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Efficient Tomography of Microwave Photonic Cluster States

Yoshiki Sunada1,*,†, Shingo Kono2, Jesper Ilves1, Takanori Sugiyama3, Yasunari Suzuki4,5, Tsuyoshi Okubo6, Shuhei Tamate2, Yutaka Tabuchi2, and Yasunobu Nakamura1,2

  • *Contact author: ysunada@stanford.edu
  • †Present address: Department of Applied Physics, Stanford University, Stanford, California 94305, USA.
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PRX Quantum 7, 010323 – Published 3 February, 2026

DOI: https://doi.org/10.1103/3vnw-yjd5

Abstract

Entanglement among a large number of qubits is a crucial resource for many quantum algorithms. Such many-body states have been efficiently generated by entangling a chain of itinerant photonic qubits in the optical or microwave domain. However, it has remained challenging to fully characterize the generated many-body states by experimentally reconstructing their exponentially large density matrices. Here, we develop an efficient tomography method based on the matrix-product-operator formalism and demonstrate it on a cluster state of up to 35 microwave photonic qubits by reconstructing its 235×235 density matrix. The full characterization enables us to detect the performance degradation of our photon source which occurs only when generating a large cluster state. This tomography method is generally applicable to various physical realizations of entangled qubits and provides an efficient benchmarking method for guiding the development of high-fidelity sources of entangled photons.

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Popular Summary

Entanglement is a key feature of quantum mechanics and an essential resource for quantum algorithms. Recent experimental innovations have made it possible to efficiently entangle a large number of itinerant photons. However, it has remained challenging to measure and fully characterize the generated entanglement because it needs to be pinpointed inside the exponentially large quantum state space. In this work, we propose and experimentally demonstrate a full and efficient characterization method for a chain of entangled photons.

We use the fact that the entangling process can be expressed using a tensor network, which is a diagrammatic language developed to efficiently describe a quantum many-body system. This fact allows us to fully characterize the entangled state by estimating only a linear number of parameters with respect to the length of the photon chain. We also develop an efficient procedure for estimating these parameters from the local correlations of the photon chain, which can be measured efficiently using a small number of measurement settings. To demonstrate this procedure, we use a superconducting qubit to entangle a chain of 35 microwave photonic qubits, measure their correlations, and reconstruct their quantum many-body state, which is described by a matrix with 34 billion rows and 34 billion columns.

The methods developed in this work are applicable to other entangling protocols for microwave photons as well as for optical photons, which are another promising system for generating and utilizing large-scale entanglement. Our characterization method can be used to efficiently benchmark and troubleshoot such protocols and help them reach the scale and accuracy required for practical applications.

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PRX Quantum 7, 010323– Published 3 February, 2026
  • Received 8 October 2024
  • Accepted 10 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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