Krylov complexity has it all

This paper demonstrates that Krylov complexity fully encodes the dynamics of quantum operators by providing a recursive algorithm to derive Lanczos coefficients from its Taylor expansion, thereby establishing it as a complete characterization of operator evolution distinct from spread complexity.

The Gist

Imagine you are watching a complex dance performance. You want to understand the entire story of how the dancers move, interact, and spread out across the stage over time. In the world of quantum physics, this "dance" is the evolution of an operator (a mathematical tool representing a physical quantity) as time passes.

For a long time, physicists have known that you can describe this dance in a few different, equivalent ways. It's like having a map, a GPS track, and a list of step-by-step instructions; if you have one, you can mathematically reconstruct the others. These known "maps" include:

• Lanczos coefficients: The specific "rules" or "weights" that dictate how the dance steps connect.
• Return amplitude: How likely the dancer is to return to their starting spot.
• Spectral density: A frequency profile of the movement.

The Big Discovery
This paper, written by Wolfgang Mück, introduces a new "map" to this list: Krylov complexity.

Think of Krylov complexity as a measure of the "size" of the stage the dancer has explored. If the dancer stays in one corner, the complexity is low. If they run all over the stage, the complexity is high.

The paper's main claim is simple but powerful: If you know the Krylov complexity (the size of the explored stage) at every moment in time, you know everything about the dance. You can mathematically reverse-engineer the exact rules (the Lanczos coefficients) that govern the movement, just as if you had the original instruction manual.

How It Works: The Recipe
To prove this, the author created a specific "recipe" or algorithm.

1. The Input: You take the Krylov complexity curve and look at its shape right at the very beginning (at time $t=0$). You break this shape down into a series of simple building blocks (a Taylor expansion).
2. The Process: Using a step-by-step recursive method (like solving a puzzle where each piece reveals the next), the author shows how to calculate the exact "rules" of the dance (the Lanczos coefficients) from those building blocks.
3. The Result: You end up with the complete set of rules that define the system's dynamics.

The Twist: Why It Doesn't Work for "Spread Complexity"
The paper also addresses a similar concept called Spread Complexity, which measures how a quantum state (like a single particle) spreads out, rather than how an operator evolves.

The author explains why the same "recipe" fails here.

• The Analogy: Imagine the Krylov complexity is a dance where the dancer only moves forward or backward on a straight line. The rules are simple and one-dimensional.
• The Problem: Spread complexity is like a dance where the dancer can also spin or move sideways (introducing a "phase" or imaginary component).
• The Missing Piece: If you only look at the "size" of the spread (the complexity), you lose information about the sideways spinning. It's like trying to guess the full choreography of a dancer just by measuring how far they are from the center; you can't tell if they are spinning left or right.
• The Solution: To decode the spread complexity, you would need extra information, such as a second measurement (like the "variance" or how much the spread fluctuates). Without that extra clue, the recipe is incomplete.

In Summary
This paper establishes a "proof of principle": Krylov complexity is a complete story. It contains every detail needed to reconstruct the entire history of an operator's evolution. While a similar concept for quantum states (spread complexity) is missing a piece of the puzzle, the author shows exactly what that missing piece would look like.

The author notes that while this mathematical recipe works in theory, putting it into practice on a computer might face some stability challenges, which would need further investigation. But fundamentally, the door is open: knowing the "size" of the quantum exploration is enough to know the "rules" of the universe's dance.

Technical Summary

Technical Summary: Krylov Complexity as a Complete Characterization of Operator Dynamics

Problem Statement
The dynamics of a quantum operator $O(t)$ in the Heisenberg picture can be represented in multiple equivalent ways, including the set of Lanczos coefficients, the return amplitude, and the spectral density. While it is established that these quantities independently contain the full information regarding the operator's evolution, the status of Krylov complexity, $K(t)$, as a complete descriptor remained to be formally demonstrated. Specifically, the paper addresses whether knowing the time-dependent Krylov complexity of an operator is sufficient to uniquely reconstruct the underlying dynamics (i.e., the Lanczos coefficients) without additional input. The paper also seeks to clarify the distinction between Krylov complexity (for operators) and spread complexity (for quantum states), particularly regarding the existence of a recursive reconstruction algorithm for the latter.

Methodology
The author employs the Recursion Method and the Lanczos algorithm to map the time evolution of an operator $O(t)$ onto a hopping model on a semi-infinite one-dimensional chain. In this framework:

1. Operator Evolution: The operator evolves under the Liouvillian $L = [H, \cdot]$, generating a Krylov basis $\{O_n\}$ where $L O_n = b_{n+1} O_{n+1} + b_n O_{n-1}$. The coefficients $b_n$ are the Lanczos coefficients.
2. Wavefunction Mapping: The projection of the time-evolved operator onto the Krylov basis defines wavefunctions $\phi_n(t) = (O_n | O(t))$, which satisfy a system of coupled differential equations (Eq. 1).
3. Taylor Expansion Analysis: The author expands the wavefunctions $\phi_n(t)$ and the resulting Krylov complexity $K(t) = \sum n \phi_n(t)^2$ in Taylor series around $t=0$.
4. Recursive Construction: By analyzing the coefficients $B_p$ of the Taylor expansion of $K(t)$, the paper constructs an explicit recursive algorithm. This algorithm isolates the dependence of $B_p$ on the Lanczos coefficients $\Delta_p = b_p^2$, demonstrating that $\Delta_p$ can be solved for using $B_p$ and previously determined coefficients.

Key Contributions and Results

1. Equivalence Proof: The paper establishes that Krylov complexity $K(t)$ contains the entire information about the dynamics of a quantum operator. It extends the list of equivalent quantities characterizing operator dynamics to include $K(t)$.
2. Explicit Algorithm: A recursive algorithm (detailed in Table 1) is constructed to calculate the Lanczos coefficients $\Delta_p$ directly from the Taylor expansion coefficients $B_p$ of $K(t)$. The derivation shows that the term containing $\Delta_p$ in the expression for $B_p$ is isolated from terms involving higher-order coefficients, allowing for step-by-step reconstruction.
3. Distinction from Spread Complexity: The paper demonstrates that a similar recursive algorithm cannot exist for spread complexity (defined for quantum states) without additional dynamical input.
   • For operators (Krylov complexity), the dynamics are governed by real coefficients $b_n$, leading to a system where $K(t)$ depends solely on these coefficients.
   • For states (spread complexity), the dynamics involve a Hamiltonian with both diagonal ($a_n$) and off-diagonal ($b_n$) terms in the hopping equation. The spread complexity $K(t)$ depends on the norm $|\phi_n(t)|^2$, which mixes real and imaginary parts of the wavefunctions. Consequently, the Taylor expansion of $K(t)$ for states contains even powers of $t$ but depends on a combination of $a_n$ and $b_n$ that cannot be disentangled by $K(t)$ alone.
4. Conditions for Reconstruction: The author notes a caveat: the reconstruction algorithm assumes the input coefficients $B_p$ genuinely represent a Krylov complexity. If the algorithm terminates at a finite dimension (where $\Delta_P = 0$), subsequent coefficients must satisfy specific constraints to be valid.

Significance and Claims
The paper claims to provide a "proof of principle" that Krylov complexity serves as a complete characterization of operator evolution in quantum systems. By showing that the Lanczos coefficients can be uniquely recovered from the Taylor expansion of $K(t)$, the work validates the use of Krylov complexity as a standalone measure for operator dynamics, equivalent to the spectral density or return amplitude.

Regarding spread complexity, the paper modestly concludes that while $K(t)$ for a state cannot uniquely determine the Hamiltonian dynamics on its own, the combination of spread complexity and a second moment (such as the variance or $K_2(t)$) might suffice to recursively determine the full set of Lanczos coefficients. The paper does not propose new experimental applications but rather clarifies the theoretical underpinnings and limitations of using complexity measures to characterize quantum dynamics. Practical numerical stability of the proposed algorithm is identified as a subject for future investigation.