Note

This is the documentation for the current state of the development branch of Qiskit Experiments. The documentation or APIs here can change prior to being released.

T2* Ramsey Characterization

The purpose of the $T_2^{\ast }$ Ramsey experiment is to determine two of the qubit’s properties: Ramsey or detuning frequency and $T_2^{\ast }$. In this experiment, we would like to get a more precise estimate of the qubit’s frequency given a rough estimate. The difference between the frequency used for the control rotation pulses and the qubit transition frequency is called the detuning frequency. This part of the experiment is called a Ramsey Experiment. $T_2^{\ast }$ represents the rate of decay toward a mixed state, when the qubit is initialized to the $|1⟩$ state. It is the dephasing time or the transverse relaxation time of the qubit on the Bloch sphere as a result of both energy relaxation and pure dephasing in the transverse plane. Unlike $T_2$, which is measured by T2Hahn, $T_2^{\ast }$ is sensitive to inhomogenous broadening.

Since the detuning frequency is relatively small, we add a phase gate to the circuit to enable better measurement. The actual frequency measured is the sum of the detuning frequency and the user induced oscillation frequency (osc_freq parameter).

import numpy as np
import qiskit
from qiskit_experiments.library import T2Ramsey

The circuits used for the experiment comprise the following steps:

1. Hadamard gate

2. Delay

3. RZ gate that rotates the qubit in the x-y plane

4. Hadamard gate

5. Measurement

The user provides as input a series of delays (in seconds) and the oscillation frequency (in Hz). During the delay, we expect the qubit to precess about the z-axis. If the p gate and the precession offset each other perfectly, then the qubit will arrive at the $|0⟩$ state (after the second Hadamard gate). By varying the extension of the delays, we get a series of oscillations of the qubit state between the $|0⟩$ and $|1⟩$ states. We can draw the graph of the resulting function, and can analytically extract the desired values.

qubit = 0
# set the desired delays
delays = list(np.arange(1e-6, 50e-6, 2e-6))

# Create a T2Ramsey experiment. Print the first circuit as an example
exp1 = T2Ramsey((qubit,), delays, osc_freq=1e5)

print(exp1.circuits()[0])

     ┌────┐┌─────────────────┐┌─────────┐ ░ ┌────┐ ░ ┌─┐
  q: ┤ √X ├┤ Delay(1e-06[s]) ├┤ Rz(π/5) ├─░─┤ √X ├─░─┤M├
     └────┘└─────────────────┘└─────────┘ ░ └────┘ ░ └╥┘
c: 1/═════════════════════════════════════════════════╩═
                                                      0 

We run the experiment on a simulated backend using Qiskit Aer with a pure T1/T2 relaxation noise model.

Note

This tutorial requires the qiskit-aer and qiskit-ibm-runtime packages to run simulations. You can install them with python -m pip install qiskit-aer qiskit-ibm-runtime.

# A T1 simulator
from qiskit_ibm_runtime.fake_provider import FakePerth
from qiskit_aer import AerSimulator
from qiskit_aer.noise import NoiseModel

# Create a pure relaxation noise model for AerSimulator
noise_model = NoiseModel.from_backend(
    FakePerth(), thermal_relaxation=True, gate_error=False, readout_error=False
)

# Create a fake backend simulator
backend = AerSimulator.from_backend(FakePerth(), noise_model=noise_model)

The resulting graph will have the form: $f(t)=a{e}^{-t/T_2\ast }\cdot \cos (2\pi ft+\varphi )+b$ where t is the delay, $T_2^{\ast }$ is the decay factor, and f is the detuning frequency.

# Set scheduling method so circuit is scheduled for delay noise simulation
exp1.set_transpile_options(scheduling_method='asap')

# Run experiment
expdata1 = exp1.run(backend=backend, shots=2000, seed_simulator=101)
expdata1.block_for_results()  # Wait for job/analysis to finish.

# Display the figure
display(expdata1.figure(0))

# Print results
display(expdata1.analysis_results(dataframe=True))

	name	experiment	components	value	quality	backend	run_time	chisq	unit
3b378751	Frequency	T2Ramsey	[Q0]	(9.995+/-0.008)e+04	good	aer_simulator_from(fake_perth)	None	0.64604	Hz
273eaef5	T2star	T2Ramsey	[Q0]	0.000102+/-0.000005	good	aer_simulator_from(fake_perth)	None	0.64604	s

Providing initial user estimates

The user can provide initial estimates for the parameters to help the analysis process. Because the curve is expected to decay toward $0.5$, the natural choice for parameters $A$ and $B$ is $0.5$. Varying the value of $\varphi$ will shift the graph along the x-axis. Since this is not of interest to us, we can safely initialize $\varphi$ to 0. In this experiment, t2ramsey and f are the parameters of interest. Good estimates for them are values computed in previous experiments on this qubit or a similar values computed for other qubits.

user_p0={
    "A": 0.5,
    "T2star": 20e-6,
    "f": 110000,
    "phi": 0,
    "B": 0.5
        }
exp_with_p0 = T2Ramsey((qubit,), delays, osc_freq=1e5)
exp_with_p0.analysis.set_options(p0=user_p0)
exp_with_p0.set_transpile_options(scheduling_method='asap')
expdata_with_p0 = exp_with_p0.run(backend=backend, shots=2000, seed_simulator=101)
expdata_with_p0.block_for_results()

# Display fit figure
display(expdata_with_p0.figure(0))

# Print results
display(expdata_with_p0.analysis_results(dataframe=True))

	name	experiment	components	value	quality	backend	run_time	chisq	unit
5788db8d	Frequency	T2Ramsey	[Q0]	(9.995+/-0.008)e+04	good	aer_simulator_from(fake_perth)	None	0.64604	Hz
00aecfc2	T2star	T2Ramsey	[Q0]	0.000102+/-0.000005	good	aer_simulator_from(fake_perth)	None	0.64604	s

See also

• API documentation: T2Ramsey