np.arange(numelem) # the x locations for the groups
width = 0.5 # the width of the bars
if ax is None:
return_fig = True
fig, ax = plt.subplots(figsize=figsize)
else:
return_fig = False
fig = ax.get_figure()
ax.grid(zorder=0, linewidth=1, linestyle="--")
ax.bar(ind, values, width, color=color, zorder=2)
ax.axhline(linewidth=1, color="k")
# add some text for labels, title, and axes ticks
ax.set_ylabel("Coefficients", fontsize=14)
ax.set_xticks(ind)
ax.set_yticks([-1, -0.5, 0, 0.5, 1])
ax.set_xticklabels(labels, fontsize=14, rotation=70)
ax.set_xlabel("Pauli", fontsize=14)
ax.set_ylim([-1, 1])
ax.set_facecolor("#eeeeee")
for tick in ax.xaxis.get_major_ticks() + ax.yaxis.get_major_ticks():
tick.label1.set_fontsize(14)
ax.set_title(title, fontsize=16)
if return_fig:
matplotlib_close_if_inline(fig)
if filename is None:
return fig
else:
return fig.savefig(filename)
def n_choose_k(n, k):
"""Return the number of combinations for n choose k.
Args:
n (int): the total number of options .
k (int): The number of elements.
Returns:
int: returns the binomial coefficient
"""
if n == 0:
return 0
return reduce(lambda x, y: x * y[0] / y[1], zip(range(n - k + 1, n + 1), range(1, k + 1)), 1)
def lex_index(n, k, lst):
"""Return the lex index of a combination..
Args:
n (int): the total number of options .
k (int): The number of elements.
lst (list): list
Returns:
int: returns int index for lex order
Raises:
VisualizationError: if length of list is not equal to k
"""
if len(lst) != k:
raise VisualizationError("list should have length k")
comb = [n - 1 - x for x in lst]
dualm = sum(n_choose_k(comb[k - 1 - i], i + 1) for i in range(k))
return int(dualm)
def bit_string_index(s):
"""Return the index of a string of 0s and 1s."""
n = len(s)
k = s.count("1")
if s.count("0") != n - k:
raise VisualizationError("s must be a string of 0 and 1")
ones = [pos for pos, char in enumerate(s) if char == "1"]
return lex_index(n, k, ones)
def phase_to_rgb(complex_number):
"""Map a phase of a complexnumber to a color in (r,g,b).
complex_number is phase is first mapped to angle in the range
[0, 2pi] and then to the HSL color wheel
"""
angles = (np.angle(complex_number) + (np.pi * 5 / 4)) % (np.pi * 2)
rgb = colorsys.hls_to_rgb(angles / (np.pi * 2), 0.5, 0.5)
return rgb
[documentos]@deprecate_arg("rho", new_alias="state", since="0.15.1")
@_optionals.HAS_MATPLOTLIB.require_in_call
@_optionals.HAS_SEABORN.require_in_call
def plot_state_qsphere(
state,
figsize=None,
ax=None,
show_state_labels=True,
show_state_phases=False,
use_degrees=False,
*,
rho=None,
filename=None,
):
"""Plot the qsphere representation of a quantum state.
Here, the size of the points is proportional to the probability
of the corresponding term in the state and the color represents
the phase.
Args:
state (Statevector or DensityMatrix or ndarray): an N-qubit quantum state.
figsize (tuple): Figure size in inches.
ax (matplotlib.axes.Axes): An optional Axes object to be used for
the visualization output. If none is specified a new matplotlib
Figure will be created and used. Additionally, if specified there
will be no returned Figure since it is redundant.
show_state_labels (bool): An optional boolean indicating whether to
show labels for each basis state.
show_state_phases (bool): An optional boolean indicating whether to
show the phase for each basis state.
use_degrees (bool): An optional boolean indicating whether to use
radians or degrees for the phase values in the plot.
Returns:
:class:`matplotlib:matplotlib.figure.Figure` :
A matplotlib figure instance if the ``ax`` kwarg is not set
Raises:
MissingOptionalLibraryError: Requires matplotlib.
VisualizationError: if input is not a valid N-qubit state.
QiskitError: Input statevector does not have valid dimensions.
Examples:
.. plot::
:include-source:
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_state_qsphere
qc = QuantumCircuit(2)
qc.h(0)
qc.cx(0, 1)
state = Statevector(qc)
plot_state_qsphere(state)
.. plot::
:include-source:
# You can show the phase of each state and use
# degrees instead of radians
from qiskit.quantum_info import DensityMatrix
import numpy as np
from qiskit import QuantumCircuit
from qiskit.visualization import plot_state_qsphere
qc = QuantumCircuit(2)
qc.h([0, 1])
qc.cz(0,1)
qc.ry(np.pi/3, 0)
qc.rx(np.pi/5, 1)
qc.z(1)
matrix = DensityMatrix(qc)
plot_state_qsphere(matrix,
show_state_phases = True, use_degrees = True)
"""
from matplotlib import gridspec
from matplotlib import pyplot as plt
from matplotlib.patches import Circle
import seaborn as sns
from scipy import linalg
from .bloch import Arrow3D
rho = DensityMatrix(state)
num = rho.num_qubits
if num is None:
raise VisualizationError("Input is not a multi-qubit quantum state.")
# get the eigenvectors and eigenvalues
eigvals, eigvecs = linalg.eigh(rho.data)
if figsize is None:
figsize = (7, 7)
if ax is None:
return_fig = True
fig = plt.figure(figsize=figsize)
else:
return_fig = False
fig = ax.get_figure()
gs = gridspec.GridSpec(nrows=3, ncols=3)
ax = fig.add_subplot(gs[0:3, 0:3], projection="3d")
ax.axes.set_xlim3d(-1.0, 1.0)
ax.axes.set_ylim3d(-1.0, 1.0)
ax.axes.set_zlim3d(-1.0, 1.0)
ax.axes.grid(False)
ax.view_init(elev=5, azim=275)
# Force aspect ratio
# MPL 3.2 or previous do not have set_box_aspect
if hasattr(ax.axes, "set_box_aspect"):
ax.axes.set_box_aspect((1, 1, 1))
# start the plotting
# Plot semi-transparent sphere
u = np.linspace(0, 2 * np.pi, 25)
v = np.linspace(0, np.pi, 25)
x = np.outer(np.cos(u), np.sin(v))
y = np.outer(np.sin(u), np.sin(v))
z = np.outer(np.ones(np.size(u)), np.cos(v))
ax.plot_surface(
x, y, z, rstride=1, cstride=1, color=plt.rcParams["grid.color"], alpha=0.2, linewidth=0
)
# Get rid of the panes
ax.xaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.yaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
ax.zaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))
# Get rid of the spines
ax.xaxis.line.set_color((1.0, 1.0, 1.0, 0.0))
ax.yaxis.line.set_color((1.0, 1.0, 1.0, 0.0))
ax.zaxis.line.set_color((1.0, 1.0, 1.0, 0.0))
# Get rid of the ticks
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])
# traversing the eigvals/vecs backward as sorted low->high
for idx in range(eigvals.shape[0] - 1, -1, -1):
if eigvals[idx] > 0.001:
# get the max eigenvalue
state = eigvecs[:, idx]
loc = np.absolute(state).argmax()
# remove the global phase from max element
angles = (np.angle(state[loc]) + 2 * np.pi) % (2 * np.pi)
angleset = np.exp(-1j * angles)
state = angleset * state
d = num
for i in range(2**num):
# get x,y,z points
element = bin(i)[2:].zfill(num)
weight = element.count("1")
zvalue = -2 * weight / d + 1
number_of_divisions = n_choose_k(d, weight)
weight_order = bit_string_index(element)
angle = (float(weight) / d) * (np.pi * 2) + (
weight_order * 2 * (np.pi / number_of_divisions)
)
if (weight > d / 2) or (
(weight == d / 2) and (weight_order >= number_of_divisions / 2)
):
angle = np.pi - angle - (2 * np.pi / number_of_divisions)
xvalue = np.sqrt(1 - zvalue**2) * np.cos(angle)
yvalue = np.sqrt(1 - zvalue**2) * np.sin(angle)
# get prob and angle - prob will be shade and angle color
prob = np.real(np.dot(state[i], state[i].conj()))
prob = min(prob, 1) # See https://github.com/Qiskit/qiskit-terra/issues/4666
colorstate = phase_to_rgb(state[i])
alfa = 1
if yvalue >= 0.1:
alfa = 1.0 - yvalue
if not np.isclose(prob, 0) and show_state_labels:
rprime = 1.3
angle_theta = np.arctan2(np.sqrt(1 - zvalue**2), zvalue)
xvalue_text = rprime * np.sin(angle_theta) * np.cos(angle)
yvalue_text = rprime * np.sin(angle_theta) * np.sin(angle)
zvalue_text = rprime * np.cos(angle_theta)
element_text = "$\\vert" + element + "\\rangle$"
if show_state_phases:
element_angle = (np.angle(state[i]) + (np.pi * 4)) % (np.pi * 2)
if use_degrees:
element_text += "\n$%.1f^\\circ$" % (element_angle * 180 / np.pi)
else:
element_angle = pi_check(element_angle, ndigits=3).replace("pi", "\\pi")
element_text += "\n$%s$" % (element_angle)
ax.text(
xvalue_text,
yvalue_text,
zvalue_text,
element_text,
ha="center",
va="center",
size=12,
)
ax.plot(
[xvalue],
[yvalue],
[zvalue],
markerfacecolor=colorstate,
markeredgecolor=colorstate,
marker="o",
markersize=np.sqrt(prob) * 30,
alpha=alfa,
)
a = Arrow3D(
[0, xvalue],
[0, yvalue],
[0, zvalue],
mutation_scale=20,
alpha=prob,
arrowstyle="-",
color=colorstate,
lw=2,
)
ax.add_artist(a)
# add weight lines
for weight in range(d + 1):
theta = np.linspace(-2 * np.pi, 2 * np.pi, 100)
z = -2 * weight / d + 1
r = np.sqrt(1 - z**2)
x = r * np.cos(theta)
y = r * np.sin(theta)
ax.plot(x, y, z, color=(0.5, 0.5, 0.5), lw=1, ls=":", alpha=0.5)
# add center point
ax.plot(
[0],
[0],
[0],
markerfacecolor=(0.5, 0.5, 0.5),
markeredgecolor=(0.5, 0.5, 0.5),
marker="o",
markersize=3,
alpha=1,
)
else:
break
n = 64
theta = np.ones(n)
colors = sns.hls_palette(n)
ax2 = fig.add_subplot(gs[2:, 2:])
ax2.pie(theta, colors=colors[5 * n // 8 :] + colors[: 5 * n // 8], radius=0.75)
ax2.add_artist(Circle((0, 0), 0.5, color="white", zorder=1))
offset = 0.95 # since radius of sphere is one.
if use_degrees:
labels = ["Phase\n(Deg)", "0", "90", "180 ", "270"]
else:
labels = ["Phase", "$0$", "$\\pi/2$", "$\\pi$", "$3\\pi/2$"]
ax2.text(0, 0, labels[0], horizontalalignment="center", verticalalignment="center", fontsize=14)
ax2.text(
offset, 0, labels[1], horizontalalignment="center", verticalalignment="center", fontsize=14
)
ax2.text(
0, offset, labels[2], horizontalalignment="center", verticalalignment="center", fontsize=14
)
ax2.text(
-offset, 0, labels[3], horizontalalignment="center", verticalalignment="center", fontsize=14
)
ax2.text(
0, -offset, labels[4], horizontalalignment="center", verticalalignment="center", fontsize=14
)
if return_fig:
matplotlib_close_if_inline(fig)
if filename is None:
return fig
else:
return fig.savefig(filename)
@_optionals.HAS_MATPLOTLIB.require_in_call
def generate_facecolors(x, y, z, dx, dy, dz, color):
"""Generates shaded facecolors for shaded bars.
This is here to work around a Matplotlib bug
where alpha does not work in Bar3D.
Args:
x (array_like): The x- coordinates of the anchor point of the bars.
y (array_like): The y- coordinates of the anchor point of the bars.
z (array_like): The z- coordinates of the anchor point of the bars.
dx (array_like): Width of bars.
dy (array_like): Depth of bars.
dz (array_like): Height of bars.
color (array_like): sequence of valid color specifications, optional
Returns:
list: Shaded colors for bars.
Raises:
MissingOptionalLibraryError: If matplotlib is not installed
"""
import matplotlib.colors as mcolors
cuboid = np.array(
[
# -z
(
(0, 0, 0),
(0, 1, 0),
(1, 1, 0),
(1, 0, 0),
),
# +z
(
(0, 0, 1),
(1, 0, 1),
(1, 1, 1),
(0, 1, 1),
),
# -y
(
(0, 0, 0),
(1, 0, 0),
(1, 0, 1),
(0, 0, 1),
),
# +y
(
(0, 1, 0),
(0, 1, 1),
(1, 1, 1),
(1, 1, 0),
),
# -x
(
(0, 0, 0),
(0, 0, 1),
(0, 1, 1),
(0, 1, 0),
),
# +x
(
(1, 0, 0),
(1, 1, 0),
(1, 1, 1),
(1, 0, 1),
),
]
)
# indexed by [bar, face, vertex, coord]
polys = np.empty(x.shape + cuboid.shape)
# handle each coordinate separately
for i, p, dp in [(0, x, dx), (1, y, dy), (2, z, dz)]:
p = p[..., np.newaxis, np.newaxis]
dp = dp[..., np.newaxis, np.newaxis]
polys[..., i] = p + dp * cuboid[..., i]
# collapse the first two axes
polys = polys.reshape((-1,) + polys.shape[2:])
facecolors = []
if len(color) == len(x):
# bar colors specified, need to expand to number of faces
for c in color:
facecolors.extend([c] * 6)
else:
# a single color specified, or face colors specified explicitly
facecolors = list(mcolors.to_rgba_array(color))
if len(facecolors) < len(x):
facecolors *= 6 * len(x)
normals = _generate_normals(polys)
return _shade_colors(facecolors, normals)
def _generate_normals(polygons):
"""Takes a list of polygons and return an array of their normals.
Normals point towards the viewer for a face with its vertices in
counterclockwise order, following the right hand rule.
Uses three points equally spaced around the polygon.
This normal of course might not make sense for polygons with more than
three points not lying in a plane, but it's a plausible and fast
approximation.
Args:
polygons (list): list of (M_i, 3) array_like, or (..., M, 3) array_like
A sequence of polygons to compute normals for, which can have
varying numbers of vertices. If the polygons all have the same
number of vertices and array is passed, then the operation will
be vectorized.
Returns:
normals: (..., 3) array_like
A normal vector estimated for the polygon.
"""
if isinstance(polygons, np.ndarray):
# optimization: polygons all have the same number of points, so can
# vectorize
n = polygons.shape[-2]
i1, i2, i3 = 0, n // 3, 2 * n // 3
v1 = polygons[..., i1, :] - polygons[..., i2, :]
v2 = polygons[..., i2, :] - polygons[..., i3, :]
else:
# The subtraction doesn't vectorize because polygons is jagged.
v1 = np.empty((len(polygons), 3))
v2 = np.empty((len(polygons), 3))
for poly_i, ps in enumerate(polygons):
n = len(ps)
i1, i2, i3 = 0, n // 3, 2 * n // 3
v1[poly_i, :] = ps[i1, :] - ps[i2, :]
v2[poly_i, :] = ps[i2, :] - ps[i3, :]
return np.cross(v1, v2)
def _shade_colors(color, normals, lightsource=None):
"""
Shade *color* using normal vectors given by *normals*.
*color* can also be an array of the same length as *normals*.
"""
from matplotlib.colors import Normalize, LightSource
import matplotlib.colors as mcolors
if lightsource is None:
# chosen for backwards-compatibility
lightsource = LightSource(azdeg=225, altdeg=19.4712)
def mod(v):
return np.sqrt(v[0] ** 2 + v[1] ** 2 + v[2] ** 2)
shade = np.array(
[np.dot(n / mod(n), lightsource.direction) if mod(n) else np.nan for n in normals]
)
mask = ~np.isnan(shade)
if mask.any():
norm = Normalize(min(shade[mask]), max(shade[mask]))
shade[~mask] = min(shade[mask])
color = mcolors.to_rgba_array(color)
# shape of color should be (M, 4) (where M is number of faces)
# shape of shade should be (M,)
# colors should have final shape of (M, 4)
alpha = color[:, 3]
colors = (0.5 + norm(shade)[:, np.newaxis] * 0.5) * color
colors[:, 3] = alpha
else:
colors = np.asanyarray(color).copy()
return colors
def state_to_latex(
state: Union[Statevector, DensityMatrix], dims: bool = None, convention: str = "ket", **args
) -> str:
"""Return a Latex representation of a state. Wrapper function
for `qiskit.visualization.array_to_latex` for convention 'vector'.
Adds dims if necessary.
Intended for use within `state_drawer`.
Args:
state: State to be drawn
dims (bool): Whether to display the state's `dims`
convention (str): Either 'vector' or 'ket'. For 'ket' plot the state in the ket-notation.
Otherwise plot as a vector
**args: Arguments to be passed directly to `array_to_latex` for convention 'ket'
Returns:
Latex representation of the state
"""
if dims is None: # show dims if state is not only qubits
if set(state.dims()) == {2}:
dims = False
else:
dims = True
prefix = ""
suffix = ""
if dims:
prefix = "\\begin{align}\n"
dims_str = state._op_shape.dims_l()
suffix = f"\\\\\n\\text{{dims={dims_str}}}\n\\end{{align}}"
operator_shape = state._op_shape
# we only use the ket convetion for qubit statevectors
# this means the operator shape should hve no input dimensions and all output dimensions equal to 2
is_qubit_statevector = len(operator_shape.dims_r()) == 0 and set(operator_shape.dims_l()) == {2}
if convention == "ket" and is_qubit_statevector:
latex_str = _state_to_latex_ket(state._data, **args)
else:
latex_str = array_to_latex(state._data, source=True, **args)
return prefix + latex_str + suffix
@deprecate_func(
additional_msg="For similar functionality, see sympy's ``nsimplify`` and ``latex`` functions.",
since="0.23.0",
)
def num_to_latex_ket(raw_value: complex, first_term: bool, decimals: int = 10) -> Optional[str]:
"""Convert a complex number to latex code suitable for a ket expression
Args:
raw_value: Value to convert
first_term: If True then generate latex code for the first term in an expression
decimals: Number of decimal places to round to (default: 10).
Returns:
String with latex code or None if no term is required
"""
if np.around(np.abs(raw_value), decimals=decimals) == 0:
return None
return _num_to_latex(raw_value, first_term=first_term, decimals=decimals, coefficient=True)
@deprecate_func(
additional_msg="For similar functionality, see sympy's ``nsimplify`` and ``latex`` functions.",
since="0.23.0",
)
def numbers_to_latex_terms(numbers: List[complex], decimals: int = 10) -> List[str]:
"""Convert a list of numbers to latex formatted terms
The first non-zero term is treated differently. For this term a leading + is suppressed.
Args:
numbers: List of numbers to format
decimals: Number of decimal places to round to (default: 10).
Returns:
List of formatted terms
"""
first_term = True
terms = []
for number in numbers:
term = num_to_latex_ket(number, first_term, decimals)
if term is not None:
first_term = False
terms.append(term)
return terms
def _numbers_to_latex_terms(numbers: List[complex], decimals: int = 10) -> List[str]:
"""Convert a list of numbers to latex formatted terms
The first non-zero term is treated differently. For this term a leading + is suppressed.
Args:
numbers: List of numbers to format
decimals: Number of decimal places to round to (default: 10).
Returns:
List of formatted terms
"""
first_term = True
terms = []
for number in numbers:
term = _num_to_latex(number, decimals=decimals, first_term=first_term, coefficient=True)
terms.append(term)
first_term = False
return terms
def _state_to_latex_ket(
data: List[complex], max_size: int = 12, prefix: str = "", decimals: int = 10
) -> str:
"""Convert state vector to latex representation
Args:
data: State vector
max_size: Maximum number of non-zero terms in the expression. If the number of
non-zero terms is larger than the max_size, then the representation is truncated.
prefix: Latex string to be prepended to the latex, intended for labels.
decimals: Number of decimal places to round to (default: 10).
Returns:
String with LaTeX representation of the state vector
"""
num = int(np.log2(len(data)))
def ket_name(i):
return bin(i)[2:].zfill(num)
data = np.around(data, decimals)
nonzero_indices = np.where(data != 0)[0].tolist()
if len(nonzero_indices) > max_size:
nonzero_indices = (
nonzero_indices[: max_size // 2] + [0] + nonzero_indices[-max_size // 2 + 1 :]
)
latex_terms = _numbers_to_latex_terms(data[nonzero_indices], decimals)
nonzero_indices[max_size // 2] = None
else:
latex_terms = _numbers_to_latex_terms(data[nonzero_indices], decimals)
latex_str = ""
for idx, ket_idx in enumerate(nonzero_indices):
if ket_idx is None:
latex_str += r" + \ldots "
else:
term = latex_terms[idx]
ket = ket_name(ket_idx)
latex_str += f"{term} |{ket}\\rangle"
return prefix + latex_str
class TextMatrix:
"""Text representation of an array, with `__str__` method so it
displays nicely in Jupyter notebooks"""
def __init__(self, state, max_size=8, dims=None, prefix="", suffix=""):
self.state = state
self.max_size = max_size
if dims is None: # show dims if state is not only qubits
if set(state.dims()) == {2}:
dims = False
else:
dims = True
self.dims = dims
self.prefix = prefix
self.suffix = suffix
if isinstance(max_size, int):
self.max_size = max_size
elif isinstance(state, DensityMatrix):
# density matrices are square, so threshold for
# summarization is shortest side squared
self.max_size = min(max_size) ** 2
else:
self.max_size = max_size[0]
def __str__(self):
threshold = self.max_size
data = np.array2string(
self.state._data, prefix=self.prefix, threshold=threshold, separator=","
)
dimstr = ""
if self.dims:
data += ",\n"
dimstr += " " * len(self.prefix)
dimstr += f"dims={self.state._op_shape.dims_l()}"
return self.prefix + data + dimstr + self.suffix
def __repr__(self):
return self.__str__()
def state_drawer(state, output=None, **drawer_args):
"""Returns a visualization of the state.
**repr**: ASCII TextMatrix of the state's ``_repr_``.
**text**: ASCII TextMatrix that can be printed in the console.
**latex**: An IPython Latex object for displaying in Jupyter Notebooks.
**latex_source**: Raw, uncompiled ASCII source to generate array using LaTeX.
**qsphere**: Matplotlib figure, rendering of statevector using `plot_state_qsphere()`.
**hinton**: Matplotlib figure, rendering of statevector using `plot_state_hinton()`.
**bloch**: Matplotlib figure, rendering of statevector using `plot_bloch_multivector()`.
**city**: Matplotlib figure, rendering of statevector using `plot_state_city()`.
**paulivec**: Matplotlib figure, rendering of statevector using `plot_state_paulivec()`.
Args:
output (str): Select the output method to use for drawing the
circuit. Valid choices are ``text``, ``latex``, ``latex_source``,
``qsphere``, ``hinton``, ``bloch``, ``city`` or ``paulivec``.
Default is `'text`'.
drawer_args: Arguments to be passed to the relevant drawer. For
'latex' and 'latex_source' see ``array_to_latex``
Returns:
:class:`matplotlib.figure` or :class:`str` or
:class:`TextMatrix` or :class:`IPython.display.Latex`:
Drawing of the state.
Raises:
MissingOptionalLibraryError: when `output` is `latex` and IPython is not installed.
ValueError: when `output` is not a valid selection.
"""
config = user_config.get_config()
# Get default 'output' from config file else use 'repr'
default_output = "repr"
if output is None:
if config:
default_output = config.get("state_drawer", "repr")
output = default_output
output = output.lower()
# Choose drawing backend:
drawers = {
"text": TextMatrix,
"latex_source": state_to_latex,
"qsphere": plot_state_qsphere,
"hinton": plot_state_hinton,
"bloch": plot_bloch_multivector,
"city": plot_state_city,
"paulivec": plot_state_paulivec,
}
if output == "latex":
_optionals.HAS_IPYTHON.require_now("state_drawer")
from IPython.display import Latex
draw_func = drawers["latex_source"]
return Latex(f"$${draw_func(state, **drawer_args)}$$")
if output == "repr":
return state.__repr__()
try:
draw_func = drawers[output]
return draw_func(state, **drawer_args)
except KeyError as err:
raise ValueError(
"""'{}' is not a valid option for drawing {} objects. Please choose from:
'text', 'latex', 'latex_source', 'qsphere', 'hinton',
'bloch', 'city' or 'paulivec'.""".format(
output, type(state).__name__
)
) from err
def _bloch_multivector_data(state):
"""Return list of Bloch vectors for each qubit
Args:
state (DensityMatrix or Statevector): an N-qubit state.
Returns:
list: list of Bloch vectors (x, y, z) for each qubit.
Raises:
VisualizationError: if input is not an N-qubit state.
"""
rho = DensityMatrix(state)
num = rho.num_qubits
if num is None:
raise VisualizationError("Input is not a multi-qubit quantum state.")
pauli_singles = PauliList(["X", "Y", "Z"])
bloch_data = []
for i in range(num):
if num > 1:
paulis = PauliList.from_symplectic(
np.zeros((3, (num - 1)), dtype=bool), np.zeros((3, (num - 1)), dtype=bool)
).insert(i, pauli_singles, qubit=True)
else:
paulis = pauli_singles
bloch_state = [np.real(np.trace(np.dot(mat, rho.data))) for mat in paulis.matrix_iter()]
bloch_data.append(bloch_state)
return bloch_data
def _paulivec_data(state):
"""Return paulivec data for plotting.
Args:
state (DensityMatrix or Statevector): an N-qubit state.
Returns:
tuple: (labels, values) for Pauli vector.
Raises:
VisualizationError: if input is not an N-qubit state.
"""
rho = SparsePauliOp.from_operator(DensityMatrix(state))
if rho.num_qubits is None:
raise VisualizationError("Input is not a multi-qubit quantum state.")
return rho.paulis.to_labels(), np.real(rho.coeffs * 2**rho.num_qubits)