(7051) Quantum Simulation - Two sites

Consider a 2 site system that is represented by a qubit, such that "0" is the first site, and "1" is the second site. The dynamics is generated by \( H =c \sigma_x+(\epsilon/2) \sigma_z \). Define \( U ( t ) = e^{ - i H t} \equiv R ( \theta )\), where \( \theta = 2ct \) is the dimensionless time. This means that if we choose the units of time such that \( c=1/2 \) then \( \theta = t \). This choice of units is used in subsequent exercises.

(A1) For \( \epsilon = 0 \) use the single rotation matrix RX in order to simulate the dynamics. Plot \( P(\theta ) = |\langle 0|RX(\theta )|0\rangle|^2 \) for \( \theta \in [ 0 , T] \) with \( T=3\pi \). The figure should contain the result of the simulation (stars), compared with the analytical result (line).

(A2) Choose \( N \). Define \( \tau = T /N \). Plot \( P(\theta)=|\langle 0|RX(\tau )...RX(\tau )|0\rangle|^2 \). Check what is the largest value of \(N \) that still provides good results. Use of Barrier is required (the icon is vertical dashed line), else the IBM-compiler will try to combine gates.

(A3) For a chosen \( N \), plot \( P(\theta)=|\langle 0|(RX(\tau )RZ(\tau ))...(RX(\tau )RZ(\tau ))|0\rangle|^2 \). That is the Trotter approximation. The plot should contain the result of the simulation (stars), compared with the analytical result (line). Add a few lines for different values of \(N\).

(A4) Conclude what is the optimal choice of \( N \). It should not be too small, else the Trotter approximation is bad. It should not be too large, else the IBM computer accumulates errors due to decoherence.

In the above simulation we have considered unitary evolution \( \rho := U \rho U \). In the subsequent simulations we consider measurement and detection scenarios where the \( \rho \) of the "system" undergoes a non-unitary operation due to the coupling to an external degree of freedom ("pointer").