Harmonic oscillators are promising continuous-variable (CV) quantum resources because their infinite-dimensional Hilbert spaces allow for resource-efficient quantum computing and simulation. To reach their full potential, CV platforms need to be able to efficiently implement non-Gaussian operations. Bosonic quantum-signal-processing schemes have emerged as attractive methods to construct arbitrary non-Gaussian operations; however, these schemes are restricted to single modes, i.e., the implementation of anharmonic potentials. Here, we introduce trigonometric-gate-implemented Fourier synthesis (TGIFS), a method for implementing arbitrary non-Gaussian operations applicable to both single- and multi-mode systems, allowing the generation of both anharmonicities and nonlinear multi-mode couplings. TGIFS synthesizes a target Hamiltonian by decomposing it into a Fourier series whose terms are implemented via bosonic quantum signal processing, which uses a discrete-variable (DV) system to induce a nonlinearity in the CV system. Our hybrid CV-DV protocol allows for the direct simulation of a broad range of CV phenomena (such as those in lattice gauge theory, chemical dynamics, and quantum chaos) and provides a richer toolbox for CV circuit compilation.
Harmonic oscillators are promising continuous-variable (CV) quantum resources because their infinite-dimensional Hilbert spaces allow for resource-efficient quantum computing and simulation. To reach their full potential, CV platforms need to be able to efficiently implement non-Gaussian operations. Bosonic quantum-signal-processing schemes have emerged as attractive methods to construct arbitrary non-Gaussian operations; however, these schemes are restricted to single modes, i.e., the implementation of anharmonic potentials. Here, we introduce trigonometric-gate-implemented Fourier synthesis (TGIFS), a method for implementing arbitrary non-Gaussian operations applicable to both single- and multi-mode systems, allowing the generation of both anharmonicities and nonlinear multi-mode couplings. TGIFS synthesizes a target Hamiltonian by decomposing it into a Fourier series whose terms are implemented via bosonic quantum signal processing, which uses a discrete-variable (DV) system to induce a nonlinearity in the CV system. Our hybrid CV-DV protocol allows for the direct simulation of a broad range of CV phenomena (such as those in lattice gauge theory, chemical dynamics, and quantum chaos) and provides a richer toolbox for CV circuit compilation.
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.
Continuous-variable-discrete-variable (CV-DV) quantum simulators offer a natural route to simulating bosonic dynamics relevant to many branches of physics and chemistry. However, programmable simulation of arbitrary dynamics is an outstanding challenge. In particular, simulating anharmonic dynamics, which is ubiquitous across the physical sciences, is challenging due to the highly harmonic nature of oscillators used in CV-DV simulators. Here, we experimentally demonstrate programmable CV-DV quantum simulation of anharmonic dynamics in a range of double-well potentials, implemented in a trapped-ion system. We synthesise the time-evolution operators using a bosonic-quantum-signal-processing subroutine, which allows the potential to be tuned between experiments by controlling classical experimental parameters. We observe coherent dynamics in various double-well potentials, where a wavepacket tunnels through the potential barrier, and we suppress this effect by programmatically introducing asymmetry.
Abstract Continuous-variable (CV) quantum information processing harnesses versatile experimental tools that leverage the power of infinite-dimensional oscillators controlled by a single qubit. Increasingly available elementary Rabi gates have been proposed as a resource for implementing universal CV gates, but the requirement of many weak, non-commuting gates is a bottleneck in scaling up such an approach. In this study, we propose a resource-efficient technique using Fourier expansion to implement arbitrary non-linear phase gates in a single oscillator. This method reduces the number of sequentially required gates exponentially. These gates represented by cubic, quartic, and other arbitrary nonlinear potentials have applications in CV quantum information processing with infinite-dimensional oscillators controlled by a single qubit. Our method outperforms previous approaches and enables the experimental realization of a wide range of applications, including the development of bosonic quantum sensors, simulations, and computation using trapped ions and superconducting circuits.
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.
Hybrid oscillator-qubit processors have recently demonstrated high-fidelity control of both continuous- and discrete-variable information processing. However, most quantum algorithms remain limited to homogeneous quantum architectures. Here, we present a compiler for hybrid oscillator-qubit processors that implements state preparation and time evolution. In this setting, the compiler invokes generalized quantum signal processing (GQSP) to synthesize arbitrary analytic bosonic phase gates in a constructive manner, with circuit depth scaling as . The approximation cost scales with the Fourier bandwidth of the target bosonic phase, rather than with the degree of nonlinearity. Armed with OQ-GQSP, multi-state vibronic coupling Hamiltonian dynamics can be decomposed into state-dependent arbitrary-phase potential propagators, which also enable extension to multi-state systems through the parity-measurement technique. Compared to fully discrete encodings, our approach avoids the overhead of truncating continuous variables, resulting in linear dependence on the number of vibrational modes. We validate our method on the uracil cation, a canonical system whose accurate modeling requires anharmonic vibronic models, and estimate the cost of state preparation and time evolution.
Continuous-variable quantum systems offer a natural platform for quantum information processing and analogue quantum simulation of real-world processes. Realising these applications, however, requires anharmonic potentials with programmable functional forms, which are challenging to engineer on demand. Here, in a superconducting harmonic oscillator coupled to a transmon qubit, we construct programmable non-Gaussian phase gates e−iV(X^), corresponding to the impulsive action of an anharmonic potential V(X^), and characterise them by reconstructing their corresponding force pointwise. Using bosonic quantum signal processing, we compile target potentials into modular circuits of qubit rotations interleaved with a fixed conditional-displacement primitive. We first demonstrate a cubic phase gate whose cubic nature is jointly validated by the resulting non-Gaussian states and the force reconstructed pointwise from the gate's action on coherent probe states. We then construct a family of double-well potentials and use the reconstructed force to witness the double-well topology and tunable asymmetry. Finally, we engineer an approximate Morse potential and obtain its force profile using position-compressed probes, which extend reliable reconstruction towards the steep repulsive wall. These results establish a programmable route to constructing the potential landscape of quantum harmonic oscillators and a practical method for reconstructing its spatial structure, providing a building block for quantum simulation of anharmonic dynamics.