Qiskit: Single Systems

Qiskit: Single Systems#

Last updated on 9/20/2023
Ref: https://learning.quantum-computing.ibm.com/course/basics-of-quantum-information/single-systems

import numpy as np

Numpy array#

ket0 = np.array([1, 0])
ket1 = np.array([0, 1])
display(ket0)
display(ket1)
print(ket0)
print(ket1)

array([1, 0])

array([0, 1])

[1 0]
[0 1]

ket0/2 + ket1/2

array([0.5, 0.5])

M1 = np.array([[1, 1],
               [0, 0]])
M2 = np.array([[1, 1],
               [1, 0]])

M1/2 + M2/2

array([[1. , 1. ],
       [0.5, 0. ]])

M1 @ M2

array([[2, 1],
       [0, 0]])

M2 @ ket0

array([1, 1])

Qiskit statevector#

https://docs.quantum-computing.ibm.com/api/qiskit/qiskit.quantum_info.Statevector

from qiskit.quantum_info import Statevector

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
/tmp/ipykernel_1819/3907888485.py in <module>
----> 1 from qiskit.quantum_info import Statevector

ModuleNotFoundError: No module named 'qiskit'

Create a statevector#

u = Statevector([1/np.sqrt(2), 1/np.sqrt(2)])
v = Statevector([(1 + 2.0j)/3, -2/3])
w = Statevector([1/3, 2/3])

print(u)

Statevector([0.70710678+0.j, 0.70710678+0.j],
            dims=(2,))

Diaply a statevector#

Statevector([0.70710678+0.j, 0.70710678+0.j],
            dims=(2,))

u.draw("latex")

\[\frac{\sqrt{2}}{2} |0\rangle+\frac{\sqrt{2}}{2} |1\rangle\]

u.draw("bloch")

../_images/1032d385d699ee0497e5d72af712c02d95ecfd27eafc7bb9ad85eb6c503aca49.png

u.draw("qsphere")

../_images/0e512fb936d7e3e0b5c39f8215ca998f4b8e0dc1ac04037721486973f5035025.png

Inner product#

self.inner(other)=\(\langle self|other\rangle\).

u = Statevector([1/np.sqrt(2), 1/np.sqrt(2)])
v = Statevector([(1 + 2.0j)/3, -2/3])
w = Statevector([1/3, 2/3])

u.inner(u)

(0.9999999999999998+0j)

display(u.inner(v))
display(v.inner(u))

(-0.2357022603955158+0.4714045207910316j)

(-0.2357022603955158-0.4714045207910316j)

Check a statevector#

A valid state vector should have its Euclidean norm equal to 1.

u.is_valid()

True

uu = u/2
uu.is_valid()

False

w.is_valid()

False

Measurement of a statevector#

ket.measure()
ket.sample_count()
The quantum state is projected onto the standard basis.
The original quantum state is NOT distroyed.

v = Statevector([(1 + 2.0j) / 3, -2 / 3])
v0 = Statevector([(1 + 2.0j) / 3, 0])
v0 = v0/np.sqrt(v0.inner(v0))
v1 = Statevector([0, -2 / 3])
v1 = v1/np.sqrt(v1.inner(v1))

display(v.draw("latex"))
print(v)
display(v0.draw("latex"))
print(v0)
display(v1.draw("latex"))
print(v1)

\[(\frac{1}{3} + \frac{2 i}{3}) |0\rangle- \frac{2}{3} |1\rangle\]

Statevector([ 0.33333333+0.66666667j, -0.66666667+0.j        ],
            dims=(2,))

\[(\frac{\sqrt{5}}{5} + \frac{2 \sqrt{5} i}{5}) |0\rangle\]

Statevector([0.4472136+0.89442719j, 0.       +0.j        ],
            dims=(2,))

\[- |1\rangle\]

Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))

v.measure()

('1',
 Statevector([ 0.+0.j, -1.+0.j],
             dims=(2,)))

Statevector([ 0.33333333+0.66666667j, -0.66666667+0.j        ],
            dims=(2,))

for _ in range(10):
    out, state = v.measure()
    print(out, state)

0 Statevector([0.4472136+0.89442719j, 0.       +0.j        ],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))
0 Statevector([0.4472136+0.89442719j, 0.       +0.j        ],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))
0 Statevector([0.4472136+0.89442719j, 0.       +0.j        ],
            dims=(2,))
0 Statevector([0.4472136+0.89442719j, 0.       +0.j        ],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))
1 Statevector([ 0.+0.j, -1.+0.j],
            dims=(2,))

Plot distribution#

‘qiskit.visualization.plot_histogram’ is dprecated.

from qiskit.visualization import plot_distribution

statistics = v.sample_counts(1000)
display(statistics)
plot_distribution(statistics)

{'0': 572, '1': 428}

../_images/18c5095cf0b84234c5f153ab52959a51e7b47950eab9b3b2d75c77ab6dc1f17e.png

Qiskit operator#

O|v> = v.evolve(O)

from qiskit.quantum_info import Operator

X = Operator([[0, 1], [1, 0]])
display(X)
v = Statevector([1, 0])
display(v.draw("latex"))
v = v.evolve(X)
display(v.draw("latex"))
v = v.evolve(X)
display(v.draw("latex"))

Operator([[0.+0.j, 1.+0.j],
          [1.+0.j, 0.+0.j]],
         input_dims=(2,), output_dims=(2,))

\[ |0\rangle\]

\[ |1\rangle\]