Analog circuit for preparing a Bell state¶

Bell states are maximally entangled, two-qubit states that are ubiquitous in quantum information tasks. Here, we will create the analog quantum program to prepare the two-qubit Bell state, $$ | \psi \rangle = \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle \right) $$ First, we import the relevant libraries and OQD modules.

In [1]:

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import itertools
import warnings
warnings.filterwarnings("ignore")

from oqd_core.interface.analog.operator import *
from oqd_core.interface.analog.operation import *
from oqd_core.backend.metric import *
from oqd_core.backend.task import Task, TaskArgsAnalog

from oqd_analog_emulator.qutip_backend import QutipBackend

import numpy as np import matplotlib.pyplot as plt import seaborn as sns import itertools import warnings warnings.filterwarnings("ignore") from oqd_core.interface.analog.operator import * from oqd_core.interface.analog.operation import * from oqd_core.backend.metric import * from oqd_core.backend.task import Task, TaskArgsAnalog from oqd_analog_emulator.qutip_backend import QutipBackend

Next, we define the one- and two qubit Rabi frequencies and the control Hamiltonians.

In [2]:

X = PauliX()
Y = PauliY()
Z = PauliZ()
I = PauliI()

n = 2  # number of qubits

# 1-qubit & 2-qubit Rabi frequencies
w1 = 2 * np.pi * 1
w2 = 2 * np.pi * 0.1

X = PauliX() Y = PauliY() Z = PauliZ() I = PauliI() n = 2 # number of qubits # 1-qubit & 2-qubit Rabi frequencies w1 = 2 * np.pi * 1 w2 = 2 * np.pi * 0.1

In [3]:

Hii = AnalogGate(hamiltonian=I @ I)
Hxi = AnalogGate(hamiltonian=X @ I)
Hyi = AnalogGate(hamiltonian=Y @ I)
Hxx = AnalogGate(hamiltonian=X @ X)
Hmix = AnalogGate(hamiltonian=-1 * (I @ X))
Hmxi = AnalogGate(hamiltonian=-1 * (X @ I))
Hmyi = AnalogGate(hamiltonian=-1 * (Y @ I))

Hii = AnalogGate(hamiltonian=I @ I) Hxi = AnalogGate(hamiltonian=X @ I) Hyi = AnalogGate(hamiltonian=Y @ I) Hxx = AnalogGate(hamiltonian=X @ X) Hmix = AnalogGate(hamiltonian=-1 * (I @ X)) Hmxi = AnalogGate(hamiltonian=-1 * (X @ I)) Hmyi = AnalogGate(hamiltonian=-1 * (Y @ I))

Now, we apply these Hamiltonians sequentially on the two qubits to prepare the entangled state and measure the whole system.

In [4]:

circuit = AnalogCircuit()

# Hadamard
circuit.evolve(duration=(3 * np.pi) / 2, gate=Hii)
circuit.evolve(duration=np.pi / 2, gate=Hxi)
circuit.evolve(duration=np.pi / 4, gate=Hmyi)

# CNOT
circuit.evolve(duration=np.pi / 4, gate=Hyi)
circuit.evolve(duration=np.pi / 4, gate=Hxx)
circuit.evolve(duration=np.pi / 4, gate=Hmix)
circuit.evolve(duration=np.pi / 4, gate=Hmxi)
circuit.evolve(duration=np.pi / 4, gate=Hmyi)
circuit.evolve(duration=np.pi / 4, gate=Hii)

circuit.measure()

circuit = AnalogCircuit() # Hadamard circuit.evolve(duration=(3 * np.pi) / 2, gate=Hii) circuit.evolve(duration=np.pi / 2, gate=Hxi) circuit.evolve(duration=np.pi / 4, gate=Hmyi) # CNOT circuit.evolve(duration=np.pi / 4, gate=Hyi) circuit.evolve(duration=np.pi / 4, gate=Hxx) circuit.evolve(duration=np.pi / 4, gate=Hmix) circuit.evolve(duration=np.pi / 4, gate=Hmxi) circuit.evolve(duration=np.pi / 4, gate=Hmyi) circuit.evolve(duration=np.pi / 4, gate=Hii) circuit.measure()

Finally, we can emulate the circuit evolution using a classical emulation and track the $Z_1$ and $Z_2$ expectation values.

In [5]:

args = TaskArgsAnalog(
    n_shots=1000,
    fock_cutoff=4,
    metrics={
        "Z_0": Expectation(operator=(Z @ I)),
        "Z_1": Expectation(operator=(I @ Z)),
    },
    dt=1e-2,
)
task = Task(program=circuit, args=args)

args = TaskArgsAnalog( n_shots=1000, fock_cutoff=4, metrics={ "Z_0": Expectation(operator=(Z @ I)), "Z_1": Expectation(operator=(I @ Z)), }, dt=1e-2, ) task = Task(program=circuit, args=args)

In [6]:

backend = QutipBackend()
results = backend.run(task=task)

backend = QutipBackend() results = backend.run(task=task)

Finally, we visualize and plot the results for the circuit evolution and the resultant state.

In [7]:

fig, ax = plt.subplots(1, 1, figsize=[6, 3])
colors = sns.color_palette(palette="crest", n_colors=4)

for k, (name, metric) in enumerate(results.metrics.items()):
    ax.plot(results.times, metric, label=f"$\\langle {name} \\rangle$", color=colors[k])
ax.legend()

fig, ax = plt.subplots(1, 1, figsize=[6, 3]) colors = sns.color_palette(palette="crest", n_colors=4) for k, (name, metric) in enumerate(results.metrics.items()): ax.plot(results.times, metric, label=f"$\\langle {name} \\rangle$", color=colors[k]) ax.legend()

Out[7]:

<matplotlib.legend.Legend at 0x11dc1b230>

In [8]:

fig, axs = plt.subplots(4, 1, sharex=True, figsize=[5, 9])

state = np.array([basis.real + 1j * basis.imag for basis in results.state])
bases = ["".join(bits) for bits in itertools.product("01", repeat=n)]
counts = {basis: results.counts.get(basis, 0) for basis in bases}

ax = axs[0]
ax.bar(x=bases, height=np.abs(state) ** 2, color=colors[0])
ax.set(ylabel="Probability")


ax = axs[1]
ax.bar(x=bases, height=list(counts.values()), color=colors[1])
ax.set(ylabel="Count")

ax = axs[2]
ax.bar(x=bases, height=state.real, color=colors[2])
ax.set(ylabel="Amplitude (real)")

ax = axs[3]
ax.bar(x=bases, height=state.imag, color=colors[3])
ax.set(xlabel="Basis state", ylabel="Amplitude (imag)", ylim=[-np.pi, np.pi])

fig, axs = plt.subplots(4, 1, sharex=True, figsize=[5, 9]) state = np.array([basis.real + 1j * basis.imag for basis in results.state]) bases = ["".join(bits) for bits in itertools.product("01", repeat=n)] counts = {basis: results.counts.get(basis, 0) for basis in bases} ax = axs[0] ax.bar(x=bases, height=np.abs(state) ** 2, color=colors[0]) ax.set(ylabel="Probability") ax = axs[1] ax.bar(x=bases, height=list(counts.values()), color=colors[1]) ax.set(ylabel="Count") ax = axs[2] ax.bar(x=bases, height=state.real, color=colors[2]) ax.set(ylabel="Amplitude (real)") ax = axs[3] ax.bar(x=bases, height=state.imag, color=colors[3]) ax.set(xlabel="Basis state", ylabel="Amplitude (imag)", ylim=[-np.pi, np.pi])

Out[8]:

[Text(0.5, 0, 'Basis state'),
 Text(0, 0.5, 'Amplitude (imag)'),
 (-3.141592653589793, 3.141592653589793)]