T2* Ramsey Characterization¶

The purpose of the $T_{2}^{*}$ Ramsey experiment is to determine two of the qubit’s properties: Ramsey or detuning frequency and $T_{2}^{*}$ . 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. $T_{2}^{*}$ 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, $T_{2}^{*}$ 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:

Hadamard gate
Delay
RZ gate that rotates the qubit in the x-y plane
Hadamard gate
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: $f (t) = a^{- t / T_{2} *} \cdot \cos (2 π f t + ϕ) + b$ where t is the delay, $T_{2}^{*}$ 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
for result in expdata1.analysis_results():
    print(result)

AnalysisResult
- name: @Parameters_T2RamseyAnalysis
- value: CurveFitResult:
 - fitting method: least_squares
 - number of sub-models: 1
  * F_cos_decay(x) = amp * exp(-x / tau) * cos(2 * pi * freq * x + phi) + base
 - success: True
 - number of function evals: 55
 - degree of freedom: 20
 - chi-square: 12.823450300966433
 - reduced chi-square: 0.6411725150483216
 - Akaike info crit.: -6.690006883626211
 - Bayesian info crit.: -0.5956277592852075
 - init params:
  * amp = 0.5
  * tau = 0.0001128424076839831
  * freq = 120071.20598218754
  * phi = -1.5707963267948966
  * base = 0.4766116941529235
 - fit params:
  * amp = 0.491368728646375 ± 0.004916006342544435
  * tau = 0.0001020165082319563 ± 4.464271894146877e-06
  * freq = 99947.5512727167 ± 82.7343978625094
  * phi = 0.004163082478029033 ± 0.012641175702185295
  * base = 0.501137727889661 ± 0.0018216496206444265
 - correlations:
  * (freq, phi) = -0.8232512482515973
  * (amp, tau) = -0.8137656484569767
  * (freq, base) = -0.17127251168225768
  * (amp, freq) = -0.11530229927895952
  * (tau, phi) = -0.1124312557427985
  * (amp, base) = -0.019555140905848853
  * (tau, base) = 0.03653816712062805
  * (tau, freq) = 0.06145064025302784
  * (amp, phi) = 0.16161922673992699
  * (phi, base) = 0.16371035036138787
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: Frequency
- value: (9.995+/-0.008)e+04
- χ²: 0.6411725150483216
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: T2star
- value: 0.000102+/-0.000004
- χ²: 0.6411725150483216
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False

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 $ϕ$ will shift the graph along the x-axis. Since this is not of interest to us, we can safely initialize $ϕ$ 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
for result in expdata_with_p0.analysis_results():
    print(result)

AnalysisResult
- name: @Parameters_T2RamseyAnalysis
- value: CurveFitResult:
 - fitting method: least_squares
 - number of sub-models: 1
  * F_cos_decay(x) = amp * exp(-x / tau) * cos(2 * pi * freq * x + phi) + base
 - success: True
 - number of function evals: 74
 - degree of freedom: 20
 - chi-square: 12.823450300967393
 - reduced chi-square: 0.6411725150483696
 - Akaike info crit.: -6.690006883624335
 - Bayesian info crit.: -0.5956277592833317
 - init params:
  * amp = 0.5
  * tau = 0.0001128424076839831
  * freq = 80106.72996087316
  * phi = 0.0
  * base = 0.4766116941529235
 - fit params:
  * amp = 0.49136873033031153 ± 0.004916006360304091
  * tau = 0.0001020165068145614 ± 4.4642717842585194e-06
  * freq = 99947.55117665333 ± 82.7343993349782
  * phi = 0.004163094706704786 ± 0.012641175763416525
  * base = 0.5011377282284685 ± 0.0018216496199839913
 - correlations:
  * (freq, phi) = -0.823251250482674
  * (amp, tau) = -0.8137656489368544
  * (freq, base) = -0.17127251032704133
  * (amp, freq) = -0.11530230066487511
  * (tau, phi) = -0.11243125938387093
  * (amp, base) = -0.019555140018146396
  * (tau, base) = 0.03653816660984814
  * (tau, freq) = 0.061450642595947895
  * (amp, phi) = 0.16161922789886046
  * (phi, base) = 0.16371034794195535
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: Frequency
- value: (9.995+/-0.008)e+04
- χ²: 0.6411725150483696
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: T2star
- value: 0.000102+/-0.000004
- χ²: 0.6411725150483696
- quality: good
- extra: <3 items>
- device_components: ['Q0']
- verified: False

T2* Ramsey Characterization¶

Providing initial user estimates¶

See also¶