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Physica Scripta

Continuous-variable quantum computing on a trapped ion: neural network applications

Alexandre C. Ricardo, Gubio G. de Lima, Amanda G. Valério, Tiago de Souza Farias, Celso J. Villas-Bôas

Published Nov 19, 2025• 1 citation
Publisher

Abstract

Abstract Continuous-variable quantum computing (CVQC) uses quantum states with continuous degrees of freedom, such as the quadratures of bosonic modes, to encode information, promising efficient solutions to complex problems. Trapped-ion systems provide a robust platform with long coherence times and precise qubit control, enabling the manipulation of quantum information through its motional and electronic degrees of freedom. In this theoretical work, we quantitatively analyze the fidelity of a Kerr operation in a single trapped-ion system, calibrating the interaction time using a motional-state Rabi frequency correction. Through numerical simulations, we validate Gaussian and non-Gaussian CVQC operations, achieving fidelities exceeding 99% in all cases, including non-Gaussian operations. Furthermore, we demonstrate two key applications: first, the implementation of a continuous-variable quantum neural network for regression problems, which exhibits low error rates; and second, its application to state preparation tasks, achieving a fidelity of 90% for the given example. These results open new opportunities to execute quantum computing algorithms on continuous variables with trapped ions, such as quantum machine learning algorithms.

DOI
10.1088/1402-4896/ae2164

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Abstract Continuous-variable quantum computing (CVQC) uses quantum states with continuous degrees of freedom, such as the quadratures of bosonic modes, to encode information, promising efficient solutions to complex problems. Trapped-ion systems provide a robust platform with long coherence times and precise qubit control, enabling the manipulation of quantum information through its motional and electronic degrees of freedom. In this theoretical work, we quantitatively analyze the fidelity of a Kerr operation in a single trapped-ion system, calibrating the interaction time using a motional-state Rabi frequency correction. Through numerical simulations, we validate Gaussian and non-Gaussian CVQC operations, achieving fidelities exceeding 99% in all cases, including non-Gaussian operations. Furthermore, we demonstrate two key applications: first, the implementation of a continuous-variable quantum neural network for regression problems, which exhibits low error rates; and second, its application to state preparation tasks, achieving a fidelity of 90% for the given example. These results open new opportunities to execute quantum computing algorithms on continuous variables with trapped ions, such as quantum machine learning algorithms.
Continuous-variable quantum computing utilizes continuous parameters of a quantum system to encode information, promising efficient solutions to complex problems. Trapped-ion systems provide a robust platform with long coherence times and precise qubit control, enabling the manipulation of quantum information through its motional and electronic degrees of freedom. In this work, quantum operations that can be generated in trapped-ion systems are employed to investigate applications aimed at state preparation in continuous-variable quantum systems.
Continuous-variable (CV) quantum computing has shown great potential for building neural network models. These neural networks can have different levels of quantum-classical hybridization depending on the complexity of the problem. Previous work on CV neural network protocols required the implementation of non-Gaussian operators in the network. These operators were used to introduce non-linearity, an essential feature of neural networks. However, these protocols are hard to execute experimentally. We built a CV hybrid quantum-classical neural network protocol that can be realized experimentally with current photonic quantum hardware. Our protocol uses Gaussian gates only with the addition of ancillary qumodes. We implemented non-linearity through repeat-until-success measurements on ancillary qumodes. To test our neural network, we studied canonical machine learning and quantum computer problems in a supervised learning setting -- state preparation, curve fitting, and classification problems. We achieved high fidelity in state preparation of single-photon (99.9%), cat (99.8%), and Gottesman-Kitaev-Preskill (93.9%) states, a well-fitted curve in the presence of noise at a cost of less than 1%, and more than 95% accuracy in classification problems. These results bode well for real-world applications of CV quantum neural networks.
We explore the feasibility of gate-based hybrid quantum computing using both discrete (qubit) and continuous (qumode) variables on trapped-ion platforms. Trapped-ion systems have demonstrated record one- and two-qubit gate fidelities and long qubit coherence times, while qumodes, which can be represented by the collective vibrational modes of the ion chain, have remained relatively unexplored for their use in computing. Using numerical simulations, we show that high-fidelity hybrid gates and measurement operations can be achieved for existing trapped-ion quantum platforms. As an exemplary application, we consider quantum simulations of the Jaynes-Cummings-Hubbard model, which is given by a one-dimensional chain of interacting spin and boson degrees of freedom. Using classical simulations, we study its real-time evolution and develop a suitable variational quantum algorithm for ground state preparation. Our results motivate further studies of hybrid quantum computing in this context, which may lead to direct applications in condensed matter and fundamental particle and nuclear physics.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.
Using discrete and continuous variable subsystems, hybrid approaches to quantum information could enable more quantum computational power for the same physical resources. Here, we propose a hybrid scheme that can be used to generate the necessary Gaussian and non-Gaussian operations for universal continuous variable quantum computing in trapped ions. This scheme utilizes two linear spin-motion interactions to generate a broad set of nonlinear effective spin-motion interactions including one- and two-mode squeezing, beam splitter, and trisqueezing operations in trapped ion systems. We discuss possible experimental implementations using laser-based and laser-free approaches.
Audio classification is a key application in deep learning, and quantum machine learning (QML) provides new possibilities by capturing complex patterns in signals through parameterized quantum circuits. However, existing qubit-based models struggle to efficiently represent continuous-valued data common in audio processing, limiting their scalability and expressivity. Continuous Variable Quantum Computing (CVQC) offers a natural framework for processing real-valued data such as mel-spectrograms, addressing the limitations of qubit-based systems while introducing nonlinearity through non-gaussian gates to ensure universality. In CVQC, each qumode represents an infinite-dimensional Hilbert space, enabling compact yet expressive information representation. This work introduces a two-qumode Hybrid Continuous Variable Quantum Neural Network (HCVQNN) for audio classification that integrates classical neural layers with a CV quantum circuit for advanced feature extraction and encoding. Experiments conducted using PennyLane demonstrate that HCVQNN achieves comparable audio classification accuracy while requiring fewer resources compared to classical deep networks and qubit-based quantum models.