Quantum signal processing (QSP) has emerged as a promising framework to manipulate and determine properties of quantum systems. QSP not only unifies most existing quantum algorithms but also provides tools to discover new ones. Quantum signal processing is applicable to single-qubit or multiqubit systems that can be ``qubitized'' so one can exploit the SU(2) structure of system evolution within special invariant two-dimensional subspaces. In the context of quantum algorithms, this SU(2) structure is artificially imposed on the system through highly nonlocal evolution operators that are difficult to implement on near-term quantum devices. In this work, we propose QSP protocols for the infinite-dimensional Onsager Lie algebra, which is relevant to the physical dynamics of quantum devices that can simulate the transverse-field Ising model. To this end, we consider QSP sequences in the Heisenberg picture, allowing us to exploit the emergent SU(2) structure in momentum space and ``synthesize'' QSP sequences for the Onsager algebra. Our results demonstrate a concrete connection between QSP techniques and noisy intermediate scale quantum protocols. We provide examples and applications of our approach in diverse fields ranging from space-time dual quantum circuits and quantum simulation to quantum control.
Quantum signal processing (QSP) has emerged as a promising framework to manipulate and determine properties of quantum systems. QSP not only unifies most existing quantum algorithms but also provides tools to discover new ones. Quantum signal processing is applicable to single-qubit or multiqubit systems that can be qubitized'' so one can exploit the SU(2) structure of system evolution within special invariant two-dimensional subspaces. In the context of quantum algorithms, this SU(2) structure is artificially imposed on the system through highly nonlocal evolution operators that are difficult to implement on near-term quantum devices. In this work, we propose QSP protocols for the infinite-dimensional Onsager Lie algebra, which is relevant to the physical dynamics of quantum devices that can simulate the transverse-field Ising model. To this end, we consider QSP sequences in the Heisenberg picture, allowing us to exploit the emergent SU(2) structure in momentum space and synthesize'' QSP sequences for the Onsager algebra. Our results demonstrate a concrete connection between QSP techniques and noisy intermediate scale quantum protocols. We provide examples and applications of our approach in diverse fields ranging from space-time dual quantum circuits and quantum simulation to quantum control.
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.
Abstract Quantum signal processing (QSP) is a powerful toolbox for the design of quantum algorithms and can lead to asymptotically optimal computational costs. Its realization on noisy quantum computers without fault tolerance, however, is challenging because it requires a deep quantum circuit in general. We propose a strategy to run an entire QSP protocol on noisy quantum hardware by carefully reducing overhead costs at each step. To illustrate the approach, we consider the application of Hamiltonian simulation for which QSP implements a polynomial approximation of the time evolution operator. We test the protocol by running the algorithm on the Quantinuum H1-1 trapped-ion quantum computer powered by Honeywell. In particular, we compute the time dependence of bipartite entanglement entropies for Ising spin chains and find good agreements with exact numerical simulations. To make the best use of the device, we determine optimal experimental parameters by using a simplified error model for the hardware and numerically studying the trade-off between Hamiltonian simulation time, polynomial degree, and total accuracy. Our results are the first step in the experimental realization of QSP-based quantum algorithms.
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.
In recent years there has been an increasing interest on the theoretical and experimental investigation of space-time dual quantum circuits. They exhibit unique properties and have applications to diverse fields. Periodic space-time dual quantum circuits are of special interest, due to their iterative structure defined by the Floquet operator. A very similar iterative structure naturally appears in Quantum Signal processing (QSP), which has emerged as a framework that embodies all the known quantum algorithms. However, it is yet unclear whether there is deeper relation between these two apparently different concepts. In this work, we establish a relation between a circuit defining a Floquet operator in a single period and its space-time dual defining QSP sequences for the Lie algebra sl(2,C), which is the complexification of su(2). First, we show that our complexified QSP sequences can be interpreted in terms of action of the Lorentz group on density matrices and that they can be interpreted as hybrid circuits involving unitaries and measurements. We also show that unitary representations of our QSP sequences exist, although they are infinite-dimensional and are defined for bosonic operators in the Heisenberg picture. Finally, we also show the relation between our complexified QSP and the nonlinear Fourier transform for sl(2,C), which is a generalization of the previous results on su(2) QSP.
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.
Quantum signal processing (QSP), originally developed for composite pulse sequences in nuclear magnetic resonance systems, has recently attracted attention as a unified framework for quantum algorithms. A pioneering study applied QSP to nonequilibrium control in integrable many-body systems, enabling the realization of nonequilibrium dynamics with greater flexibility than Floquet engineering. However, extending QSP to nonintegrable systems faces fundamental obstacles arising from the limited number of conserved quantities and thermalization. In this work, we propose a protocol that leverages QSP in systems exhibiting Hilbert space fragmentation (HSF). Specifically, we consider a pair-hopping model with four-fold periodic potentials that exhibits an HSF structure, thereby providing integrable and nonintegrable sectors within a single system. We analytically show that nonequilibrium dynamics can be flexibly designed through QSP engineered by these potentials in the integrable sectors. In contrast, we numerically identify signatures of thermalization in the nonintegrable sectors. Remarkably, by inserting domain walls, we achieve parallel control of multiple quantum dynamics within a single system. This approach sheds light on the control of nonequilibrium dynamics from the perspective of quantum computation by extending the scope of QSP to nonintegrable systems.