Krylov complexity has it all

arXiv:2605.28681 · hep-th, quant-ph · Submitted 2026-05-27 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Krylov complexity has it all".

Mira: Krylov complexity provides a complete characterization of an operator's dynamics by encompassing all equivalent quantities, including Lanczos coefficients, return amplitudes, and spectral densities.

Kai: First, who's behind it and why it matters.

Title and authors: Mira: Moving on to the actual substance of the paper, "Krylov complexity has it all," they summarize that this complexity measure is not just a random calculation but a complete characterization of an operator's dynamics.

Kai: They explain that this means you don't need to track every single state evolution or every possible quantity separately; the Krylov complexity contains everything relevant, like Lanczos coefficients and spectral densities.

Lev: So if we trust their argument, we can stop thinking about tracking the full history of the operator dynamics and just focus on this one complexity function instead.

Mira: They lay out a chain of reasoning starting from the Heisenberg picture, where an operator evolves as O(t) = e itLO, and then they use the Lanczos algorithm to define an ordered basis of operators known as the Krylov basis.

Kai: That hopping model connection is key because it translates that operator evolution directly into a mathematical structure on a one-dimensional chain, which is what makes the subsequent math tractable.

Lev: I’m interested in how this mapping works; if we can map dynamics onto a simple chain recurrence, that suggests the complexity measure itself is doing some heavy lifting for us.

Mira: They show that the time evolution of a state phi n(t) in this basis follows a specific recurrence relation (equation one), which has very specific Taylor expansions involving factorials, as shown in equation (three).

Kai: Those factorial terms are what allow them to relate the complexity K(t) to the Lanczos coefficients n through those summation formulas they developed.

Lev: So, in essence, they’re showing that if you have this complexity function, you can recursively reverse-engineer the fundamental parameters of the operator dynamics just by looking at its Taylor expansion coefficients.

Mira: That recursive extraction process is what makes this paper so interesting because it provides a constructive method to go from a complexity measure back to those essential dynamical ingredients like return amplitudes and spectral densities.

Kai: It’s not just a theoretical claim; they've built an explicit algorithm in Table one that shows exactly how to perform this derivation step-by-step, which is what makes this paper feel like it has real computational utility.

Lev: From an engineering standpoint, if that algorithm works robustly on the theoretical inputs, it gives us a path toward creating tools that can analyze system behavior without needing massive numerical solvers for every single time step.

Mira: The distinction they draw between Krylov complexity and spread complexity reinforces that the method is specific to operator dynamics, as spread complexity requires more input, like the a n coefficients.

Kai: So we’re seeing how this work builds on earlier ideas about operator evolution and state spreading to create a comprehensive characterization tool.

The paper's summary: Mira: The authors suggest several ways this work improves upon previous characterizations by providing an explicit recursive algorithm, which is the main improvement they highlight for establishing this equivalence.

Kai: They point out that the primary improvement is moving beyond just listing equivalent quantities to actually constructing a method to derive one set of parameters, like Lanczos coefficients, from another set, like the Taylor expansion of complexity.

Lev: I’m concerned about the practical limitations they mention regarding the non-negativity requirement; if we can't guarantee p at least zero then this method is immediately unusable for many physical systems without significant prior knowledge.

Mira: They address this by noting that if the algorithm returns a single coefficient p that is less than zero, it signals that the input function K(t) is simply incompatible with being a valid Krylov complexity measure.

Kai: That incompatibility check is really clever; it turns the derivation process into a validation step, which adds a layer of physical consistency to the entire approach.

Lev: So it’s not just about finding a number; it’s about verifying that the input function actually corresponds to a physical evolution before you try to extract anything from it.

Mira: They also make a clear distinction regarding spread complexity, showing that while the input information is similar in terms of norms, the specific structure of the Taylor expansions differs because spread complexity depends on both a n and b n.

Kai: That structural difference explains why you can’t just use the same algorithm to find both sets of parameters, which is a clear limitation they are pointing out.

Lev: That suggests that if we want to fully characterize state evolution, we might need more than just spread complexity; perhaps knowing the second spread complexity would be necessary for a complete picture.

Mira: The authors conclude by stating this recursive algorithm serves as a "proof of principle" demonstrating that Krylov complexity is indeed a complete characterization of operator evolution, provided the non-negativity conditions are met.

Kai: So, the key takeaway for us is that this paper provides the explicit construction and verification mechanism for using Krylov complexity as a total descriptor of quantum dynamics.

Lev: For running real experiments, this means we need to focus on developing stable ways to calculate those Taylor coefficients from experimental data without losing information.

The paper's improvements: Mira: To wrap things up with the paper, "Krylov complexity has it all," the authors emphasize that their explicit construction of the algorithm serves as a strong proof-of-principle for using Krylov complexity to characterize operator evolution in quantum systems.

Kai: They’ve successfully demonstrated that this measure contains all the information about the set of Lanczos coefficients, provided those coefficients are non-negative, which is a significant technical achievement.

Lev: From my perspective, the major implication is that we have a new systematic tool to analyze dynamical systems by translating complexity measures into fundamental physical parameters like the Lanczos coefficients.

Mira: It’s about showing that knowing this one quantity allows us to reconstruct the full picture, including return amplitudes and spectral densities, which is a big step in theoretical understanding.

Kai: So, we are left with this explicit recursive algorithm as a concrete method for extracting these dynamical parameters from the Taylor expansion of Krylov complexity around zero.

Lev: I think the next step for researchers is to work on the numerical stability and ensuring that those non-negativity constraints hold under realistic, noisy conditions in hardware.

Mira: That's exactly right; the practical challenges around numerical stability are something we definitely need to investigate further as this moves from theory into applied physics.

Kai: So, it seems "Krylov complexity has it all" provides a very concrete framework for how we can systematically analyze operator dynamics using these complex mathematical tools.

Conclusion: Kai: So we’ve seen how this paper, "Krylov complexity has it all," shows that this measure actually captures the entire operator dynamics, which is pretty wild to think about.

Mira: It really is a substantial result because they build this recursive algorithm from the ground up, showing how you can extract Lanczos coefficients and spectral densities just by analyzing the complexity function's Taylor expansion.

Lev: If we take that recursive extraction method seriously, it means we could potentially bypass some of the heavy numerical machinery usually needed to solve these types of operator problems on real hardware.

Mira: Exactly, and they’ve even included a test where if the algorithm spits out a negative coefficient, it tells you the input function isn't physically compatible with being a Krylov complexity.

Kai: That validation step is what makes this approach really promising for experimentalists because it gives us a way to check if the data we get from our measurements actually corresponds to a valid quantum evolution.

Lev: For error correction, if we can reliably extract these fundamental parameters, it might give us new ways to characterize the underlying noise in a system that goes beyond just standard state description.

Mira: And remember, they made a clear distinction between this and spread complexity, showing that for operator dynamics, the input information is fundamentally different and requires more constraints to fully determine.

Kai: That distinction is important because it tells us what kind of measurement or data we need to collect depending on whether we're looking at the operator itself or a specific state.

Lev: I’m just wondering if there are any known cases where the non-negativity constraint on those coefficients becomes a real bottleneck when trying to map this onto an actual physical system.

Mira: That's a fair point; we need to keep an eye on those numerical issues, because if the algorithm fails due to instability, the whole characterization falls apart.

Kai: So, in short, this paper provides a complete mathematical toolkit for characterizing operator dynamics through Krylov complexity using an explicit and verifiable recursive method.

Lev: It’s a solid theoretical foundation for understanding the structure of quantum evolution without needing to solve every single differential equation directly.

Mira: I think this work opens up a new avenue where complexity measures can serve as a complete and rigorous fingerprint for any operator in a quantum system.

Wolfgang M¨uck

Dipartimento di Fisica “Ettore Pancini”, Universita degli Studi di Napoli Federico II · Istituto Nazionale di Fisica Nucleare, Sezione di Napoli

hep-th, quant-ph

Submitted: 2026-05-27

Updated: 2026-09-10

Comments: 11 pages, 1 table, no figures, v.2: references added, v.3: extended caveat of the algorithm, one more reference, v.4: final version to appear in PRD

Journal ref: PRD 114, L061702 (2026)

DOI: 10.1103/3g6c-jx8t

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: Krylov complexity provides a complete characterization of an operator's dynamics by encompassing all equivalent quantities, including Lanczos coefficients, return amplitudes, and spectral densities.

Key concepts

Krylov Complexity: This is a measure of an operator's dynamics derived from the sum of squared norms of Krylov states. It captures all the necessary information about how an operator evolves over time by quantifying the evolution within a specific basis defined by recurrence relations.
Lanczos Coefficients: These coefficients are fundamental parameters that define the basis (Krylov basis) used to describe an operator's dynamics. The paper shows that Krylov complexity contains all these coefficients, making it a complete characterization of the system's evolution.
Return Amplitude: This is a quantity related to the time evolution of an operator in the Heisenberg picture. It is one of several equivalent quantities that fully describe the dynamics, and its relationship with Krylov complexity helps link these different dynamical descriptions together.

Terminology

Krylov Complexity — 57× in this paper · explained in 3 episodes
Verification — 2× in this paper · explained in 2 episodes
Chaotic Dynamics — 1× in this paper · explained in 2 episodes
Consistency — 1× in this paper · explained in 3 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes

Summary

Krylov complexity provides a complete characterization of an operator's dynamics by encompassing all equivalent quantities, including Lanczos coefficients, return amplitudes, and spectral densities. This work establishes this equivalence by constructing an explicit recursive algorithm to derive Lanczos coefficients from the Taylor expansion of Krylov complexity around zero.

The Equivalence of Dynamical Information

The paper demonstrates that knowing the Krylov complexity of an operator is equivalent to knowing its dynamics. This equivalence stems from a chain of reasoning based on the Heisenberg picture, where an operator evolves as O(t) = eitL O, and the Lanczos algorithm provides a basis (the Krylov basis) defined by recurrence relations. The quantities that independently contain the entire information about this dynamics are listed as:

the set of Lanczos coefficients
the return amplitude ϕ0(t) or its Laplace transform
the spectral density associated with the orthogonal polynomials
the moments of the spectral density, because they also appear as coefficients in the Taylor expansion of the return amplitude.

Deriving Lanczos Coefficients from Krylov Complexity

The central contribution is an explicit recursive algorithm designed to calculate Lanczos coefficients from the Taylor expansion of Krylov complexity around t = 0. This process begins by relating the time evolution in the Krylov basis to a hopping model on a one-dimensional chain, defined by:

)&∂tϕn(t) = −bn+1ϕn+1(t) + bnϕn−1(t), ϕn(0) = δn,0. (1)

The Krylov complexity is defined as the sum of the squared norms of the Krylov states: K(t) = X∞ n=0 nϕn(t)2, (5). By substituting Taylor expansions for the solutions ϕn(t) and performing necessary summations reorderings, one obtains a form for K(t): K(t) = X∞ p=0 Bp t2p (2p)! (6). The coefficients Bp are then related to the Lanczos coefficients ∆n = b squared n / h n squared and other terms through complex summation formulas.

The Recursive Algorithm for Verification

The algorithm presented in Table 1 allows one to recursively calculate the Lanczos coefficients ∆p from the Taylor series coefficients Bp of Krylov complexity. The steps involve:

)&h1 = 1/2 B1, ∆1 = h1/h0, Stop, if ∆1 ≤ 0.

)&Calculate Bp,1 from (10) and Bp,2 from (12).

)&hp = Bp − Bp,1 − Bp,2 / 2(2p-2) p-1, ∆p = hp/hp−1, Stop, if ∆p ≤ 0.

This procedure serves a dual purpose: it can determine the Lanczos coefficients or test whether a given function K(t) is compatible with a Krylov complexity. A crucial caveat is that the construction requires the condition ∆p ≥ 0 ∀p. If the algorithm returns a single ∆p < 0, it indicates incompatibility.

Distinction from Spread Complexity

The paper explicitly discusses why this recursive algorithm does not apply to spread complexity, Ks(t) = X∞ n=0 nϕn(t)2, (13). While both complexities depend on the norm ϕn(t)2, their Taylor expansions differ because Ks(t) depends only on even powers of t. The paper notes that the input information for spread complexity is exactly the same amount of input information as Krylov complexity K(t). However, an algorithm to determine both an and bn from the same type of input cannot exist because the Krylov algorithm uniquely determines ∆n = b squared n (with a n = 0), whereas spread complexity involves both a n and b n. The paper suggests that determining both might require additional input, such as the second spread complexity, K2(t).

Conclusion

The explicit construction of the algorithm serves as a proof of principle for using Krylov complexity as a complete characterization of operator evolution in quantum systems. It confirms that Krylov complexity contains the full information about the entire set of Lanczos coefficients, provided they are non-negative. The practical issues such as numerical stability remain subjects for future investigation.

The gist

Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density.

How it works

Improvements for AI systems

Here are the specific improvements that can be made to AI systems based on the concepts presented in this scientific paper, along with what those improved systems could achieve:

The core contribution of this paper is establishing that Krylov complexity, a measure derived from operator dynamics and represented by a recursive structure (related to Lanczos coefficients), contains the complete information about an operator's time evolution. The proposed algorithm allows one to extract these fundamental dynamical parameters recursively.

Here are the specific improvements:

Individual/Operator Dynamics Characterization via Krylov Complexity:
Recursive Parameter Extraction for Quantum System Modeling:
Distinguishing Operator vs. State Dynamics and Complexity Analysis:
Developing Predictive Models for Chaotic/Complex Systems (e.g., Quantum Chaos, High-Energy Physics):

Specific Capabilities of the Improved AI System:

Specific Improvements Detailed:

Individual/Operator Dynamics Characterization via Krylov Complexity:

The AI system can take the time evolution of a quantum operator (or a high-dimensional system's governing Hamiltonian) and calculate its Krylov complexity, using the Taylor expansion coefficients as input. The system then recursively applies the algorithm described in Table 1 to extract all fundamental dynamical parameters:

It can explicitly determine the set of Lanczos coefficients, which define the operator's action on its Krylov basis.
It can calculate the return amplitude and spectral density associated with that operator's evolution, effectively reconstructing a comprehensive description of its dynamics from just the complexity measure.

Recursive Parameter Extraction for Quantum System Modeling:

The AI system can be used as a universal solver for complex quantum dynamics by implementing the recursive algorithm (Table 1). Instead of solving complex differential equations directly, it takes an arbitrary function representing complexity, computes its Taylor coefficients, and recursively derives the underlying Lanczos coefficients.

It can invert the relationship: Given a target complexity function (or a set of input coefficients), it determines if those inputs are physically compatible with a valid quantum evolution by checking if the derived Lanczos coefficients satisfy physical constraints (e.g., ensuring all calculated Lanczos coefficients, or related quantities, remain non-negative where required).
It can serve as an automated compatibility tester for proposed dynamical models.

Distinguishing Operator vs. State Dynamics and Complexity Analysis:

The AI system can differentiate between the dynamics of an operator (governed by the Liouvillian) and those of a quantum state (governed by the spread complexity). By comparing the recursive extraction process for Krylov complexity with that for spread complexity, it can identify necessary additional inputs.

For operator dynamics, it extracts both real and imaginary components of evolution (related to hopping models).
For state dynamics, it identifies what is required beyond just spread complexity (e.g., the need for second spread complexity, or specific correlations like the 'a' coefficients mentioned in the text) to fully determine all relevant parameters.

Developing Predictive Models for Chaotic/Complex Systems:

The system can be trained on complex systems (like those studied in holographic theories mentioned in the references) to predict their long-term behavior based solely on a complexity measure.

It can predict phase transitions or chaotic regimes by analyzing the stability and termination points of the recursive Lanczos coefficient calculation (where a coefficient becomes zero or negative).
It can be used to test and validate holographic proposals (like the Krylov–momentum proposal), by checking if a given complexity function corresponds to a physically realizable set of Lanczos coefficients.

Abstract

This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around t=0. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a ``proof of principle'' for using Krylov complexity as a complete characterization of operator evolution in quantum systems.