Bloch Sphere Binary Trees: A method for the visualization of sets of multi-qubit systems pure states

TL;DR

This paper introduces a novel visualization method called Bloch Sphere Binary Trees for representing sets of multi-qubit pure states, combining Schmidt decomposition and Bloch spheres to enhance understanding of quantum state evolution.

Contribution

The work presents a new mapping technique and recursive algorithm for visualizing multi-qubit pure states on a Binary Tree of Bloch Spheres, enabling better analysis of quantum systems.

Findings

Allows visualization of multi-qubit state sets on a single image

Provides insights into quantum state periodicity and entanglement

Open source Python implementation available

Abstract

Understanding the evolution of a multi-qubit quantum system, or elucidating what portion of the Hilbert space is occupied by a quantum dataset becomes increasingly hard with the number of qubits. In this context, the visualisation of sets of multi-qubit pure quantum states on a single image can be helpful. However, the current approaches to visualization of this type only allow the representation of a set of single qubits (not allowing multi-qubit systems) or a just a single multi-qubit system (not suitable if we care about sets of states), sometimes with additional restrictions, on symmetry or entanglement for example. [1{3]. In this work we present a mapping that can uniquely represent a set of arbitrary multi-qubit pure states on what we call a Binary Tree of Bloch Spheres. The backbone of this technique is the combination of the Schmidt decomposition and the Bloch sphere representation. We illustrate how this can be used in the context of understanding the time evolution of quantum states, e.g. providing immediate insights into the periodicity of the system and even entanglement properties. We also provide a recursive algorithm which translates from the computational basis state representation to the binary tree of Bloch spheres representation. The algorithm was implemented together with a visualization library in Python released as open source.