(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).