⚛️ high-energy theory

Krylov Complexity for Time-Dependent Hamiltonians

This paper investigates Krylov spread complexity for time-dependent quantum systems by formulating the problem within Floquet theory for periodic driving and extending the framework to general time-dependent cases using both globally and piecewise truncated Magnus expansions to ensure reliable analysis when standard expansions fail.

Original authors: Amin Faraji Astaneh, Parsa Kafashi

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Amin Faraji Astaneh, Parsa Kafashi

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a tiny, magical toy box (a quantum system) that changes its rules every second. Sometimes the rules change in a perfect, repeating rhythm, like a song on repeat. Other times, the rules shift in a messy, unpredictable way, like a sudden storm. Scientists want to know: how complicated does the toy box get as it plays? They call this measurement "Krylov complexity." It's basically a score that tells us how wildly the toy box's state spreads out into the vast universe of possibilities.

For a long time, scientists could only calculate this score easily when the rules stayed the same forever. But when the rules change over time, it's like trying to solve a puzzle where the pieces keep morphing shape. The math gets messy because the order in which you do things matters (if you push the box then spin it, it's different than spinning then pushing).

The Rhythmic Rhythm: When the Rules Repeat

First, the authors looked at systems that dance to a steady beat (periodically driven systems). Think of a drummer hitting a drum every second. Because the pattern repeats, you can pretend the whole song is just one giant, static beat. This is called Floquet theory. It lets you freeze the motion and calculate the complexity score as if the music never changed.

But here's the catch: sometimes, figuring out that "frozen beat" is impossible to write down on paper. It's like trying to describe the exact flavor of a complex soup without a recipe. To fix this, the authors used a clever trick called the Magnus expansion. Imagine you are trying to guess the flavor of that soup by tasting it in tiny, bite-sized sips. You take a sip, guess the main flavor, then take another sip to catch the subtle spices, and so on.

They tested this on a simple two-level system (like a light switch that can be on or off) being wiggled by a laser.

Circularly Polarized Drive: When the laser spins in a circle, the "sip" method worked perfectly, giving them a clear recipe for the complexity.
Linearly Polarized Drive: When the laser just wiggles back and forth, the soup is trickier. The authors showed that by taking many tiny sips (using the Magnus expansion) and adding up the flavors, they could get a very good approximation. They found that the complexity depends on special math functions (Bessel and Struve functions) that act like secret ingredients. They even showed that if you stop taking sips too early, your guess might be wrong, but if you take enough, the error shrinks rapidly.

The Chaotic Storm: When the Rules Don't Repeat

Next, they tackled the messy, non-repeating systems. Imagine the toy box is being shaken by a hand that moves randomly. You can't freeze the motion here because there is no repeating beat.

The authors tried the old "sip" method (Global Magnus expansion) on a whole hour of shaking. They found that if you shake the box for too long, your single big guess becomes terrible. The "flavor" of the soup changes too much for one recipe to cover the whole hour.

The Solution: The Piecewise Approach
So, they invented a new strategy: Piecewise Magnus expansion. Instead of trying to guess the flavor of the whole hour at once, they chopped the hour into tiny, manageable minutes.

They calculated the "flavor" for the first minute.
Then they used that result as the starting point for the second minute.
They kept chaining these tiny, accurate guesses together to build the whole story.

They tested this on a harmonic oscillator (think of a spring bouncing up and down) where the stiffness of the spring changed over time.

The Sudden Quench: First, they tested a scenario where the spring stiffness changed instantly (like snapping a rubber band). Here, the old method worked perfectly because the rules stopped changing after the snap.
The Soft Quench: Then, they tested a scenario where the spring stiffness changed slowly and smoothly over time (like gently stretching a rubber band). This is where the old method failed, but the Piecewise method shined.

They ran simulations (computer experiments) with different speeds of stretching:

Fast Stretch: The complexity shot up and then bounced around wildly, just like a spring that was snapped.
Slow Stretch: The complexity grew smoothly and settled down, because the system had time to adjust (this is called "adiabatic").

What They Found (and What They Didn't)

The main finding is that this Piecewise Magnus method is a reliable tool for calculating complexity in systems where the rules change over time, even when the old methods break down.

What they ruled out: They showed that relying on a single, global guess for long periods of time is unreliable. If you try to use the old "one big recipe" method for a long time, the error grows until the answer is useless.
How sure are they? They proved the math works for simple cases (like the sudden snap) and demonstrated through simulations that it works for the smooth, slow changes. They didn't just guess; they showed the numbers match up with what physics expects.

The Takeaway

If you want to know how complicated a quantum system gets when its rules are constantly shifting, don't try to solve the whole movie at once. Break it down into scenes. Calculate the complexity for each tiny scene, and then stitch them together. The authors have provided a practical "stitching kit" (the Piecewise Magnus expansion) that lets scientists track this complexity growth in a wide variety of time-dependent systems, from rhythmic lasers to chaotic springs.

They also mentioned that other scientists have tried different ways to solve this, like mapping the problem to a chain of magnets or using extra dimensions in math, but this new "stitching" approach offers a fresh, practical way to handle the messiness of time. It's not a magic wand that solves everything instantly, but it's a very sturdy ladder for climbing the mountain of time-dependent complexity.

Technical Summary: Krylov Complexity for Time-Dependent Hamiltonians

Problem Statement
Krylov complexity (or spread complexity) has emerged as a significant bottom-up measure of quantum complexity, quantifying how an initial state explores Hilbert space under time evolution. While extensively studied for time-independent Hamiltonians, its application to time-dependent Hamiltonians $H(t)$ presents conceptual and technical challenges. The primary obstacle is the non-commutativity of the Hamiltonian at different times, $[H(t_1), H(t_2)] \neq 0$ , which necessitates time-ordering in the evolution operator $U(t, t_0) = \mathcal{T} \exp[-i \int_{t_0}^t H(s) ds]$ . This structure prevents the direct application of standard Krylov constructions, which are naturally formulated for time-independent generators. The paper addresses the need for practical methods to compute Krylov complexity in such time-dependent settings, particularly when exact solutions are unavailable.

Methodology
The authors propose a framework utilizing the Magnus expansion to approximate the time-evolution operator, thereby enabling the definition of an effective time-independent generator for Krylov complexity calculations. The methodology is divided into two main regimes:

Periodically Driven Systems (Floquet Theory):
For systems with $H(t+T) = H(t)$ , the authors utilize Floquet theory, where the one-period evolution operator is defined by a Floquet Hamiltonian $H_F$ via $U(t_0+T, t_0) = e^{-iH_F T}$ . When $H_F$ cannot be obtained in closed form, the Floquet–Magnus expansion is employed as a systematic perturbative approximation. The authors analyze the convergence of this expansion, noting that while it is reliable for short times or high-frequency driving, it may lose accuracy over long intervals due to secular terms or heating effects.
General Time-Dependent Systems (Piecewise Magnus Expansion):
For non-periodic systems where a global Floquet description does not exist, the authors introduce a piecewise Magnus expansion. This approach partitions the total time interval $[0, t]$ into smaller subintervals $[t_j, t_{j+1}]$ such that the local convergence condition of the Magnus expansion is satisfied on each step. The full evolution is reconstructed as an ordered product of local propagators, each approximated by a truncated Magnus series. This method allows for the tracking of effective Hamiltonians over arbitrarily long times, even when a global expansion would diverge or become inaccurate.

Key Contributions and Results

Two-Level Periodically Driven Systems:
The framework is applied to driven two-level systems.
- Circularly Polarized Drive: The problem is solved exactly via a rotating-frame transformation, yielding a static effective Hamiltonian and an analytical expression for stroboscopic Krylov complexity.
- Linearly Polarized Drive: In the high-frequency regime, the authors derive an effective stroboscopic Hamiltonian using the Floquet–Magnus expansion. They demonstrate that the leading order involves a Bessel-function renormalization of the static term, while the second-order correction introduces a transverse term involving both Bessel and Struve functions. This transverse term, absent in the average-Hamiltonian approximation, is responsible for weak transitions and contributes to the complexity growth. The authors also provide rigorous bounds for the truncation error of the Magnus series, showing exponential improvement with order within the convergence regime.
Quadratic Oscillators and Symplectic Evolution:
The authors extend the analysis to harmonic oscillators with time-dependent frequencies, where dynamics are governed by symplectic transformations.
- Sudden Quench Benchmark: For a sudden frequency quench, the global Magnus expansion terminates exactly at first order (as commutators vanish), providing an exact analytical benchmark for the Krylov complexity.
- Soft Quench (Piecewise Approach): For smooth, finite-time frequency changes ("soft quenches"), the global expansion is insufficient. The authors implement the piecewise Magnus expansion numerically. They show that the Krylov complexity grows during the ramp and exhibits bounded oscillations for fast quenches, while slow (adiabatic) quenches result in a smooth approach to a constant final value with negligible post-ramp oscillations.
Krylov Complexity for Squeezed Vacua:
A specific result derived in the appendices establishes that for a squeezed vacuum state evolving from a reference vacuum, the Krylov complexity is directly proportional to the square of the squeezing parameter: $C_K(t) = \frac{1}{2} \sinh^2 r(t) = \frac{1}{2} |\nu(t)|^2$ , where $\nu(t)$ is the Bogoliubov coefficient.

Significance and Claims
The paper claims to provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems. Its primary significance lies in bridging the gap between the theoretical definition of Krylov complexity and its computability in time-dependent scenarios. By combining Floquet theory with the Magnus expansion, the authors offer a systematic way to define stroboscopic complexity for periodic systems. Furthermore, the introduction of the piecewise Magnus expansion addresses the limitations of global approximations, offering a reliable method for non-periodic systems where global convergence fails.

The authors position their work as complementary to existing approaches, such as mapping Floquet evolution to effective 1D models or using extended Hilbert space formalisms. They emphasize that their Magnus-based framework is particularly useful for systems where the effective Hamiltonian can be constructed perturbatively or locally, allowing for the extraction of complexity without solving the full time-ordered evolution. The paper concludes that these methods are robust, as evidenced by their agreement with known analytical limits (sudden quench) and physical expectations (adiabatic behavior), and suggests future extensions to operator complexity and generalized Krylov complexity.

The Rhythmic Rhythm: When the Rules Repeat

The Chaotic Storm: When the Rules Don't Repeat

What They Found (and What They Didn't)

The Takeaway

More like this