Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leverageddigitalquantum resources, such as discrete-variable (DV) systems like qubits, rather thananalogquantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals— specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.
Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leverageddigitalquantum resources, such as discrete-variable (DV) systems like qubits, rather thananalogquantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals— specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.
This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.
Transferring the information stored in the expansion coefficients of a multi-qubit state to the coefficients of a continuous-variable state is an important protocol for communicating quantum information. It was shown in previous work how to transfer an n-qubit state to a single qumode in O(2n) time. We show that by transferring this state to m qumodes, the runtime can be improved to O(2n/m). Furthermore, we demonstrate how multi-qumode state transfer can be used as a subroutine for approximately realizing the n-qubits quantum Fourier transform on m-qumode with runtime scaling O(m2n/m/ε+m2), accelerating qubit quantum Fourier transform using qumodes. This work presents a scalable approach to convert discrete and continuous quantum information between an arbitrary number of qubits and qumodes. It represents a crucial step forward in mixed analog-digital quantum signal processing for computing, sensing, and communication.
Quantum Signal Processing (QSP) emerges as a groundbreaking paradigm, exploiting the principles of quantum mechanics to revolutionize the analysis, manipulation, and interpretation of signals. This paper introduces a novel framework for QSP, delineating its theoretical foundations and potential to surpass classical signal processing capabilities. The research delves into the development of quantum algorithms that exhibit superior efficiency in the frequency domain analysis and the processing of temporally entangled signal structures. A pivotal aspect of this work is the introduction of quantum Fourier transforms as a mechanism to achieve exponential speed-ups in signal decomposition. Furthermore, the paper explores the implementation of quantum error correction techniques to enhance the robustness of signal processing in the presence of quantum noise and decoherence. The practicality of QSP is demonstrated through simulated quantum circuits, providing a blueprint for future quantum computing hardware applications. The implications of this research are profound, suggesting a transformative impact on fields ranging from secure communications to biomedical imaging. By harnessing the entanglement and superposition properties inherent to quantum systems, QSP is poised to redefine the limits of what is computationally feasible within information processing.
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.
Quantum systems can be fundamentally classified into discrete-variable (DV) and continuous-variable (CV) systems. The degree of quantum advantage achieved with a quantum system is often dictated by the scalability, robustness, and operational feasibility-factors that are closely tied to the nature of the variables employed. Notably, DV and CV systems are largely complementary in these factors. Consequently, integrating DV and CV variables holds significant promise for overcoming the inherent challenges associated with relying solely on one type of variable system. In this study, we build a bidirectional quantum analog-to-digital converter (QADC/QDAC), which facilitates deterministic and reversible quantum information transfer between CV and DV systems. This QADC/QDAC is hardware-efficient, and the error decreases exponentially with the number of qubits, thereby allowing high-fidelity information transfer with a few qubits. With this photonic QADC/QDAC, we demonstrate efficient CV mode tomography leveraging DV tomography techniques and achieve deterministic generation of non-Gaussian CV mode from qubits. Our findings address a critical barrier in constructing hybrid quantum processors and open up new avenues for scientific exploration previously inaccessible to single-variable quantum systems. A quantum analog-to-digital converter is built between photonic wavefront and qubits, which is bidirectional, fully optical, efficient and deterministic-thereby unifying the complementary advantages of discrete-variable and continuous-variable platforms within one architecture.
Making sense of the world happens through receiving and processing signals, whether it be sound, light, or other waves that interact with us. The fields of nuclear magnetic resonance and digital signal processing each built a rigorous language for manipulating and extracting information from signals of particular types. Quantum signal processing (QSP) connects these two traditions, providing a unified algebraic framework for understanding how quantum systems are transformed by both classical and quantum signals. QSP provides a systematic framework for characterizing possible transformations. QSP was developed for qubit systems, but physical quantum systems are rarely two-level and the full richness of high-dimensional Hilbert spaces has remained largely unexplored. This thesis extends and applies QSP in two directions: from qubits to high-dimensional systems such as finite-level qudits and infinite-dimensional oscillators and from static signals to signals that include continuous-time noise processes. We will also show how QSP for qudits leads to novel experimental applications in trapped-ion systems. One type of high-dimensional system is a qubit coupled to an oscillator, and a qubit-oscillator qudit can be formed and universally controlled using only a limited set of interactions. In this qubit-oscillator space, a bosonic version of QSP allows novel incarnations of quantum algorithms and sensing. The quantum Fourier transform can be performed by clever conversion of information in a many-qubit system into a single oscillator, and an interferometric protocol uses binary decisions from a qubit to sense a signal affecting the oscillator. Another type of high-dimensional system is a qudit made from the electronic levels of an atom. This thesis presents a new construction of Grover’s algorithm using a qudit in an atomic hyperfine manifold, along with a calibration method that finds accurate QSP pulse parameters for the experiment. For decision algorithms, QSP in qudits enables theoretically perfect decision-making for quantum channel discrimination problems with certain symmetries. In the face of noise, temporal QSP is developed to extend the canonical framework to continuous-time stochastic noise. Combined with Kubo linear-response theory, a noise filter function description emerges which formalizes the optimization of atomic clock performance. These results from qudit, bosonic, and temporal QSP establish QSP as a natural language for high-dimensional quantum control and sensing.