Quantum Simulation of Schrödinger Equation

Authors: Goga Suknidze, Kira Chaika

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Abstract

In this work, we use the Qiskit library to simulate the time evolution of a quantum particle in one dimension under different potential profiles. The quantum simulation is benchmarked against classical numerical solutions of the same Schrödinger equation to evaluate how accurately the quantum approach reproduces the expected dynamics. We analyze how the accuracy of the simulation depends on the number of qubits and system discretization by comparing fidelity, wavepacket center and width errors, global $L^{2}$ error accumulation, and momentum discretization effects associated with the Nyquist limit.

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1. Introduction

Quantum computing offers a fundamentally different way of processing information compared to classical computational methods. One of its most natural applications is the simulation of quantum systems, since classical simulation becomes increasingly difficult as system complexity grows due to the exponential growth of the Hilbert space [4, 5, 3]. Since quantum system dynamics are governed by the Schrödinger equation, developing quantum algorithms for solving it is of particular interest.

In this work, we use the Qiskit framework to simulate the time evolution of a one-dimensional quantum particle and benchmark the results against corresponding classical numerical solutions. Our approach is based on gate-based quantum simulation using Trotter decomposition and the Quantum Fourier Transform (QFT) to switch between position and momentum representations.

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2. Simulation Methodology

2.1 General Approach

The dynamics of the physical system under consideration is governed by the time-dependent Schrödinger equation [2]:

$$i\hbar\frac{\partial\psi(x,t)}{\partial t}=\left(-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V(x)\right)\psi(x,t)$$

We adopt natural units where $m=1$ and $\hbar=1$. The initial state is defined by a localized Gaussian wavepacket:

$$\psi(x,0)=\frac{1}{(2\pi\sigma^{2})^{1/4}}\exp\left(-\frac{(x-\mu)^{2}}{4\sigma^{2}}+ik_{0}x\right)$$

We evaluate this system under two distinct potentials: a free particle, $V(x)=0$, and a harmonic oscillator dictated by the quadratic potential $V(x)=\frac{1}{2}m\omega^{2}x^{2}.$

The general solution can be written as:

$$\psi(x,t+\Delta t)=e^{-iH\Delta t/\hbar}\psi(x,t)$$

Because the Hamiltonian consists of non-commuting kinetic and potential operators, $H=K+V$, we utilize a split-operator Trotterization method to simulate temporal propagation [6]:

$$e^{-iH\Delta t}\approx e^{-iV\Delta t/2}e^{-iK\Delta t}e^{-iV\Delta t/2}$$

This approximation isolates the operators into bases where they are diagonal. The potential energy operator is diagonal in coordinate space, whereas the kinetic energy operator is diagonal in momentum space. Therefore, the evolution can be written as:

$$K=\frac{P^{2}}{2m}$$

$$\psi(x,t+\Delta t)=e^{-iV(x)\frac{\Delta t}{2}}\mathcal{F}^{-1}e^{-i\frac{p^{2}}{2m}\Delta t}\mathcal{F}e^{-iV(x)\frac{\Delta t}{2}}\psi(x,t)$$

2.2 Classical Simulation Method

To implement the split-operator method classically, the continuous spatial grid is discretized into $N=2^{i}$ uniform points $x_{j}=x_{\min}+j\Delta x$. Consequently the continuous wavefunction maps onto the discretized spatial grid as a complex vector:

$$\psi(t)=(\psi(x_{0},t),\psi(x_{1},t),...,\psi(x_{N-1},t))^{T}$$

with normalization $\sum_{j=0}^{N-1}|\psi(x_{j})|^{2}\Delta x=1$. The momentum grid is defined as:

$$k_{n}=\frac{2\pi}{L}\tilde{n}$$

$$\overline{n}=\begin{cases}n,&n<\frac{N}{2},\ n-N,&n\ge\frac{N}{2}.\end{cases}$$

This definition maps the discrete momentum indices into positive and negative frequency components. We use the Discrete Fourier Transform (DFT) [7]:

$$\tilde{\psi}_{n}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}\psi_{j}e^{-i2\pi jn/N}$$

and the Inverse Discrete Fourier Transform (IDFT):

$$\psi_{j}=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}\tilde{\psi}_{n}e^{i2\pi jn/N}$$

The classical time evolution of the system was numerically evaluated using Python, and the obtained results agree with the expected behavior of a free particle and a harmonic oscillator.

Figure 1: Classical numerical evolution.

2.3 Gate-Based Quantum Simulation Using Qiskit

To compare this approach with the classical method, the key difference lies in the representation of the system. While the classical simulation describes the wavefunction on a discretized spatial grid, the quantum gate-based approach encodes the same information into the computational basis states of a quantum register.

The total number of discretization points depends on the number of qubits used in the simulation, $N = 2^n$ qubits, and the spatial grid is defined as $x_{j}=x_{\min}+j\Delta x$, $j=0,1,...,N-1$. Each computational basis state corresponds to one discretized spatial point, $|j\rangle\leftrightarrow x_{j}$.

The quantum state of the system is represented as $|\psi\rangle=\sum_{j=0}^{N-1}a_{j}|j\rangle$, where the amplitudes are determined from the discretized wavefunction as $a_{j}=\psi(x_{j})\sqrt{\Delta x}$ and satisfy the normalization condition:

$$\sum_{j=0}^{N-1}|\psi(x_{j})|^{2}\Delta x=1\Rightarrow\sum_{j=0}^{N-1}|a_{j}|^{2}=1$$

Since the kinetic energy operator is diagonal in momentum space, the Quantum Fourier Transform (QFT) is used to transform the state between position and momentum representations during the gate-based time evolution [3]:

$$QFT_{N}|x\rangle=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}e^{\frac{2\pi ixk}{N}}|k\rangle$$

Within the Qiskit environment [1], a single Trotter evolution step is constructed as a circuit sequence. The spatial register is initialized, followed by diagonal phase gates representing the potential $e^{-iV(x)\Delta t/2}.$ The QFT shifts the state to the momentum basis, where the kinetic phase $e^{-ik^{2}\Delta t/2m}$ is applied, and the Inverse QFT returns the system to the spatial basis.

Figure 2: Quantum gate-based evolution.

3. Results

To analyze the accuracy of the quantum gate-based simulation, we compared the obtained quantum results with the corresponding classical numerical solution. First, we studied how the accuracy depends on the number of qubits by calculating the wavepacket center error, width error, and global $L^{2}$ error:

$$E_{L^{2}}=|\psi_{q}-\psi_{c}|^{2}$$

Figure 3: Center, width, and global L^2 errors between quantum and classical simulations as functions of qubit number.

The state fidelity was also calculated to measure how strongly the quantum state overlaps with the corresponding classical solution. Fidelity values close to one indicate that the two states are almost identical:

$$F(t)=|\langle\psi_{\text{quantum}}(t)|\psi_{\text{classical}}(t)\rangle|^{2}$$

Figure 4: Fidelity between quantum and classical states during time evolution.

Finally, we analyzed the role of momentum discretization and the Nyquist limit. Since the spatial grid is finite, the momentum values that can be represented are also limited, with $k_{\max}=\frac{\pi}{\Delta x}$. If the initial momentum becomes too large for the chosen grid resolution, the simulation accuracy decreases due to aliasing effects.

Figure 5: Simulation breakdown near Nyquist limit.

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4. Conclusion

In this work, we implemented a gate-based quantum simulation of the one-dimensional time-dependent Schrödinger equation using the Qiskit framework and compared the obtained results with corresponding classical numerical solutions. The simulations reproduced the expected behavior of free-particle and harmonic oscillator evolution, showing strong agreement between quantum and classical methods.

The obtained results demonstrate that the accuracy of the simulation depends strongly on the number of qubits and system discretization. Increasing the number of qubits improves the spatial and momentum resolution, while boundary and discretization effects become the dominant source of inaccuracies at larger evolution times and momentum values.

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References

• [1] H. Abraham et al., Qiskit: An Open-source Framework for Quantum Computing, Zenodo, 2021.
• [2] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd Edition, Cambridge University Press, 2018.
• [3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• [4] R. P. Feynman, Simulating Physics with Computers, International Journal of Theoretical Physics, Vol. 21, 1982.
• [5] S. Lloyd, Universal Quantum Simulators, Science, Vol. 273, 1996.
• [6] H. F. Trotter, On the Product of Semi-Groups of Operators, Proceedings of the American Mathematical Society, Vol. 10, 1959.
• [7] W. H. Press et al., Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007.