Non-Gaussian operations are essential for most bosonic quantum technologies. Yet, realizable non-Gaussian gates are rather limited in type and generally suffer from accuracy-duration trade-offs. In this work, we propose to use quantum signal processing (QSP) techniques to engineer non-Gaussian gates on hybrid qumode-qubit systems. For systems with dispersive coupling, our scheme can generate a new non-Gaussian gate that produces a phase shift depending on the modulus of the boson number. This gate reproduces the selective number-dependent arbitrary phase (SNAP) gates under certain parameter choices, but with higher accuracy within a short, fixed and excitation-independent interaction time. The gate unlocks new applications, for example, in entangling logical qudits and deterministically generating multi-component cat states. Additionally, our versatile QSP formalism can be extended to systems with other interactions, and also engineer non-unitary operations, such as noiseless linear amplification and generalized-parity measurement.
Non-Gaussian operations are essential for most bosonic quantum technologies. Yet, realizable non-Gaussian gates are rather limited in type and generally suffer from accuracy-duration trade-offs. In this work, we propose to use quantum signal processing (QSP) techniques to engineer non-Gaussian gates on hybrid qumode-qubit systems. For systems with dispersive coupling, our scheme can generate a new non-Gaussian gate that produces a phase shift depending on the modulus of the boson number. This gate reproduces the selective number-dependent arbitrary phase (SNAP) gates under certain parameter choices, but with higher accuracy within a short, fixed and excitation-independent interaction time. The gate unlocks new applications, for example, in entangling logical qudits and deterministically generating multi-component cat states. Additionally, our versatile QSP formalism can be extended to systems with other interactions, and also engineer non-unitary operations, such as noiseless linear amplification and generalized-parity measurement.
This repository contains the Wolfram Mathematica source code associated with the article “Engineering non-Gaussian bosonic gates through quantum signal processing” by Pak-Tik Fong and Hoi-Kwan Lau. The notebooks implement numerical calculations and simulations for engineering non-Gaussian bosonic operations using quantum signal processing (QSP). They include the construction of QSP rotation sequences for mod-kk gates and noiseless linear amplification (NLA) under dispersive coupling. Specifically, the code compares QSP-based implementations with conventional multitone-driving schemes for dispersive- and Jaynes-Cummings-coupled qumode-qubit systems, including the simulations used to generate Figs. 2, 5(a), and 5(b) of the paper. The notebooks were developed using Wolfram Mathematica 12.0.
Realizing coherent quantum computation requires precise and robust manipulation of quantum systems through quantum control protocols. Most quantum control techniques rely on heuristic methods for designing the driving pulses that steer the system towards a target state. Such methods are often based on brute-force optimization and offer limited understanding of the solution landscape. In contrast, quantum algorithms offer a rich body of analytical methods with rigorous error guarantees for implementing unitary and non-unitary transformations, which suggests a promising direction for developing new approaches to quantum control. Among various such algorithms, quantum signal processing (QSP) has emerged as a powerful framework for quantum algorithm design, implementation, and optimization. However, its potential for quantum control remains largely unexplored. In this work, we establish QSP-Control, an analytical framework for quantum control of qubit-oscillator dynamics. We focus on dispersively coupled qubit-oscillator systems and employ the QSP formalism to mitigate unwanted nonlinear effects arising from cross-Kerr interactions. In addition, we develop constructions for precise manipulation of Fock states by designing Fock-state-selective operators, based on structural parallels between the Jaynes-Cummings interaction and QSP. These findings demonstrate how several practically relevant problems in quantum control can be mapped to forms amenable to QSP, offering both a systematic design framework and an interpretable perspective on quantum control.
Quantum signal processing (QSP) transforms a unitary parametrized by a classical variable θ into one governed by a polynomial function f ( θ ) . Though quantum mechanics is linear, such highly nonlinear transformations arise naturally from the curvature of the qubit Bloch sphere. This capability underpins many quantum algorithms and enables robust control by decreasing sensitivity to errors in a control parameter, or quantum sensing by increasing sensitivity to variations one seeks to measure. Here, we extend QSP from classical, commuting control parameters to noncommuting quantum operators. We call this new multivariate class . Rather than treating the noncommutativity of quantum control variables as a complication, non-Abelian QSP exploits their richer commutator algebra as a resource for efficient control. A natural setting is hybrid oscillator-qubit systems realized in superconducting and trapped-ion processors, where the control variables can be the noncommuting position and momentum of a quantum harmonic oscillator. We introduce the Gaussian-controlled rotation (GCR), a canonical instance of non-Abelian QSP that cancels the Gaussian quantum-fluctuation error of an oscillator-controlled qubit rotation, matching the robustness of classical composite pulses with a much shorter circuit and an order-of-magnitude lower error. GCR enables fully analytical circuits for preparing squeezed, cat, GKP, and Fock states with performance comparable to state-of-the-art numerically optimized protocols. Building on this primitive, we develop an analytical framework for universal control of GKP bosonic codes, in which logical readout doubles as piecewise gate teleportation, yielding the first high-fidelity, error-mitigated single- and two-qubit logical gates for finite-energy GKP that surpass prior schemes even in the noiseless regime. The same framework supplies midcircuit error detection and extends in closed form to arbitrary lattices, qudits, and multimode codes relevant to the superconducting-circuit experiments now demonstrating beyond-break-even GKP memory. We further apply these ideas to close a key gap in oscillator-assisted quantum phase estimation. Together, these results show that the non-Abelian structure of quantum control operators can provide a practical resource for robust and efficient quantum control, establishing non-Abelian QSP as both a new theoretical framework and an experimentally accessible control paradigm.
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.
Iterative CZ-gate-based protocol for squeezed Schrödinger cat state engineering
Squeezed optical Schrödinger cat states constitute a key resource for both fundamental tests of quantum theory and up-to-date quantum technologies. We propose a measurement-assisted gate for the generation and manipulation of the cat states. In this scheme, an ancilla in the non-Gaussian small-amplitude (in general, squeezed) Schrödinger cat state and the target oscillator initially prepared in a squeezed vacuum (or coherent) state are subjected to a quantum nondemolition (QND) entangling operation followed by projective homodyne measurement. The proposed gate enables generation of high-fidelity squeezed Schrödinger cat states with controllable size and squeezing with tunable fidelity/success-probability trade-off. We also introduce an iterative, homodyne-conditioned CZ-based protocol for cat-state amplification. The parameter regimes required to achieve the desired fidelity and the success probability are analyzed. The approach is well suited for applications in measurement-based quantum computing and hybrid quantum networks where non-Gaussian resources enhance computational and communication capabilities.
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.