# Measuring T1 and T2 constants on IBM Q

We have been asked to measure relaxation and dephasing times T1 and T2 on the IBM Q using the composer only, Qiskit not allowed. I am a bit confused about the way to do so. Can someone explain the idea behind and how to implement the measurement in QASM?

I cannot give you a complete answer(I am not too familiar with the IBM quantum tools) however I might be able to give you a few hints from a NMR/EPR perspective.

In magnetic resonance T2 is commonly measured by generating a spin coherence, and refocusing at progressively longer times then measuring a spin echo.

In quantum gate language that would be: prepare in state $$|0\rangle$$, Hadamard gate, $$X$$ or $$Y$$ gate, Hadamard gate, measure. And progressively add identity operations between the $$H-X$$ and $$X-H$$. Alternatively you can add progressively more $$X$$ gates and only measure at the odd numbered ones. Though I suppose the latter method would introduce gate errors into your measurement.

And T1 in magnetic resonance is usually measured by an inversion recovery: so prepare qubit in state $$|0\rangle$$, invert with $$X$$ or $$Y$$, and measure at progressively longer delays by using identity operations as delays.

• Thank you! helped me! – Aviv Azran Feb 19 '20 at 16:51

T1 (relaxation time)

T1 is constant connected with spontaneous relaxation from state $$|1\rangle$$ to state $$|0\rangle$$. So firstly apply $$X$$ gate on a qubit to change its state from $$|0\rangle$$ to $$|1\rangle$$, then apply a few $$I$$ gates to make a delay and then measure results. Record a probability of measuring state $$|1\rangle$$ (i.e. relaxation did not occur).

T2 (dephasing time)

T2 is connected with change in phase, for example state $$|+\rangle$$ changes spontaneously to $$|-\rangle$$. To prepare $$|+\rangle$$ apply $$H$$ gate on qubit in state $$|0\rangle$$. Then apply several $$I$$ gates to make a delay. After that apply again $$H$$ gate and do measurement. Record probability of measuring state $$|0\rangle$$ (i.e. phase change did not occur).

T1 and T2 calculation

Both decoherence processes are described by exponential decay law:

$$P(t) = \mathrm{e}^{-\frac{t}{T}}$$

where $$T$$ is eiher T1 or T2 constant, $$t$$ is time between setting qubit to either state $$|1\rangle$$ for T1 or $$|+\rangle$$ for T2 and its measurment. Having probabilities that a qubit is in state either $$|1\rangle$$ and $$|+\rangle$$ and knowing $$t$$ you can easily calculate

$$T = -\frac{t}{\ln P(t)}.$$

Although it is not problem to construct such circuits on IBM Q, I realized that the problem is how to obtain time $$t$$. After simulation you get results with time the simulation actually run on a quantum processor. It seems logical to divide this time with number of shots to get length of one shot and hence $$t$$. I did so on Melbourne processor but it seems that some other operations take place among shots which lengthen time $$t$$. As a result you can not get actual time $$t$$. If you put lenght of simulation to the formula above, resulting T1 is in order of miliseconds which does not make sense.