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arXiv:2606.22832

Linear optical Bell state measurement for rotation-symmetric cat codes

Issa Oe, Suguru Endo, Rui Asaoka

quant-phSubmitted Jun 22, 2026• Announced Jun 23, 2026• 0 citations• Revised Jun 23, 2026 (v2)
v1 · Jun 22, 2026v2 · Jun 23, 2026v3 · Jul 1, 2026
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Abstract

Rotation-symmetric cat (RS-cat) codes are a bosonic-code platform for quantum information processing, combining finite-energy realizability with robustness against photon loss through their discrete rotational symmetry. For applications in long-distance quantum communication and fusion-based quantum computation (FBQC), efficient Bell state measurement (BSM) is a key primitive. In this work, we consider a BSM protocol for RS-cat codes using only a half beam splitter (HBS) and photon-number-resolving detectors (PNRDs). By exploiting the characteristic photon-number structure induced by the discrete rotational symmetry of RS-cat codes, our protocol extracts both photon-number modulo and phase information for Bell-state discrimination. We show that, under ideal loss-free conditions, the proposed BSM protocol becomes deterministic for arbitrary symmetry order $N$ for sufficiently large amplitudes $α$. We further numerically evaluate the success probability under photon loss and identify the loss regime in which higher-order RS-cat codes provide an advantage. Finally, we show that post-selection can enhance the success probability.

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Rotation-symmetric cat (RS-cat) codes are a bosonic-code platform for quantum information processing, combining finite-energy realizability with robustness against photon loss through their discrete rotational symmetry. For applications in long-distance quantum communication and fusion-based quantum computation (FBQC), efficient Bell state measurement (BSM) is a key primitive. In this work, we consider a BSM protocol for RS-cat codes using only a half beam splitter (HBS) and photon-number-resolving detectors (PNRDs). By exploiting the characteristic photon-number structure induced by the discrete rotational symmetry of RS-cat codes, our protocol extracts both photon-number modulo and phase information for Bell-state discrimination. We show that, under ideal loss-free conditions, the proposed BSM protocol becomes deterministic for arbitrary symmetry order NNN for sufficiently large amplitudes ααα. We further numerically evaluate the success probability under photon loss and identify the loss regime in which higher-order RS-cat codes provide an advantage. Finally, we show that post-selection can enhance the success probability.
Abstract Fault-tolerant fusion-based photonic quantum computing (FBQC) greatly relies on entangling two-photon measurements, called fusions. These fusions can be realized using linear-optical projective Bell-state measurements (BSMs). These linear-optical BSMs are limited to a success probability of 50%, greatly reducing the performance of FBQC schemes. The performance of FBQC can be improved using boosting, thus achieving higher success probabilities by adding additional resources. Here, we realize a boosted BSM using a 4 × 4 multiport splitter and an additional entangled photon pair, allowing for a success probability of up to 75%. In our experiment, we obtain a success probability for our boosted BSM of (69.3 ± 0.3)%, clearly exceeding the 50% limit. We further demonstrate the significance of our boosted BSM for FBQC, showing a threefold increase in robustness to photon loss and a significant reduction of the logical error rates.
Fault tolerance is essential for scalable quantum technologies and is enabled by quantum error-correction codes. Bell-state measurements (BSMs) are a fundamental building block for modern quantum technologies such as measurement-based quantum computation and fusion-based quantum computation, as well as quantum networks. Therefore, performing BSMs on error-corrected qubits is a necessary step for achieving fault tolerance in these applications. In this work, we realise a logical BSM using linear optics, based on a two-qubit repetition code, an instance of a quantum parity code that allows detection of bit-flip errors, and experimentally achieve a mean success probability of (70.8 +/- 0.4)%. While standard linear-optical BSMs are fundamentally limited to a maximum success probability of 50%, this increased success probability enables higher secure key rates in quantum communication and facilitates the generation of large graph states for quantum computation. Since fault-tolerant schemes require error-correction codes regardless, this improvement comes at no additional resource overhead. Our results demonstrate that error-correction codes can be used to surpass the linear-optics limit of BSMs, which is an important step towards practical, fault-tolerant, and scalable photonic quantum technologies.
Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger SU(2)SU(2)SU(2) symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.
The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat and binomial codes achieve the breakeven point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, the state preparation fidelity needs to be improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and postprocessing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. Then, we consider two types of stabilization: photon number and phase stabilization. Photon number stabilization can be performed by leveraging the rotation operators. On the other hand, the phase errors can be suppressed via the extension towards phase direction by leveraging the recently proposed projective squeezing method. To implement our method, we use an ancilla qubit and a randomly generated controlled gate between the bosonic code states and the ancilla qubit. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing.
Bell measurements (BMs) are ubiquitous in quantum information and technology. They are basic elements for quantum commmunication, computation, and error correction. In particular, when performed on logical qubits encoded in physical photonic qubits, they allow for a read-out of stabilizer syndrome information to enhance loss tolerance in qubit-state transmission and fusion. However, even in an ideal setting without photon loss, BMs cannot be done perfectly based on the simplest experimental toolbox of linear optics. Here we demonstrate that any logical BM on stabilizer codes can always be mapped onto a single physical BM perfomed on any qubit pair from the two codes. As a necessary condition for the success of a logical BM, this provides a general upper bound on its success probability, especially ruling out the possibility that the stabilizer information obtainable from only partially succeeding, physical linear-optics BMs could be combined into the full logical stabilizer information. We formulate sufficient criteria to find schemes for which a single successful BM on the physical level will always allow to obtain the full logical information by suitably adapting the subsequent physical measurements. Our approach based on stabilizer group theory is generally applicable to any stabilizer code, which we demonstrate for quantum parity, five-qubit, standard and rotated planar surface, tree, and seven-qubit Steane codes. Our schemes attain the general upper bound for all these codes, while this bound had previously only been reached for the quantum parity code.
Bell state measurements (BSMs) play a significant role in quantum information and quantum computing, in particular in fusion-based quantum computing (FBQC). The FBQC model is a framework for universal quantum computing, provided that we are able to perform entangling measurements, called fusions, on qubits within small entangled resource states. Here, we analyze the usage of different linear-optical BSM circuits as fusions in the FBQC schemes and numerically evaluate hardware requirements for fault-tolerance in this framework. We examine and compare the performance of several BSM circuits with varying additional resources and estimate the requirements on losses for every component of the linear-optical realization of fusions under which errors in fusion networks caused by these losses can be corrected. Our results show that fault-tolerant quantum computing in the FBQC model is possible with currently achievable levels of optical losses in an integrated photonic implementation, provided that we can create and detect single photons of the resource states with a total marginal efficiency higher than 0.973.