A Quantum of Learning: Using Quaternion Algebra to Model Learning on Quantum Devices

TL;DR

The paper tackles the challenge of learning on quantum devices by modeling quantum computation and measurement with quaternion algebra and deriving an adaptive learning algorithm via HR-calculus. It introduces a quaternion-valued representation of quantum states and an augmented quaternion space that yields linear, unitary transformations for gates and a tractable gradient-based learning rule. Key contributions include a quaternion-based measurement model, an HR-gradient descent procedure with explicit update equations, and a numerical demonstration on an 8-qubit system showing convergence and accurate probability learning. The framework provides convergence criteria and observability considerations, offering a principled path toward scalable quaternion-based quantum learning with potential practical impact on quantum information processing.

Abstract

This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines. To this end, the division algebra of quaternions is used to derive an effective model for representing computation and measurement operations on qubits. In turn, the derived model, serves as the foundation for formulating an adaptive learning problem on principal quantum learning units, thereby establishing quantum information processing units akin to that of neurons in classical approaches. Then, leveraging the modern HR-calculus, a comprehensive training framework for learning on quantum machines is developed. The quaternion-valued model accommodates mathematical tractability and establishment of performance criteria, such as convergence conditions.

Figures (4)

• Figure 1: Schematic showing the right-hand rotation of $q_{\text{pre}}$ around $\eta$ by an angle of $\theta$ to arrive at $q_{\text{post}}$.
• Figure 2: Bloch sphere representation of qubit $|\psi\rangle$ modelled as the quaternion $q=\imath\sin(\theta)\cos(\phi)+\jmath\sin(\theta)\sin(\phi)+ \kappa\cos(\theta)$.
• Figure 3: The cost, $10\log\left(\frac{1}{8}\sum^{8}_{s=1}\mathsf{d}\left(\hat{d}_{s,n},d_{s,n}\right)\right)$, is shown for $10^{3}$ independent trials of the derived algorithm. The cost across all trials lie within the light red region, while the solid red line showing the average over all trials.
• Figure 4: Likelihoods of $\zeta'$ and its estimate.