SIMULATION OF PARTICLE IN FINITE SQUARE
POTENTIAL USING IBM QUANTUM

Overview

Simulations of particles in finite square potential were performed on quantum circuit constructed based on time-evolution operator from wavefunction derived from time dependent Schroedinger equation. This research determines the probability distribution of particles in a finite square potential and its energy estimation by applying Quantum Phase Estimation algorithm. Quantum circuits were constructed with Qiskit and simulations were performed on a quantum simulator provided by IBM Quantum. The simulation results show changes in the probability distribution over time indicate time evolution occurs. This shows the ability of the quantum circuit to simulate the system. Different potentials and sigma values show no significant changes in the probability distribution. However, quantum circuit constructed with QPE algorithm cannot obtain proper energy estimation due to nonconvergance value in higher iterations.

Time-Evolution Operator

Time-Evolution Operator was obtained by deriving Time-Dependent Schroedinger Equations (TDSE). The solution of TDSE is a wavefunction in analytical form, manipulated using quantum algorithm, Quantum Fourier Transform (QFT) and third order Trotter Decomposition. The result was a wavefunction in quantum representation known as Time-Evolution Operator as follows:

Above equation then used to construct quantum circuits using IBM Quantum by initializing kinetic energy form and potential form to produce quantum gates.

Quantum Circuits

Probability distributions of particle in finite square potential were obtained by simulating Time-Evolution Operator quantum circuits as follows:

The energy of the sistem was calculated from estimated phase by applying Quantum Phase Estimation (QPE) algorithm on Time-Evolution Operator. The quantum circuit consists of two quantum registers: first quantum register acts as work register for estimating phase, and second quantum register acts as simulation register for simulating the Time-Evolution Operator circuit.

Parameters and Initializations

Gaussian wavefunctions were used as inputs to initialize the Time-Evolution Operator circuits. The circuits simulate even wavefunction $|\psi_0\rangle = e^{-\sigma x^2}$ and odd wavefunction $|\psi_0\rangle = x e^{-\sigma x^2}$, and then determined how it evolves over time.

The sistem was calculated in the region $x=-0.5$ to $x=0.5$, and the potential barriers locate at $x \le -0.25$ and $x \ge 0.25$.

Results

Energy Estimation

Quantum circuit constructed with QPE algorithm cannot obtain proper energy estimation due to nonconvergance value in higher iterations with 4, 5, and 6 qubits as shown in the figure below.

This nonconvergance is caused by some reasons. First, there might be error and imperfection of the constructed circuit, especially in the connection between time-evolution operator in simulation register and working register. Second, the QPE Algorithm implemented in this circuit could not work properly due to the precense of noise and decoherence on the quantum gates. Furthermore, rotational gates used in this circuit caused readout error.

Probability distribution with 4, 5 and 6 qubits

The number of probable states from a quantum system can be influenced by the number of qubits used in the simulation. The more qubit used, the more probable states can be observed. This contributes to the complexity of the time-evolution operator and longer computation time. For example, by using 4 qubits results in 4 states, while using 6 qubits results in $2^6$ states. By this reason, a system with 6 qubits produces more accurate result than a system with lesser qubits.

Probability distribution of a particle with even initial wavefunction calculated using 4, 5, and 6 qubits with 5, 20, and 50 iterations are respectively shown below.

As shown in the figures above, the probability distribution evolved with the increase in iteration. Probability distribution with 4 qubits (upper left), 5 qubits (upper right), and 6 qubits (bottom) show the same general pattern, smoother as more number of qubits used in the simulation.

Probability distribution with different sigma values

Sigma values $\sigma = 10$ and $\sigma = 25$ used in the wavefunctions produce similar probability distributions, indicate that the chosen sigma values have small influences to the systems.

Probability distribution with different potential values

System that simulated with different potential values; $V_1=10$, $V_1=50$, $V_1=100$, and $V_1=150$ show no significant changes in the probability distributions as shown below. This indicates that the chosen potential values have no important effect on the system.

However, systems with potential values $V_1=500$ and $V_1=1000$ show extremely different probability distributions compared to the probability distributions simulated with lower potentials as shown below. This is due to the systems behave as infinite potential well that have too high potential values that can be considered as infinite.