Harmonic flow field representations of quantum bits and gates

TL;DR

This paper introduces a method to visualize quantum states and operations as 2D vector fields, offering intuitive insights into entanglement and quantum algorithms through classical fluid flow analogies.

Contribution

It presents a novel mapping of multi-qubit states to 2D vector fields using complex rational functions, enabling visual and conceptual understanding of quantum phenomena.

Findings

Separable states correspond to symmetric flow patterns.

Entangled states are represented by complex superpositions of vector fields.

The approach visualizes quantum algorithms like Fourier transform and Shor's algorithm.

Abstract

We describe a general procedure for mapping arbitrary $n$-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multi-qubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and non-standard computation.