States#
# import quantum_simulation_recipe as qsr
# from quantum_simulation_recipe import spin_ham
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector, DensityMatrix
from qiskit.visualization import plot_bloch_multivector, plot_histogram, plot_state_city
from qiskit.quantum_info import SparsePauliOp, commutator, anti_commutator, entropy, Statevector, random_statevector, random_density_matrix, partial_trace
import numpy as np
from numpy.linalg import norm
Pure states#
Product states#
## create a statevector from label such as '11'
product_vec = Statevector.from_label('1'*2) # 0, 1, r, l
print('product_vec: ', product_vec)
product_vec: Statevector([0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],
dims=(2, 2))
## to_dict
psi = Statevector.from_label('-0')
print(psi.to_dict())
{'00': (0.7071067811865475+0j), '10': (-0.7071067811865475+0j)}
## state to projector operator
print(psi.to_operator())
print(np.outer(psi.data, psi.data.conj()))
Operator([[ 0.5+0.j, 0. +0.j, -0.5-0.j, 0. +0.j],
[ 0. +0.j, 0. +0.j, 0. -0.j, 0. +0.j],
[-0.5+0.j, 0. +0.j, 0.5+0.j, 0. +0.j],
[ 0. +0.j, 0. +0.j, 0. -0.j, 0. +0.j]],
input_dims=(2, 2), output_dims=(2, 2))
[[ 0.5+0.j 0. +0.j -0.5-0.j 0. +0.j]
[ 0. +0.j 0. +0.j 0. -0.j 0. +0.j]
[-0.5+0.j 0. +0.j 0.5+0.j 0. +0.j]
[ 0. +0.j 0. +0.j 0. -0.j 0. +0.j]]
Overlap#
n = 2
psi0 = random_statevector([2, 2])
psi1 = Statevector.from_label('0'*n)
print('psi0: ', psi0)
psi0.inner(psi1)
psi0: Statevector([ 0.1383967 +0.2600865j , -0.20928601-0.00830353j,
0.11559883-0.30354354j, 0.61472335-0.6212449j ],
dims=(2, 2))
(0.13839669998761342-0.26008649665534855j)
Visualization#
# https://docs.quantum.ibm.com/api/qiskit/qiskit.visualization.plot_bloch_multivector
from qiskit import QuantumCircuit
from qiskit.visualization import plot_bloch_multivector
qc = QuantumCircuit(2)
qc.h(0)
qc.x(1)
state = Statevector(qc)
plot_bloch_multivector(state)
# You can make the bars more transparent to better see the ones that are behind
# if they overlap.
import numpy as np
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, figsize=(8, 4))
# https://docs.quantum.ibm.com/api/qiskit/qiskit.visualization.plot_state_paulivec
# 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")
Random states#
## Haar random pure state
from qiskit.quantum_info import random_statevector
random_state = random_statevector(2**2)
print(random_state)
print(random_state.data)
Statevector([-0.37338293-0.05732617j, -0.31570027+0.20323476j,
-0.04746843-0.04889054j, -0.39519009+0.74532485j],
dims=(2, 2))
[-0.37338293-0.05732617j -0.31570027+0.20323476j -0.04746843-0.04889054j
-0.39519009+0.74532485j]
#### 1-design
Evolution#
Measure#
n = 3
random_state = random_statevector([2]*n)
print(random_state.data)
sample_count = random_state.sample_counts(100)
print(sample_count)
# plot histogram of sample count
from qiskit.visualization import plot_histogram
plot_histogram(sample_count)
[ 0.2692275 -0.20166337j 0.34713242+0.35930287j 0.06224331-0.27153787j
-0.04900098+0.16502855j 0.45275193-0.18436086j 0.24568119-0.41552992j
-0.04925681-0.03146001j -0.19897181+0.12248696j]
{'000': 9, '001': 20, '010': 13, '011': 3, '100': 25, '101': 21, '111': 9}
Mixed state#
## random densitry matrix
from qiskit.quantum_info import random_density_matrix
random_density_matrix(4)
DensityMatrix([[ 0.54496788-1.03339759e-18j, 0.12792975+1.48022386e-01j,
-0.09790559-7.79656825e-03j, -0.23076562+3.38692632e-02j],
[ 0.12792975-1.48022386e-01j, 0.21852742-4.61091820e-19j,
-0.02891472+4.34432728e-03j, -0.06771913+5.96049968e-02j],
[-0.09790559+7.79656825e-03j, -0.02891472-4.34432728e-03j,
0.07696603+2.00420617e-19j, 0.07215407-4.65205368e-02j],
[-0.23076562-3.38692632e-02j, -0.06771913-5.96049968e-02j,
0.07215407+4.65205368e-02j, 0.15953868+1.29406880e-18j]],
dims=(2, 2))
convert a pure state to density matrix#
init_state = Statevector.from_label('0'*2)
init_dm = DensityMatrix(init_state).to_operator()
init_dm
Operator([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],
input_dims=(2, 2), output_dims=(2, 2))
Entangled state#
Bell state#
sv = Statevector([1/np.sqrt(2), 0, 0, -1/np.sqrt(2)])
sv.draw(output='hinton')
GHZ, W states#
# GHZ state
ghz_state = Statevector.from_label('0000') + Statevector.from_label('1111')
# verify
print(ghz_state.is_valid())
# normalize
normalized_ghz = ghz_state/np.linalg.norm(ghz_state)
normalized_ghz.is_valid()
False
True
Stabilizer state#
https://docs.quantum.ibm.com/api/qiskit/qiskit.quantum_info.StabilizerState#stabilizerstate
from qiskit.quantum_info import StabilizerState
stabilizer_list = ["ZXX", "-XYX", "+ZYY"]
stab = StabilizerState.from_stabilizer_list(stabilizer_list)
stab
StabilizerState(['+ZXX', '-XYX', '+ZYY'])
print(stab.clifford)
print('purity: ', stab.purity())
Clifford: Stabilizer = ['+ZXX', '-XYX', '+ZYY'], Destabilizer = ['+IXZ', '-XIZ', '-XXZ']
purity: 1.0
Mixed state#
Pauli representation#
Plot the Pauli-vector representation of a quantum state as bar graph.
The Pauli-vector of a density matrix ρ is defined by the expectation of each possible tensor product of single-qubit Pauli operators (including the identity), that is …. This function plots the coefficients Tr(σρ) as bar graph.
# 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'])
