Sharif University Researchers Detail New Complexity Growth Methods

Researchers at Sharif University of Technology in Tehran, Iran are detailing new methods to measure how an initially simple state or operator explores Hilbert space under time evolution, with a crucial distinction: these methods account for Hamiltonians that change over time. While prior work largely focused on static systems, the team’s approach offers tools for analyzing a broader class of time-dependent quantum dynamics. They present a piecewise Magnus expansion as a reliable method when the globally truncated Magnus expansion loses convergence or accuracy, indicating limitations in existing modeling techniques. By combining Floquet theory and the Magnus expansion, the researchers offer a systematic approximation for periodically driven quantum systems, even when a closed-form solution for the Floquet Hamiltonian isn’t available; their results, they report, provide practical tools for analyzing complexity growth. Sharif University of Technology researchers are developing methods for computing Krylov complexity in systems undergoing continuous change, building upon established theoretical frameworks. This work notably combines Floquet theory with the Magnus expansion to analyze Krylov complexity in periodically driven quantum systems. Beyond periodic drives, the team extends this framework to general time-dependent Hamiltonians, dividing the time interval into smaller segments to improve accuracy. “This allows the Krylov complexity to be extracted from a sequence of controlled local approximations,” they state, offering tools for analyzing complexity growth in a wider range of quantum systems. These results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems.

Periodically Driven Systems and Floquet Theory Researchers are increasingly focused on Krylov complexity, a bottom-up measure of quantum complexity that quantifies how an initially simple state or operator explores Hilbert space. Extending this concept to systems where the Hamiltonian itself evolves presents significant hurdles. While prior investigations largely centered on static systems, the team is developing analytical approaches to tackle time-dependent scenarios, building on the established framework of Floquet theory. For periodically driven systems, where a Hamiltonian repeats its behavior after a fixed time interval, Floquet theory offers a pathway to an effective, time-independent description. The central object in this approach is the one-period evolution operator, which can be expressed as the exponential of the Floquet Hamiltonian. However, obtaining a closed-form solution for this Floquet Hamiltonian is often impossible, necessitating approximation techniques.

The team leverages the Magnus expansion, a perturbative method for approximating time-evolution operators, to circumvent this limitation. The researchers emphasize that the standard Magnus expansion isn’t always sufficient. They are developing a piecewise Magnus approach to address scenarios where the global expansion loses convergence or accuracy. This involves dividing the total time into smaller intervals, approximating the evolution within each interval, and then combining these local approximations.

The team notes the Floquet Hamiltonian isn’t unique, differing by multiples of 2π, but these variations do not affect measurable quantities like transition probabilities or Krylov complexity. The full time evolution also includes intra-period dynamics, which can contribute to instantaneous complexity even when the stroboscopic Floquet Hamiltonian appears static. This pursuit addresses a significant gap in existing quantum analysis, as previous investigations largely centered on static, unchanging systems.

The team’s approach focuses on accurately modeling complexity growth in these dynamic scenarios, recognizing that traditional techniques often fall short when Hamiltonians do not commute, meaning the order in which they are applied matters. The researchers demonstrate that this piecewise method provides a reliable method when the globally truncated Magnus expansion loses convergence or accuracy. Building on established theoretical foundations, the team leverages Floquet theory for periodically driven systems, which is particularly useful in high-frequency regimes where the driving period is short compared to the system’s intrinsic timescales. This work diverges from much prior investigation, which largely centered on static, unchanging systems.

The team’s approach addresses a critical need for analytical tools capable of handling the intricacies of time-dependent quantum mechanics, a realm where traditional methods often falter. This suggests existing analytical techniques possess limitations, necessitating an alternative approach in specific circumstances. Beyond periodic drives, the framework extends to general time-dependent Hamiltonians; the researchers demonstrate that the piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy.

The team’s results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems. This focus diverges from much earlier research centered on static, unchanging systems, demanding new approaches to analyzing complexity growth. Researchers are particularly interested in scenarios where obtaining a direct solution for the system’s behavior proves intractable.

The team’s approach leverages Floquet theory, a framework for analyzing periodically driven systems, and combines it with the Magnus expansion. The piecewise expansion doesn’t rely on a complete, closed-form solution for the Floquet Hamiltonian, instead dividing the time evolution into smaller, manageable segments. Beyond periodic driving, one may also consider general time-dependent Hamiltonians for which no global Floquet description exists. For such systems, a global Magnus expansion may be useful at short times, but it can lose convergence or accuracy over long time intervals. To address this limitation, the team developed a piecewise Magnus approach. The idea is to divide the full time interval into short subintervals, approximate the evolution on each subinterval by a local effective Hamiltonian, and reconstruct the full time evolution as an ordered product of these local evolutions. This allows the Krylov complexity to be extracted from a sequence of controlled local approximations. Their results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems. In these instances, the team demonstrates how the Magnus expansion can provide a systematic approximation, allowing for the extraction of Krylov complexity. Beyond purely periodic drives, the researchers extended their framework to encompass more general time-dependent Hamiltonians. They found that while the globally truncated Magnus expansion can be useful at short times, it may lose convergence or accuracy over long time intervals. Instead, a piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy. Their results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems. A central tenet of their methodology involves leveraging Floquet theory for periodically driven systems. This allows researchers to define a stroboscopic Krylov complexity using the Floquet Hamiltonian, an effective time-independent generator of the system’s evolution.

The team notes that the Floquet Hamiltonian is not unique, differing by terms proportional to the identity which shift quasienergies uniformly and do not affect transition probabilities. Source: https://arxiv.org/abs/2607.10454 Stay currentSee today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: