Introducing Quantum Computing into Statistical Physics: Random Walks and the Ising Model with Qiskit

TL;DR

To address the undergrad physics education gap, this work develops two modules that fuse quantum computing with statistical physics by using Qiskit in Jupyter notebooks to teach a quantum random walk and a quantum-Ising model. The quantum walk module employs a Hadamard coin and a binary position encoding to realize a unitary walk whose distribution shows ballistic spreading with standard deviation $\sigma \propto n$, contrasting with the classical Gaussian diffusion; the Ising module reinterprets Metropolis-like spin updates as quantum circuits, encoding energy contributions via controlled gates and measuring configurations to obtain equilibrium statistics. The work demonstrates concrete methods to encode energy, probability, and dynamics in quantum circuits, including 1D Metropolis progression and a 2D lattice decomposed into two layers for tractability, highlighting both the value and the limits of quantum simulations for classical statistical problems. The resulting notebooks and experiments serve as hands-on pedagogy that builds quantum intuition while reinforcing core statistical physics concepts, and point to extensions such as quantum annealing, real-device experiments, and noise-aware learning.

Abstract

Quantum computing offers a powerful new perspective on probabilistic and collective behaviors traditionally taught in statistical physics. This paper presents two classroom-ready modules that integrate quantum computing into the undergraduate curriculum using Qiskit: the quantum random walk and the Ising model. Both modules allow students to simulate and contrast classical and quantum systems, deepening their understanding of concepts such as superposition, interference, and statistical distributions. We outline the quantum circuits involved, provide sample code and student activities, and discuss how each example can be used to enhance student engagement with statistical physics. These modules are suitable for integration into courses in statistical mechanics, modern physics, or as part of an introductory unit on quantum computing.

Figures (8)

• Figure 1: Diagram of a classical random walk in one dimension. The walker starts at position 0 and moves left or right with equal probability at each step.
• Figure 2: Comparison of probability distributions from classical and quantum random walks. The figure is intended as a qualitative illustration: interference effects in the quantum walk lead to pronounced peaks, in contrast with the smooth, binomial-like spread of the classical walk. Specific simulation details are provided in the accompanying Jupyter notebookJupyter.
• Figure 3: Diagram of a quantum random walk in one dimension. The Hadamard gate creates a superposition state at each step, allowing the walker to explore multiple paths simultaneously. Here, the site index $x$ is represented by a multi-qubit register rather than a single qubit, allowing binary encoding of positions. Thus, the quantum state $\vert x \rangle$ refers to the state of multiple qubits forming a quantum register.
• Figure 4: Quantum circuit (3 qubits) to implement movement to the right in binary form using a CCNOT, a CNOT, and an X gate.
• Figure 5: Quantum circuit (3 qubits) to implement movement to the left in binary form using a CCNOT, a CNOT, and an X gate.