Hybrid quantum systems that combine discrete-variable (DV) and continuous-variable (CV) architectures represent a promising direction in quantum information science. However, transferring concepts, information and states between such fundamentally different platforms entails both practical and theoretical challenges. The formalisms of these two “universes” differ significantly, and many notions, although sharing the same names, possess distinct properties and physical interpretations. In this work, we construct a bridge between DV and CV systems by means of the tomographic probability representation of quantum states complemented by the Jordan--Schwinger and Holstein--Primakoff maps. While both maps are well known at the operator level, their action on the classical counterparts of quantum states, namely tomograms and other probability representations, has not been addressed in the literature. To the best of our knowledge, this work provides the first explicit demonstration of how the Jordan--Schwinger and Holstein--Primakoff maps act on tomographic probability distributions and Wigner functions, thereby establishing a direct correspondence between the classical measurement statistical descriptions of CV and DV quantum systems.Our tomographic mapping enables a direct transfer of measurement data between different quantum architectures by acting as an intrinsic data-compression kernel. It allows one to obtain the tomogram of a target representation directly from experimentally acquired data in another, without reconstructing the density matrix. This provides a unified framework for transferring and comparing quantum information across heterogeneous quantum hardware platforms, facilitating hybrid protocols, device benchmarking, and the validation of error-correction schemes that rely on mappings between finite- and infinite-dimensional systems.
Hybrid quantum systems that combine discrete-variable (DV) and continuous-variable (CV) architectures represent a promising direction in quantum information science. However, transferring concepts, information and states between such fundamentally different platforms entails both practical and theoretical challenges. The formalisms of these two “universes” differ significantly, and many notions, although sharing the same names, possess distinct properties and physical interpretations. In this work, we construct a bridge between DV and CV systems by means of the tomographic probability representation of quantum states complemented by the Jordan--Schwinger and Holstein--Primakoff maps. While both maps are well known at the operator level, their action on the classical counterparts of quantum states, namely tomograms and other probability representations, has not been addressed in the literature. To the best of our knowledge, this work provides the first explicit demonstration of how the Jordan--Schwinger and Holstein--Primakoff maps act on tomographic probability distributions and Wigner functions, thereby establishing a direct correspondence between the classical measurement statistical descriptions of CV and DV quantum systems.Our tomographic mapping enables a direct transfer of measurement data between different quantum architectures by acting as an intrinsic data-compression kernel. It allows one to obtain the tomogram of a target representation directly from experimentally acquired data in another, without reconstructing the density matrix. This provides a unified framework for transferring and comparing quantum information across heterogeneous quantum hardware platforms, facilitating hybrid protocols, device benchmarking, and the validation of error-correction schemes that rely on mappings between finite- and infinite-dimensional systems.
Quantum technologies are rapidly emerging as a foundation for next-generation information processing, communication, and sensing. In particular, optical platforms stand out as a promising avenue for scalable and efficient architectures. Here, information can be encoded either in discrete variables (DV)—such as polarization, path, frequency, or angular momentum—or in continuous variables (CV) of the optical field, such as quadratures. We develope a new formalism for the joint description of multiphoton and multimode light undergoing Gaussian transformations, and we use it to explore two DV-CV hybrid protocols. The first unifies and extends distinct forms of boson sampling, merging DV scattershot boson sampling with CV Gaussian boson sampling, thereby enabling quantum advantage demonstrations with squeezed photons. The second protocol deals with the Bell state measurement (BSM) for qudits, which are a higher-dimensional generalization of the qubit. Bell measurements play a key role in fusion-based quantum computation entanglement distribution and secret key sharing. We therefore introduce high-dimensional BSMs through linear interferometry combined with multiple single-mode squeezers, outperforming state-of-the-art theoretical methods in linear optics without using auxiliary states. This result shows how bridging the DV and CV paradigms provides a framework in which their combined strengths can outperform their standalone applications and overcome their individual limitations.
Hybrid continuous-variable (CV)-discrete-variable (DV) quantum systems present a promising direction for quantum computing by combining the high dimensional encoding capabilities of qumodes with the control offered by DV qubits on the coupled qumodes. There have been exciting recent progresses on hybrid CV-DV quantum computing, including variational algorithms, error correction, compiler-level optimizations for Hamiltonian simulation, etc. However, there is a lack of a standardized CV-DV benchmark suite for assessing various emerging hardware platforms and evaluating software optimizations on hybrid CV-DV circuits. In this work, we introduce a simulation and benchmarking framework for hybrid CV-DV circuits, implemented using Bosonic Qiskit-a tool specifically designed to model CV-DV systems, along with QuTip for functional correctness verification. We construct and characterize representative CV-DV benchmarks, including cat state generation, GKP state generation, CV-DV state transfers, hybrid quantum Fourier transform, variational quantum algorithms, Hamiltonian simulation, and Shor's algorithm. To assess circuit complexity and scalability, we define a feature map organized into two categories: general features (e.g., qubit/qumode count, gate counts) and CV-DV-specific features (e.g., Wigner negativity, energy, truncation cost). These metrics enable evaluation of both classical simulability and hardware resource requirements. Our results, including one benchmark on real hardware, demonstrate that hybrid CV-DV architectures are not only viable but well-suited for a range of computational tasks, from optimization to Hamiltonian simulation. This framework lays the groundwork for systematic evaluation and future development of hybrid quantum systems.
Transferring quantum information between different types of quantum hardware is crucial for integrated quantum technology. In particular, converting information between continuous-variable (CV) and discrete-variable (DV) devices enables many applications in quantum networking, quantum sensing, quantum machine learning, and quantum computing. This paper addresses the transfer of CV-encoded information between CV and DV devices. We present a resource-efficient method for encoding CV states and implementing CV gates on DV devices, as well as two measurement-based protocols for transferring CV states between CV and DV devices. The success probability of the transfer protocols depends on the measurement outcome and can be increased to near-deterministic values by adding ancillary qubits to the DV devices.
Hybrid quantum systems combining discrete-variable (DV) and continuous-variable (CV) subsystems arise across a broad range of physical platforms and enable quantum information processing beyond purely qubit-based designs. Their classical simulation, however, is challenging since computational costs of both CV and DV systems exponentially increase. In this paper, we introduce a simulation framework for CV--DV hybrid systems based on linear combination of Gaussian (LCoG) decomposition. We decompose the hybrid density matrix into blocks in a DV basis and represent each block as an LCoG expansion. Within the representation, Gaussian operations on the bosonic subsystem independently act on each Gaussian function. We derive analytic formulas describing how complex Gaussian functions transform under controlled Gaussian operations. We further extend the framework to controlled Gaussian channels with state-dependent Gaussian noise. The method avoids a Fock-space truncation for the CV subsystem, and the cost of updating each Gaussian term scales polynomially with the number of modes. The framework provides a general computational tool for analyzing hybrid quantum algorithms, quantum simulation protocols, and non-Gaussian state generation schemes. As an example, we simulate Gottesman--Kitaev--Preskill (GKP) state generation based on a cavity-QED system.
The wave-particle duality of light has led to two different encodings of optical quantum information. One approach, referred to as the discrete-variable (DV) one, relies on finite dimensional quantum systems, while the other one, referred to as the continuous-variable (CV) one, is based on wave-like states belonging to an infinite dimensional Hilbert space. These two encodings have historically been separated, but recently, new hybrid protocols which aim at combining the advantages of the two encodings have emerged [1] . In that prospect our recent demonstration of the measurement-induced generation of hybrid entanglement between discrete- and continuous-variable quantum states [2] , [3] , located at distant places and connected by a lossy channel, has opened the way to the implementation of hybrid protocols and heterogeneous quantum networks. Following the first protocols based on this resource, such as the remote preparation of continuous-variable qubits [4] , or a violation of an Einstein-Podolsky-Rosen steering inequality [5] , we will here report our latest experimental results on entanglement swapping and quantum teleportation protocols using hybrid entangled states based on different encodings [6] .
Transferring quantum information between different types of quantum hardware is crucial for integrated quantum technology. In particular, converting information between continuous variable (CV) and discrete variable (DV) devices enables many applications in quantum networking, quantum sensing, quantum machine learning, and quantum computing. This paper addresses the transfer of CV-encoded information between CV and DV devices. We present a resource-efficient method for encoding CV states and implementing CV gates on DV devices, as well as two measurement-based protocols for transferring CV states between CV and DV devices. The success probability of the transfer protocols depends on the measurement outcome and can be increased to near-deterministic values by adding ancillary qubits to the DV devices.