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Quantum Circuit $C^*$-algebra Net

Yuka Hashimoto, Ryuichiro Hataya

TL;DR

This work addresses integrating $C^*$-algebra nets with quantum circuits by formulating a quantum circuit $C^*$-algebra net that parameterizes gates as $C^*$-algebra elements on an $m$-state system. By introducing noncommutative interactions among $d$ parallel circuits, the model enables cross-circuit information sharing, which enhances generalization in image classification and improves the quality of data encodings for downstream quantum machine learning tasks. Empirical results on MNIST-family datasets demonstrate that interaction-enabled nets outperform non-interacting baselines and that encoded quantum states can effectively support quantum kernels and QE-based downstream tasks. The work also discusses potential extensions to decentralized learning, nonlinear activations, and optimization on the Stiefel manifold, illustrating a pathway toward integrating algebraic neural nets with quantum hardware.

Abstract

This paper introduces quantum circuit $C^*$-algebra net, which provides a connection between $C^*$-algebra nets proposed in classical machine learning and quantum circuits. Using $C^*$-algebra, a generalization of the space of complex numbers, we can represent quantum gates as weight parameters of a neural network. By introducing additional parameters, we can induce interaction among multiple circuits constructed by quantum gates. This interaction enables the circuits to share information among them, which contributes to improved generalization performance in machine learning tasks. As an application, we propose to use the quantum circuit $C^*$-algebra net to encode classical data into quantum states, which enables us to integrate classical data into quantum algorithms. Numerical results demonstrate that the interaction among circuits improves performance significantly in image classification, and encoded data by the quantum circuit $C^*$-algebra net are useful for downstream quantum machine learning tasks.

Quantum Circuit $C^*$-algebra Net

TL;DR

This work addresses integrating -algebra nets with quantum circuits by formulating a quantum circuit -algebra net that parameterizes gates as -algebra elements on an -state system. By introducing noncommutative interactions among parallel circuits, the model enables cross-circuit information sharing, which enhances generalization in image classification and improves the quality of data encodings for downstream quantum machine learning tasks. Empirical results on MNIST-family datasets demonstrate that interaction-enabled nets outperform non-interacting baselines and that encoded quantum states can effectively support quantum kernels and QE-based downstream tasks. The work also discusses potential extensions to decentralized learning, nonlinear activations, and optimization on the Stiefel manifold, illustrating a pathway toward integrating algebraic neural nets with quantum hardware.

Abstract

This paper introduces quantum circuit -algebra net, which provides a connection between -algebra nets proposed in classical machine learning and quantum circuits. Using -algebra, a generalization of the space of complex numbers, we can represent quantum gates as weight parameters of a neural network. By introducing additional parameters, we can induce interaction among multiple circuits constructed by quantum gates. This interaction enables the circuits to share information among them, which contributes to improved generalization performance in machine learning tasks. As an application, we propose to use the quantum circuit -algebra net to encode classical data into quantum states, which enables us to integrate classical data into quantum algorithms. Numerical results demonstrate that the interaction among circuits improves performance significantly in image classification, and encoded data by the quantum circuit -algebra net are useful for downstream quantum machine learning tasks.
Paper Structure (17 sections, 9 equations, 2 figures, 2 tables)

This paper contains 17 sections, 9 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^*$-algebra
  4. $C^*$-algebra net
  5. Quantum circuit $C^*$-algebra net
  6. Quantum circuit $C^*$-algebra net without interactions
  7. Quantum circuit $C^*$-algebra net with interactions
  8. Connection with commutative and noncommutative $C^*$-algebra nets
  9. Applications
  10. Encoding data to quantum states
  11. Encoding time-series data to a quantum state series
  12. Experiments
  13. Quantum circuit $C^*$-algebra net
  14. Data encoding by quantum circuit $C^*$-algebra net
  15. Conclusion and discussion
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Overview of quantum circuit $C^*$-algebra net for encoding classical data to quantum states
  • Figure 2: Overview of quantum circuit $C^*$-algebra net for encoding classical time-series data to quantum states

Theorems & Definitions (3)

  • Definition 1: $C^*$-algebra
  • Example 1: Commutative $C^*$-algebra
  • Example 2: Noncommutative $C^*$-algebra