(7052) Quantum Simulation - Repeated Measurement

Consider a 2 state system, represented by a qubit \( q_{sys} \). The system is prepared at \( (x=0) \). The system can flip state coherently. The dynamics is generated by \( H = c \sigma_x \). Without loss of generality the units of time can be chosen such that \( c=1/2 \), hence the time is like angle. A detector is used in order to measure whether the system is at destination state \( (x=1) \). For a repeated measurement scenario we define \( p(t) \) as the probability to find the particle at the monitored site, and accordingly we define the polarization as \( M(t) = 1-2p(t) \). For a repeated detection scenario we define \( f(t) \) as the probability to detect the particle at time \( t \), and \( F(t) \) as the survival probability. Note that \( f(t_n) = F(t_{n{-}1}) - F(t_n) \), meaning that \( F(t) \) is the inverse-cumulative distribution. For the purpose of presentation one might prefer to plot \( M(t) \) instead of \( p(t) \), and \( F(t) \) instead of \( f(t) \).

The measurement device (so called "pointer") is represented by a 2nd qubit \( q_{meas} \). The CX operation is required for the performance of the measurement. This is followed by an IBM readout operation. The outcome is stored in a classical bit \( c_n \), where \( n=1,2,3,...,N \) labels the bit that is used to store the result of the \( n \)-th measurement. In order to allow repeated measurements, a reset operation on the pointer should be performed after the readout operation. Statistically speaking, after each measurement the state of the system (i.e. the state of the \( q_{sys} \) qubit) becomes a mixture of \(x=0\) and \(x=1\).

(B0) An IBM experiment provides the distribution of \( c = (c_1,c_2,...c_N) \). Note that there are \(2^N\) possible sequences, and therefore most of the sequences will have zero count. The output file of IBM documents only sequences with non-zero count. Explain how to extract \( p(t_n) \) and \( f(t_n) \) from the outcome of an IBM experiment, and write a routine for that.

(B1) Build a circuit that simulates a repeated measurement experiment. The simulation consists of \( N \) time steps. It is enough to consider evolution up to \( T=2\pi \). Each time step has duration \( \tau = T/N \), which is like small-angle rotation. For simulation of coherent evolution the measurement is performed only at the last step. For simulation of supervised evolution the measurement is performed at each step.

(B2) Perform IBM experiment using your circuits. Select large enough \( N \) to demonstrate the Zeno effect. Extract \( F(t) \) and \( M(t) \) from the outcome of the experiment. Repeat the experiment with a simpler circuit where the readout is done directly on the "system" without using a pointer. (Such shortcut cannot be implemented in ex7053).

(B3) Compare the the results of the experiment with the analytical predictions of ex7045. In your plots use symbols for the results of the experiment, and lines for the the theoretical prediction. Test different values of \( N \).