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Generalized Krylov Complexity

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For a state evolved by several symmetry generators, the paper defines a generalized Krylov complexity based on a block-structured basis.

desk verdict Non-weighted block construction is sound; the weighted version has an unstated compatibility condition for non-Abelian algebras. read the letter →

arxiv 2507.23739 v2 pith:QUPKZ34V submitted 2025-07-31 hep-th quant-ph

Amin Faraji Astaneh , Niloofar Vardian This is my paper
classification hep-thquant-ph
keywords generalizedKrylovcomplexitynetworkspreadLiealgebrageneratorsblockorthogonalizationNielsenweightedquantumstate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends Krylov complexity from evolutions generated by a single Hamiltonian to evolutions generated by several operators at once. It replaces the one-dimensional Krylov chain with a network of orthogonal blocks, where block $n$ is spanned by products of the generators that apply exactly $n$ generators to the reference state, and it defines generalized Krylov complexity as the weighted average of the block number. The paper computes closed-form results for abelian examples and for $SU(2)$, and it introduces a weighted version that assigns different costs to different generator directions, connecting the construction to geometric circuit complexity. If the construction holds, any continuous symmetry of a model yields a state-complexity measure computed directly from its Lie algebra.

What carries the argument

The load-bearing object is the block-decomposed Krylov basis and the associated generalized Krylov operator $\hat K = \sum_n n P_n$. Blocks are defined by total generator degree $n = \sum k_i$; each block is built by orthogonalizing the vectors $T_1^{k_1}\cdots T_N^{k_N}|\psi_0\rangle$ at fixed $n$ after removing overlaps with lower blocks, and the projection onto each block is written with a metric-tensor formula $G_n^{-1}$. For the $SU(2)$ example, the calculation is carried by the standard angular-momentum rotation matrix elements (small $d$-matrices), which give the overlap of the rotated highest-weight state with each block.

What would settle it

Compute the weighted generalized Krylov complexity for an $SU(2)$ state with unequal weights $\mu_x,\mu_y,\mu_z$. If the block decomposition is consistent, the value must be independent of whether one builds the block for $\tilde n = \mu_x+\mu_y$ from $J_x J_y|\psi_0\rangle$ or from $J_y J_x|\psi_0\rangle$ plus a correction proportional to $J_z|\psi_0\rangle$; because the correction sits in a different weighted block, an order-dependent result would show the weighted construction is not defined for non-abelian algebras.

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Extended reading notes

Core claim

The paper's central claim is that the quantity $C = \sum_{n,i} n |\varphi^i_n|^2$, with coefficients $\varphi^i_n$ obtained by expanding the evolved state in the block-orthogonalized Krylov basis $K_n = \mathrm{Span}\{T_1^{k_1}\cdots T_N^{k_N}|\psi_0\rangle : k_1+\cdots+k_N=n\}$, is a meaningful measure of state complexity for general unitary evolutions generated by a Lie algebra. The authors define a generalized Krylov operator $\hat K = \sum_n n P_n$ whose expectation value in the evolved state gives $C$, and they construct the blocks by a projection algorithm that uses metric tensors. They also define a weighted version using $\tilde n = \sum_i \mu_i k_i$ to assign different costs to different generators, and they test the framework on $U(1)\times U(1)$, the $U(1)_X\times U(1)_Y$ subgroup of $SO(10)$, and $SU(2)$, obtaining closed-form expressions such as $C = 2j\,\sin^2(|\theta|/2)\,(|\theta|^2-\theta_z^2)/|\theta|^2$ for a highest-weight $SU(2)$ state.

Load-bearing premise

The load-bearing premise is that sorting states by the weighted degree $\tilde n = \sum_i \mu_i k_i$ yields a consistent block decomposition for every Lie algebra, which is not assured for non-abelian algebras because a commutator $[T_i,T_j]$ changes the weighted degree unless the weights satisfy a compatibility condition; the paper only demonstrates weighted block decomposition for abelian groups.

Editorial extensions

If this is right

  • For a single generator, the block construction reduces to the ordinary Krylov chain, so standard Krylov and spread complexity are recovered as the $N=1$ case.
  • For any symmetry group of a model, the generalized Krylov operator turns complexity of a state under the symmetry into the expectation value of a fixed operator built from the reference state.
  • Unequal weights produce direction-sensitive complexity that can be compared with geometric cost-based complexity, giving a discrete, basis-based counterpart to a cost function on the unitary manifold.
  • For $SU(2)$ highest-weight states, the closed formula shows the complexity depends only on the rotation angle and the component of the rotation axis along the quantization direction, and it vanishes for rotations about that axis.
  • For abelian examples the block structure truncates sharply, so the complexity can be computed from low-dimensional blocks rather than the full Hilbert space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The weighted generalization is tested only on abelian groups; for a non-abelian algebra, a commutator $[T_i,T_j]$ shifts the weighted degree, so sorting blocks by $\tilde n = \sum_i \mu_i k_i$ may not give a well-defined orthogonal decomposition unless the weights satisfy a compatibility condition that semisimple algebras cannot meet.
  • A natural next test is $SU(2)$ with unequal weights: the result should be independent of the order in which products like $T_i T_j$ are assigned to blocks, and this is not guaranteed by the present algorithm.
  • The generalized Krylov operator could serve as a symmetry-resolved complexity diagnostic in thermalizing or holographic settings, where different Lie-algebra directions play different dynamical roles.
  • The connection to geometric circuit complexity could be sharpened by asking which weight functions $\mu_i$ make the Krylov-network complexity equal to the minimal circuit cost; the paper leaves that optimization question open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes a generalization of Krylov (spread) complexity to unitary evolutions generated by multiple Lie algebra generators. A block-orthogonalized Krylov basis is constructed by grouping words T_1^{k1}...T_N^{kN}|ψ0> according to total degree n and orthogonalizing each block against all previous blocks; the generalized Krylov complexity is then defined as C = Σ n Σ_i |φ^i_n|^2, i.e., the expectation value of a generalized Krylov operator. A weighted version assigns positive weights μ_i to generators and defines blocks by the weighted degree ñ = Σ μ_i k_i. The paper presents analytic evaluations for U(1)×U(1) subgroups of SO(4) and SO(10) (both unweighted and weighted) and for SU(2) in the unweighted case, and sketches connections to Nielsen complexity and open quantum systems.

Significance. The unweighted block construction is mathematically sound and can be justified via the Poincare-Birkhoff-Witt filtration; the closed-form SU(2) results (e.g., Eq. (B56) for the highest-weight state) are useful benchmarks, and the examples are genuine analytic evaluations rather than numerical fits. The weighted extension is an interesting attempt to connect Krylov complexity to geometric complexity, but its current formulation is incomplete for non-Abelian algebras, substantially narrowing the claimed scope. The paper is clearly written and the appendix calculations are detailed.

major comments (2)
  1. [Section V, Eq. (32)] The weighted block construction is not well-defined as a cost measure for non-Abelian algebras unless the weights satisfy μ_k ≤ μ_i + μ_j for every nonvanishing structure constant f^k_{ij}. If μ_k > μ_i + μ_j, a monomial of weighted degree ñ can be reordered into a term of higher weighted degree; since the algorithm subtracts only lower blocks, that higher-weight component remains in K_{ñ}, so the block labels do not reflect the intended generator costs. For example, for SU(2) with (μ_x, μ_y, μ_z) = (1,1,3), the vector (J_x J_y − J_y J_x)|ψ0> = i J_z |ψ0> lies in K_2 although J_z carries weight 3. Conversely, a genuine grading of a semisimple algebra with positive weights is only possible when all weights are equal, so nontrivial weights inevitably lead to such commutator leakage. The paper restricts all weighted examples to Abelian groups and never states any compatibility condition; the broad claim in Sections V and VI that weighted complexity handles general Lie algebras is therefore unsupported. The authors should either prove the relevant filtration property, restrict the weighted construction to Abelian algebras, or explicitly discuss the interpretation of the leakage and its effect on the complexity label.
  2. [Section V, Eq. (31)] The weighted algorithm requires that the set of weighted degrees {Σ μ_i k_i} can be sorted in increasing order with finite gaps. For generic real weights (e.g., μ_1 = 1, μ_2 = √2), this set is dense on the positive real line, so there is no 'next' block and the procedure described in Section V is not well-defined. The paper should state a discreteness condition (e.g., all μ_i are rational multiples of a common value) or otherwise define the construction over an ordered semigroup of weights that admits a well-ordered ascending sequence of blocks.
minor comments (6)
  1. [Eq. (5)] The relation d_K = N_B d_{K_n} is only valid if all blocks have equal dimension; the correct relation is d_K = Σ_i d_{K_i}, as given in Eq. (25).
  2. [Eq. (24)] The expression 'T^{K_1}_1' should read 'T^{k_1}_1'; the manuscript contains several typographical errors ('posibbly' in the Introduction, 'formageneralLiealgebra' in Section II, 'usefule' in the Acknowledgments) and should be carefully proofread.
  3. [Eq. (10) and Section V] The weighted degree is denoted n in Eq. (10) but ñ in Section V; the notation should be unified.
  4. [Eq. (22) and Ref. [47]] The projection formula (22) is a standard linear-algebra result; citing Ref. [47] (one of the authors' own papers) is unnecessary and a standard textbook reference would be more appropriate.
  5. [Appendix B1, Eq. (B33)] In the weighted SO(10) example, the block for ñ = 1 is empty because X^2 generates a null state; for clarity, explicitly state that empty blocks are omitted from the direct sum, as is done in Eq. (B33).
  6. [Eq. (B62)] The general formula for SU(2) separates the cases m > 0 and m < 0 but does not explicitly give the m = 0 case; it follows by symmetry, but stating it would improve completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: generalized Krylov complexity is an explicit definition evaluated analytically in examples; the sole self-citation supports a standard projection formula and is not load-bearing.

full rationale

The paper's central object, generalized Krylov complexity, is defined directly in Eq. (9) as a weighted sum over block labels, with coefficients obtained from an explicitly specified block-orthogonalization procedure. There is no fitted parameter, no quantity predicted from a subset of data, and no external benchmark that the paper claims to reproduce. The examples in Section IV and Appendix B are analytic evaluations of this definition using explicit Gram-Schmidt steps and Wigner d-matrices. The only citation to prior work by one of the authors, Ref. [47], is used to justify the metric-based expression for a projection operator in Eq. (22); that identity is a standard linear-algebra fact, stated with no fitted assumptions, so it does not import the target result. The weighted construction in Section V is also definitional: it assigns weights, defines blocks by Eq. (32), and evaluates the resulting sums for Abelian examples only. The reviewer-flagged concern that the weighted block construction may not be well-defined for general non-Abelian Lie algebras unless the weights satisfy a compatibility condition is a correctness or generality issue, not a circularity issue: the paper does not claim to derive the weighted construction from anything that already contains it. Accordingly, the derivation chain is self-contained and no circular step is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The framework relies on a standard filtration of the universal enveloping algebra, an unstated compatibility condition for weighted blocks, and assumptions about the cyclic state and generator normalization. No external data or fitted constants are used, so the non-weighted core is self-contained.

free parameters (2)
  • Block weight n = n (integer block index)
    The complexity in Eq. (9) assigns weight n to each block K_n. This linear choice is asserted as "the most natural generalization" but is not derived from an operational principle.
  • Generator weights μ_i = μ1=1/2, μ2=3/2 in the examples
    In Section V and the U(1)xU(1) examples, weights are chosen by hand to break symmetry between directions; no procedure for fixing them is given.
assumptions (4)
  • standard math The spans of ordered monomials T_1^{k1}...T_N^{kN} of total degree n, after subtracting lower blocks, give a well-defined filtration of the cyclic representation.
    Used in Sec. II and Appendix A to define K_n; follows from the Poincare-Birkhoff-Witt theorem but is not stated explicitly.
  • ad hoc to paper Weighted blocks are well-defined for any Lie algebra by sorting weighted degrees.
    Sec. V, Eqs. (31)-(32); this assumes weighted degree is compatible with the commutator algebra, which fails for non-Abelian algebras.
  • domain assumption The initial state |ψ0> is cyclic with respect to the generators (the Krylov space spans the relevant Hilbert space).
    The algorithm truncates when blocks vanish; the examples assume a finite-dimensional cyclic subspace.
  • domain assumption The generators {T_i} are Hermitian and trace-orthonormal.
    Sec. II and Appendix A; needed for unitary evolution and orthogonalization.

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Pith. "Pith review of Generalized Krylov Complexity." pith.science (2026). https://pith.science/paper/QUPKZ34V

@misc{pith2026250723739,
  author       = {Pith},
  title        = {Pith review of: Generalized Krylov Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUPKZ34V}},
  note         = {Machine review of arXiv:2507.23739}
}
read the original abstract

We extend the concept of Krylov complexity to include general unitary evolutions involving multiple generators. This generalization enables us to formulate a framework for generalized Krylov complexity, which serves as a measure of the complexity of states associated with continuous symmetries within a model. Furthermore, we investigate scenarios where different directions of transformation lead to varying degrees of complexity, which can be compared to geometric approaches to understanding complexity, such as Nielsen complexity. In this context, we introduce a generalized orthogonalization algorithm and delineate its computational framework, which is structured as a network of orthogonal blocks rather than a simple linear chain. Additionally, we provide explicit evaluations of specific illustrative examples to demonstrate the practical application of this framework.

Figures

Figures reproduced from arXiv: 2507.23739 by the authors.

Figure 1
Figure 1. The complexity of the initial state in the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The Krylov complexity of the initial state [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The generalized Krylov complexity of the initial state ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The complexity of the initial state |j = 3/2, m = 3/2⟩ under SU(2) evolution. Applying the procedure defined in Section II to this state yields the following block structure K0 = span{|K0,0⟩ = |j, m⟩}, dK0 = 1 K1 = span{|K1,0⟩ = |j, m + 1⟩, |K1,1⟩ = |j, m − 1⟩}, dK1 = …
Figure 5
Figure 5. Figure 5: The complexity of the initial state |j = 3/2, m = 1/2⟩ under SU(2) evolution [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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Cited by 1 Pith paper

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