$$T_2$$ measurement in superconducting qubits

$T_2$ is a characteristic time describing the decoherence rate of a qubit. This script aims to present the protocol to deduce it for the usecase of a transmon qubit.

It consists in a $\pi$ pulse to rotate the qubit into the excited state, then waits for a given time, and then plays another $\pi$ pulse followed by an align operation and then reads out the state using the readout resonator.

Config#

The configuration dictionary is in the configuration.py file and is imported into the main program file t2.py.

The configuration defines two elements: qubit and rr (the readout resonator).

The qubit quantum element defines the qubit we are measuring. The OPX is connected to a mixer via two analog output channels of the OPX, numbered 1 and 2. We also specify the LO frequency received by the mixer using the lo_frequency field of the mixInputs dictionary, and a mixer correction matrix using the mixer field.

The qubit element defines a single operation: X which plays a gaussian pulse to the I channel of the OPX - producing a rotation of the qubit about the X axis. This, of course, must be calibrated to product a proper $\pi$ pulse, e.g. with a Time-Rabi experiment.

The rr quantum element allows to measure the qubit state by measuring the resonant I and Q components of a reflected microwave signal. It defines a readout pulse which is read on input number 1 of the OPX, as set by the output entry of the rr element dictionary. Note also the time_of_flight and smearing parameters which must be defined to perform a measurement. As for the qubit we define the associated lo_frequency and mixer correction.

⚠️Note that failing to declare a digital_marker will not fail program compilation, but will prevent data from being acquired.

Program#

The QUA program T2 is built around two nested for_ loops. The external loop is used for repeated averaging steps, and the internal loop scans the parameter tau. The body of the loops plays the X operation to rotate the qubit into the |1> state. The wait statement is then used with a variable duration tau, followed by another X operation and then an align operation a measurement statement demodulating the readout into the I and Q variables. Each loop ends by a wait statement. The program ends by a recovery_delay period which is assumed to be sufficient to allow qubit to decay back in the ground state.

We run the program on the simulator for 500 clock cycles and take the simulated samples from the simulation job.

The acquired ADC stream is taken from result_handles. Note the change of name of the raw ADC steam from the name we specified in the program (raw_adc) to the name we use to get the stream (raw_adc_input1). This is an idiosyncrasy of the raw ADC interface which does not appear in other QUA data saving mechanisms.

with for_(n, 0, n < NAVG, n + 1):
with for_(tau, 4, tau < taumax, tau + dtau):
play("pi2", "qubit")
wait(tau, "qubit")
play("pi2", "qubit")
align("rr", "qubit")
measure_and_save_state("rr", I_res, Q_res, state_res, th)
wait(recovery_delay // 4, "qubit")
save(tau, tau_vec)

Post-processing#

To get an estimate of the probability to be in the excited state as a function of delay duration, we need to reshape the output streams and calculate the statistics: mean and variance of the probability to be in the excited state.

This is a convenient starting point for integrating with a custom system-specific procedure.and Qua

1. Qubit has 2 lines in from OPX e.g. mixers/modulators and direct line from OPX and 1 line out back to OPX
2. QM's Python package sits at top level along with QM's app - that is where Qua program are run