(7053) Quantum Simulation - Repeated Detection
Consider a 4 site chain. We choose units such that the hopping frequency is \( c=1/2 \).
We consider the evolution over a fixed time interval, say \( t \in [0,T] \).
The particle is initially localized at one edge of the chain \(x=0\), which we call launch-site.
A detector is positioned at the opposite edge \(x=11=3\), which we call target.
The detector monitors the target site.
The time between successive measurements is \( \tau = T/N \).
Once the particle is detected the rest of the evolution is of no interest.
We would like to get the probability distribution \( f(t) \) of the time-of-detection.
Note that \( f(t) \) is trivially related to the survival probability \( F(t) \).
(C0) Using e.g. Matlab perform calculation of the expected result.
The choice \( T=20\pi \) might be a good for getting interesting results.
Plot several \( F(t) \) lines for different values of \( N \).
Tip: apparently the best presentation procedure is to plot \( N \log(F(t)) \).
Add a dashed line for the probability to detect the particle in a single (not repeated) measurement that is performed at time \( t \).
(C1) Build a circuit block that simulates
the dynamics of a particle in an isolated 4 site chain
during a time interval \( \tau \).
(C2) Build a multi-block circuit that simulates a repeated-measurement experiment.
The state of the particle in the chain is represented by two qubits,
and the measurement device (so called "pointer") is represented by a 3rd qubit.
Tip: a CCX operation is required for the performance of the measurement.
(C3) Perform an IBM experiment, and plot the probability \( f(t) \) to detect the particle at time \( t \).
Note that given \( T \) the number of measurements \( N \) is a free parameter.
Compare the results of the simulation (stars) to the calculation (solid line).