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arXiv:2506.11236

Configuration design of multimode Gaussian operations on continuous-variable quad-rail lattice cluster states

Jun–ichi Yoshikawa, Warit Asavanant, H. Nagayoshi, Sakaguchi, Atsushi, Shota Yokoyama, Hidehiro Yonezawa, Akira Furusawa

Published Jun 12, 2025• 0 citations
arXivHTMLPDF

Abstract

Continuous-variable quad-rail lattice cluster states enable flexible quantum circuit design on their two-dimensional structure. However, how to combine basic operations on the quad-rail lattice cluster state to realize multimode operations has not been deeply discussed. Here we show a concrete configuration design to efficiently implement beamsplitter network operations on the cluster state. Furthermore, combining the beamsplitter networks, a configuration design of multimode Gaussian unitary operations is also shown. It is theoretically known that the Gaussian operations are sufficient for universal quantum computation if appropriate non-Gaussian states are injected. Our results are fundamentally important for utilizing the flexible quad-rail lattice cluster states for computations.

DOI
10.48550/arxiv.2506.11236

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Continuous-variable quad-rail lattice cluster states enable flexible quantum circuit design on their two-dimensional structure. However, how to combine basic operations on the quad-rail lattice cluster state to realize multimode operations has not been deeply discussed. Here we show a concrete configuration design to efficiently implement beamsplitter network operations on the cluster state. Furthermore, combining the beamsplitter networks, a configuration design of multimode Gaussian unitary operations is also shown. It is theoretically known that the Gaussian operations are sufficient for universal quantum computation if appropriate non-Gaussian states are injected. Our results are fundamentally important for utilizing the flexible quad-rail lattice cluster states for computations.
We consider measurement-based quantum computation that uses scalable continuous-variable cluster states with a one-dimensional topology. The physical resource, known here as the dual-rail quantum wire, can be generated using temporally multiplexed offline squeezing and linear optics or by using a single optical parametric oscillator. We focus on an important class of quantum gates, specifically Gaussian unitaries that act on single quantum modes (qumodes), which gives universal quantum computation when supplemented with multi-qumode operations and photon-counting measurements. The dual-rail wire supports two routes for applying single-qumode Gaussian unitaries: The first is to use traditional one-dimensional quantum-wire cluster-state measurement protocols. The second takes advantage of the dual-rail quantum wire in order to apply unitaries by measuring pairs of qumodes called macronodes. We analyze and compare these methods in terms of the suitability for implementing single-qumode Gaussian measurement-based quantum computation.
Due to its unique scalability potential, continuous-variable quantum optics is a promising platform for large-scale quantum computing. In particular, very large cluster states with a two-dimensional topology that are suitable for universal quantum computing and quantum simulation can be readily generated in a deterministic manner, and routes towards fault tolerance via bosonic quantum error correction are known. In this article we propose a complete measurement-based quantum computing architecture for the implementation of a universal set of gates on the recently generated two-dimensional cluster states [M. V. Larsen et al., Science 366, 369 (2019); W. Asavanant et al., Science 366, 373 (2019)]. We analyze the performance of the various quantum gates that are executed in these cluster states as well as in other two-dimensional cluster states (the bilayer-square lattice and quad-rail lattice cluster states [R. N. Alexander et al., Phys. Rev. A 94, 032327 (2016); N. C. Menicucci, Phys. Rev. A 83, 062314 (2011)]) by estimating and minimizing the associated stochastic noise addition as well as the resulting gate error probability. We compare the four different states and find that, although they all allow for universal computation, the quad-rail lattice cluster state performs better than the other three states, which all exhibit similar performance.
We describe a generalization of the cluster-state model of quantum computation to continuous-variable systems, along with a proposal for an optical implementation using squeezed-light sources, linear optics, and homodyne detection. For universal quantum computation, a nonlinear element is required. This can be satisfied by adding to the toolbox any single-mode non-Gaussian measurement, while the initial cluster state itself remains Gaussian. Homodyne detection alone suffices to perform an arbitrary multimode Gaussian transformation via the cluster state. We also propose an experiment to demonstrate cluster-based error reduction when implementing Gaussian operations.
We experimentally demonstrate the continuous-variable quantum approximate optimization algorithm (CV-QAOA) for multi-variable problems and multiple QAOA depths using a measurement-based CV quantum computing platform on a quad-rail lattice (QRL) cluster state. We propose a systematic method to map arbitrary quadratic cost functions onto the QRL architecture and examine the resulting construction in settings involving up to 100 modes. Using the programmable platform, we prepare the CV-QAOA ansatz and optimize the variational parameters via Bayesian optimization. We then investigate the performance on quadratic optimization problems and observe that increasing the depth from 1 to 2 improves performance, whereas further increases yield only limited gains. In contrast, numerical simulations under idealized conditions, assuming an infinite number of measurement shots and gradient-based optimization, indicate that the performance of CV-QAOA can improve with increasing depth, suggesting that the experimentally observed limitations primarily arise from noise accumulation and classical optimization challenges. This work provides an experimental demonstration of CV-QAOA on a programmable CV platform and establishes a foundation for future developments of variational quantum algorithms in CV systems.
Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally outscaled qubit platforms by orders of magnitude.Room-temperature implementation of CV cluster states has been achieved with quantum optics by using multimode squeezed Gaussian states.It has also been proven that fault tolerance thresholds for CV quantum computing can be reached at realistic squeezing levels.In this paper, we show that standard approaches to design and characterize CV cluster states can miss entanglement present in the system.Such hidden entanglement may be used to increase the power of a quantum computer but it can also, if undetected, hinder the successful implementation of a quantum algorithm.By a detailed analysis of the structure of Gaussian states, we derive an algorithm that reveals hidden entanglement in an arbitrary Gaussian state and optimizes its use for one-way quantum computing.
We propose an experimental design for universal continuous-variable quantum computation that incorporates recent innovations in linear-optics-based continuous-variable cluster state generation and cubic-phase gate teleportation. The first ingredient is a protocol for generating the bilayer-square-lattice cluster state (a universal resource state) with temporal modes of light. With this state, measurement-based implementation of Gaussian unitary gates requires only homodyne detection. Second, we describe a measurement device that implements an adaptive cubic-phase gate, up to a random phase-space displacement. It requires a two-step sequence of homodyne measurements and consumes a (non-Gaussian) cubic-phase state.