Walsh Functions Improve Simulation of Strongly Driven Quantum Many-Body Systems

Researchers demonstrate improved modelling of periodically driven quantum systems by employing the Walsh basis – a set of square-wave functions – instead of the conventional Fourier basis. This approach yields more accurate representations of system dynamics, particularly in strongly driven regimes, by leveraging the system’s temporal response. An extended theoretical framework, incorporating Walsh polaritons – hybridised states within the system – explains the enhanced performance, linked to reduced errors arising from state localisation. This Walsh-Floquet theory offers a naturally implementable method for manipulating quantum states on digital simulators.

The precise control of quantum systems relies heavily on applying periodic ‘kicks’ – discrete impulses used to manipulate their state. While conventionally analysed using Fourier transforms, this approach can struggle with convergence, hindering accurate modelling of complex behaviours like chaos and thermalisation. New research demonstrates a superior method employing the Walsh basis – a set of periodic square-wave functions – to describe these driven systems. This alternative basis improves the accuracy of modelling strongly driven quantum systems, revealing a connection between the system’s response and a phenomenon akin to localisation. James Walkling and Marin Bukov, both from the Max Planck Institute for the Physics of Complex Systems, detail this approach in their paper, ‘Walsh-Floquet Theory of Periodic Kick Drives’, presenting a framework with potential for enhanced implementation on digital quantum simulators.

Walsh-Floquet Theory: Advancing Quantum Control and Simulation

Recent research demonstrates a significant improvement in modelling periodically driven quantum systems through the application of Walsh functions, a basis of periodic square-wave functions, rather than the traditionally used Fourier basis. This approach, termed Walsh-Floquet theory, offers enhanced accuracy in describing the dynamics of both single and many-body systems subject to strong, discrete ‘kicks’. The study reveals that the Fourier basis often struggles to converge to accurate physical quantities when modelling these kicked systems, while the Walsh basis accurately recovers Floquet dynamics – the long-term behaviour of periodically driven systems – due to its better representation of the system’s temporal response. Researchers derive an extended Sambe space formulation and an inverse-frequency expansion specifically within the Walsh basis, providing a theoretical framework for understanding this improved performance.

This work builds upon a strong foundation in signal processing, originating with the foundational work of Nyquist and Shannon, and extends these concepts into specialized areas like Walsh functions and Hadamard transforms as tools for quantum information processing. Current research actively refines techniques for controlling quantum systems, with a notable emphasis on optimizing pulse sequences and basis set selection for achieving robust and precise control. The exploration of periodically driven systems, underpinned by Floquet theory, reveals a growing interest in engineering novel quantum states and materials through time-periodic manipulation.

Central to this advancement is the observation of single-particle localization on the frequency lattice, which correlates directly with reduced truncation errors, meaning fewer computational resources are required to achieve a given level of accuracy. The research further identifies the formation of ‘Walsh polaritons’ – hybrid states arising from strong interactions between the kicked system and Walsh modes – which are readily accessible for investigation using digital simulators. This establishes a foundation for a new theoretical framework naturally suited to implementation on digital quantum devices.

Researchers actively investigate techniques like the Schrieffer-Wolff transformation and perturbation theory to tackle complex interactions within quantum many-body systems. They also integrate quantum sensing and noise spectroscopy to characterize and mitigate environmental noise, a crucial step towards building practical quantum technologies. The application of these techniques to molecular simulation, through the development of shortcuts for adiabatic and variational algorithms, suggests a growing focus on leveraging quantum computation for materials science and drug discovery.

The ability to accurately represent and control Floquet states on digital simulators represents a significant leap forward, paving the way for exploring novel quantum phenomena and developing new quantum technologies. This approach promises to accelerate research in areas like quantum materials, many-body physics, and the development of robust quantum control protocols. By establishing a “Walsh-Floquet theory,” researchers have created a framework naturally suited for implementation on digital devices, opening doors for simulating and manipulating complex quantum systems using readily available digital tools.

This body of work demonstrates a clear and sustained investigation into the theoretical underpinnings and practical applications of quantum control and simulation. Recent publications consistently highlight the importance of efficient mathematical frameworks for representing and manipulating quantum systems, particularly those subject to periodic driving. The integration of quantum sensing and noise spectroscopy provides valuable insights into the limitations imposed by environmental noise, enabling the development of strategies for mitigating these effects.

Researchers derive an extended Sambe space formulation and an inverse-frequency expansion specifically within the Walsh basis, providing a theoretical framework for understanding this improved performance. This framework allows for a more accurate description of the system’s temporal response, leading to improved simulation results and a deeper understanding of the underlying physics. The observation of single-particle localization on the frequency lattice further enhances the efficiency of the calculations, reducing the computational resources required to achieve a given level of accuracy.

The formation of ‘Walsh polaritons’ – hybrid states arising from strong interactions between the kicked system and Walsh modes – offers a unique opportunity for investigating novel quantum phenomena. These polaritons are readily accessible for investigation using digital simulators, providing a powerful tool for exploring the complex dynamics of periodically driven quantum systems. This opens up new avenues for research in areas like quantum materials and many-body physics, potentially leading to the discovery of new quantum technologies.

The ability to accurately represent and control Floquet states on digital simulators represents a significant leap forward, paving the way for exploring novel quantum phenomena and developing new quantum technologies. This approach promises to accelerate research in areas like quantum materials, many-body physics, and the development of robust quantum control protocols. By establishing a “Walsh-Floquet theory,” researchers have created a framework naturally suited for implementation on digital devices, opening doors for simulating and manipulating complex quantum systems using readily available digital tools.

The development of shortcuts for adiabatic and variational algorithms demonstrates a commitment to improving the efficiency of quantum computations, enabling the simulation of more complex systems. This is particularly important for applications in materials science and drug discovery, where accurate simulations can accelerate the development of new materials and drugs. The integration of quantum sensing and noise spectroscopy provides valuable insights into the limitations imposed by environmental noise, enabling the development of strategies for mitigating these effects.