Fast numerical generation of Lie closure

Yutaro Iiyama

REVIEW 3 major objections 5 minor 13 references

Fast numerical generation of Lie closure

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that the linear-independence bottleneck in Lie-closure construction can be replaced by Gram-matrix inversion or by projection onto an orthonormal basis, reducing each check to O(|B|) and preserving sparse operator…

desk verdict A useful, modest DLA speedup whose central reliability claim is undercut by an unstated zero-test tolerance and two validation mismatches the authors wave off. read the letter →

arxiv 2506.01120 v1 pith:H6P7OJ46 submitted 2025-06-01 cs.CE quant-ph

Yutaro Iiyama This is my paper

classification cs.CEquant-ph MSC 15A0315A3065-04

keywords LieclosuredynamicalalgebralinearindependencecheckorthonormalizationGrammatrixsparsePaulirepresentationvariationalquantumcircuitsnumericalalgorithm

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

The paper asks how to compute the smallest Lie algebra containing a set of matrices, the dynamical Lie algebra of a parametrized quantum circuit, without being stopped by a costly linear-independence test. It proposes two replacements for the usual rank-and-SVD check: a matrix-inversion method built on the Gram matrix, and an orthonormalization method that projects each new commutator onto the existing orthonormal basis and accepts it only if the residual is nonzero. The orthonormalization check costs O(|B|) per candidate instead of the O($d^{2}$ N'^2) of rank calculation, and it never requires expanding sparse operators into dense matrices. Validation reproduces known DLA dimensions for most tested circuit families, with two small mismatches the author attributes to accumulated round-off.

What carries the argument

The load-bearing identity is the Gram-matrix lemma: the matrix of inner products of non-null, linearly independent operators is invertible, because any unitarily mixed linear combination with zero norm would be a null operator. Theorem 3.2 converts linear dependence into the coefficient formula x = $A^{{-1}}$ $\beta$, and Corollary 3.3 simplifies that to orthogonal projection when the basis is orthonormal. The orthonormalization algorithms maintain either two bases (an output basis of generators and nested commutators plus an auxiliary orthonormal basis) or just the orthonormal basis, using the projection residual h_⊥ as the independence test.

What would settle it

Run the orthonormalization algorithm on the open-boundary transverse-field Ising Hamiltonian variational ansatz with n=8 using higher-precision arithmetic, such as 128-bit floats, and count the DLA dimension; if the count returns $n^{2}$ − 1 = 63 rather than the paper's reported 64, the extra dimension is a round-off artifact, confirming that the zero-test tolerance is the fragile point.

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Extended reading notes

Core claim

The central discovery is that the linear-independence subroutine dominating Lie-closure construction can be turned into a Gram-matrix solve: for a tuple B of independent operators, h is dependent on B exactly when h = B · ($A^{{-1}}$ $\beta$), where A_ij = <B[i], B[j]> and beta_i = <B[i], h>. When B is orthonormalized, A becomes the identity, so the test reduces to forming the projection residual h_⊥ = h − proj_B h and checking that h_⊥ is nonzero. That makes each independence check O(|B|), eliminates matrix factorizations from the inner loop, and works directly on sparse representations such as sums of Pauli strings. The paper presents algorithms built on both methods and reports measured runtimes on quantum-circuit benchmarks that scale dramatically better than the rank-calculation implementation.

Load-bearing premise

The load-bearing premise is that floating-point arithmetic can reliably decide whether the leftover part of a candidate operator after projection is zero, but no tolerance is specified, and the paper's own validation shows this decision failing for two test cases because of rounding-off errors.

Editorial extensions

If this is right

DLA dimensions for hardware-efficient ansätze can be computed directly for 7 qubits, reaching dimension 16,383 in about 39 minutes with a sparse Pauli representation.
Because sparse operator representations are preserved throughout, memory use scales with the number of basis elements and their sparsity rather than with the full d^2 × |B| dense matrix.
The O(|B|) per-check cost lowers the exponent of the overall closure algorithm, pushing practical numerical Lie-closure studies to larger system sizes than the standard rank-based construction.
Studies of barren plateaus and overparameterization can compute DLA dimensions of candidate circuit families directly, without relying on analytically derived closures that exist only for special cases.

Reading between the lines

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

Beyond the paper: because the orthonormalization test depends on an implicit tolerance for when h_⊥ counts as nonzero, tracking the relative size ‖h_⊥‖/‖h‖ across iterations would expose when the basis has drifted into numerical degeneracy.
Beyond the paper: the conditioning of the Gram matrix in the inversion method could be monitored cheaply, and a re-orthonormalization pass could stabilize cases where normalized nested commutators still produce poorly conditioned inner-product matrices.
Beyond the paper: the same projection test transfers to any finite-dimensional matrix Lie algebra equipped with an inner product, not just su(d), so the implementation could be reused for other matrix algebras with sparse bases.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Lead: this is a modest but real speedup for a practically annoying subroutine—numerically building Lie closures / DLAs. The exact-arithmetic linear algebra is correct, the sparse-matrix angle is sensible, and the runtime table shows a clear improvement over rank/SVD. It deserves a serious referee, but the paper's own validation contains a warning the authors wave off too quickly.

What's new: combining a Gram-matrix inversion or an orthonormal-basis projection as the independence check inside iterative DLA construction, preserving sparse Pauli-sum representations. The pieces are textbook linear algebra; the packaging for this problem is useful and not in the cited prior work, which falls back to rank or exploits Pauli-string uniqueness. The paper is honest that the matrix-inversion variant is numerically fragile and does not implement it, and the normalization trick for stability is a good practical detail. The external validation against Larocca et al.'s DLA dimensions covers several families and shows real scaling gains.

Soft spots, in order: - The 'h_perp != 0' test has no stated tolerance or error analysis. This is not a nitpick: Table 1(d) gives DLA dimension 64 instead of 63 for n=8 and 100 instead of 99 for n=10, attributed to round-off. A falsely accepted vector becomes part of the basis and can generate new commutators, so one bad acceptance can cascade. For the large sizes the abstract promises, there is no analytic check, so the central scalability claim is not fully established by the tests that pass. - The equivalence of Algorithm 3.4 (commutators of V rather than B) is asserted with 'can be proven easily' but no proof is given. It is probably true, but the paper should show it. - No code or data is released, so the favorable runtimes are hard to reproduce independently. - Minor: the matrix-inversion method is described but never implemented or validated, so only one of the two advertised methods actually gets tested.

Bottom line: a competent, useful paper with a load-bearing numerical-reliability gap. The right fix is a tolerance/error analysis and a repair or explanation of the Ising mismatches, plus the missing proof and ideally code. I would send it to peer review with that request, and I'd not cite it as-is for numerical work until those are addressed.

Referee Report

3 major / 5 minor

Summary. The paper proposes two numerical methods for accelerating the construction of a Lie closure (dynamical Lie algebra) from a set of generator matrices. The first method, the matrix inversion method (Algorithm 3.2), checks linear independence by solving the Gram system A x = β and testing the residual against zero. The second, the orthonormalization method (Algorithms 3.3 and 3.4), maintains an orthonormal basis V and tests whether the orthogonal projection residual h − proj_V h is zero. The paper claims both methods reduce the per-check cost and memory usage relative to the standard rank/SVD method, and it validates the orthonormalization method on DLA dimensions of several quantum circuit families from Ref. [6], reporting runtime improvements. It also demonstrates a sparse Pauli-string computation for the hardware-efficient ansatz at 6 and 7 qubits, reaching Hilbert-space dimensions that would be prohibitive with dense rank-based checks.

Significance. If correct, the orthonormalization method would provide a practical, representation-agnostic way to compute Lie closures for larger quantum systems than the standard dense rank-based approach, which is directly relevant to studies of variational quantum circuits and quantum control. The paper gives an exact-arithmetic derivation (Lemma 3.1 and Theorem 3.2) that is standard and sound, and it demonstrates clear runtime gains on the validated examples. The sparse Pauli-string experiment in Table 2 is a concrete step that supports the scalability claim, and the use of previously published DLA dimensions as external benchmarks is appropriate. The main unresolved weaknesses are the unspecified numerical tolerance in the zero tests, the two incorrect dimensions in the paper's own validation table, and the unproved equivalence of Algorithm 3.4 to Algorithm 3.3; these issues need to be addressed before the central claims of numerical reliability and of enabling larger system sizes are fully established.

major comments (3)

[Section 3.2, 'One nontrivial difference...'] The assertion that Algorithm 3.4, which computes commutators of the orthonormalized basis V, generates the same closure as Algorithm 3.3, which uses the unnormalized basis B, is only stated as 'can be proven easily' and no proof is supplied. This equivalence is not entirely trivial because V elements are linear combinations of B elements with coefficients obtained during the orthogonalization process, and the order in which the loops enumerate [V[l], V[m]] may omit linear combinations that [B[l], B[m]] would generate. The authors should provide a rigorous proof that the closure of the span of the iterated commutators of V equals the closure of G, or alternatively use Algorithm 3.3 as the primary method throughout and avoid relying on the unproved equivalence.
[Section 5 (Conclusion) and Section 3.2 (memory discussion)] The conclusion states that the proposed algorithms 'consume less computer memory', but Section 3.2 explicitly notes that the orthonormalization method requires holding two bases (B and V), which for dense matrices doubles the memory footprint relative to the standard basis-only algorithm. The rank-based reference method storing a d² × |B| matrix may use comparable or more memory, but the blanket claim in Section 5 is not justified for all proposed methods. The authors should qualify the memory claim: the matrix inversion method reduces the Gram-matrix memory, while the orthonormalization method trades memory for speed unless B is discarded as in Algorithm 3.4, whose equivalence is not proven. This is a load-bearing part of the abstract's promise of reduced memory footprint.
[Section 3.2, Corollary 3.3 and Equation (3.10)] The paper claims that the orthonormalization method makes the linear-independence check 'O(|B|)' (Section 3.2), but this omits the cost of the operations themselves. Each inner product ⟨B[l], h⟩ and each scalar multiplication/accumulation in the sum in Eq. (3.10) operates on d×d matrices or on sparse representations whose size depends on the physical system. In dense form the cost per check is O(|B| d²), so the claimed improvement over the rank method's O(d² |B|²) is only a reduction in the exponent with respect to |B|, not in the system-size dependence. For sparse representations, the cost depends on the sparsity structure; the paper does not give a model for it. The scaling claims in Section 5 and the abstract should be stated with the representation-dependent constants made explicit, otherwise the complexity statements are misleading.

minor comments (5)

[Throughout] There are several typos and formatting issues, including 'F AST' in the title, 'V alidation' at the start of Section 4, and inconsistent spacing around equations; these should be corrected during revision.
[Section 3.1, Algorithm 3.2] In Algorithm 3.2, the expansion of the Gram matrix A and its inverse is described textually; for reproducibility, the authors should specify how the Schur complement or another update is computed, and whether the inverse is explicitly stored or used through a factorization.
[Section 4.1, Table 1] The runtime measurements are given only as a few data points and without error bars or specification of the number of runs; adding confidence intervals or at least noting the hardware and library versions would make the runtime comparison more reproducible.
[References] The paper refers to 'Algorithm 1 in [6]' for the standard DLA construction, but the reference list does not include the exact algorithm number; a more precise pointer would help readers.
[Section 2.2, Notation] The tuple concatenation notation in Eq. (2.5) uses '+' which is easily confused with arithmetic addition; consider using the ∥ or ⊕ operator for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained standard linear algebra with external validation.

full rationale

The paper's central derivation chain is not circular. Theorem 3.2 reduces linear-independence checking to forming a Gram matrix and testing whether the residual h - B(A^{-1} beta) is zero; Lemma 3.1 proves invertibility of the Gram matrix for linearly independent inputs, an exact standard linear-algebra statement whose assumptions do not include the target DLA dimension. Corollary 3.3 and Equation (3.10) specialize the same residual to the orthonormal case, where A = I, so the projection formula follows by construction from the same theorem rather than from any fitted quantity. No parameter is fitted to the validation data and then renamed as a prediction: the known DLA dimensions in Table 1 come from Ref. [6], an external benchmark that is not an input to the algorithms. The paper contains no self-citations by the author, and no load-bearing uniqueness claim is imported from prior work by the same authors. The only notable weakness is the unspecified floating-point zero test in Algorithms 3.2-3.4 and the reported mismatches in Table 1(d) (64 vs 63 and 100 vs 99). That is a numerical reliability and correctness-risk concern, not a circularity concern: the algorithms' logical derivation does not reduce to their outputs. The scalability conclusion may be weakened by unproven tolerance assumptions, but that does not make the derivation circular. Accordingly, a score of 0 is appropriate.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on no fitted constants, but it does rest on an unstated numerical tolerance, on the correctness of the standard closure algorithm from Ref. [6], on the unproved equivalence of Algorithm 3.4, and on the assumption that sparse representations remain efficient under commutator and projection.

free parameters (1)

numerical tolerance for zero tests
The pseudocode accepts h-perp 'not equal to 0' as independence, but floating point requires a threshold epsilon; the paper never specifies it. This threshold decides whether a commutator is added to the basis and likely caused the n=8,10 mismatches in Table 1(d).

assumptions (5)

standard math Finite-dimensional complex matrix Lie algebra with commutator [a,b]=ab-ba and inner product (1/d)tr(a-dagger b)
Sets the algebraic setting; assumed throughout Section 2.
standard math Gram matrix of linearly independent non-null operators is invertible
Lemma 3.1 proof; background linear algebra.
domain assumption Standard closure construction (Algorithm 3.1) correctly computes the Lie closure
Adopted from Ref. [6] without proof; the paper's algorithms modify only the independence check.
ad hoc to paper Algorithm 3.4, using commutators of the orthonormal basis V instead of the original basis B, generates the same closure
Asserted in Section 3.2 as 'can be proven easily'; no proof is supplied. It is true because span(V)=span(B), but it is an unproved load-bearing step for Algorithm 3.4.
domain assumption Sparse representation (Pauli sums) remains efficient under commutator and projection without expanding to dense matrices
The sparse feasibility claim in the Abstract, Section 3, and Table 2 depends on commutator and inner product operations being cheap on sparse operators; no complexity or memory bound is proven.

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Cite this review

Pith. "Pith review of Fast numerical generation of Lie closure." pith.science (2026). https://pith.science/paper/H6P7OJ46

@misc{pith2026250601120,
  author       = {Pith},
  title        = {Pith review of: Fast numerical generation of Lie closure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6P7OJ46}},
  note         = {Machine review of arXiv:2506.01120}
}

read the original abstract

Finding the Lie-algebraic closure of a handful of matrices has important applications in quantum computing and quantum control. For most realistic cases, the closure cannot be determined analytically, necessitating an explicit numerical construction. The standard construction algorithm makes repeated calls to a subroutine that determines whether a matrix is linearly independent from a potentially large set of matrices. Because the common implementation of this subroutine has a high complexity, the construction of Lie closure is practically limited to trivially small matrix sizes. We present efficient alternative methods of linear independence check that simultaneously reduce the computational complexity and memory footprint. An implementation of one of the methods is validated against known results. Our new algorithms enable numerical studies of Lie closure in larger system sizes than was previously possible.

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Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

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Larocca, P

M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo , Diagnosing barren plateaus with tools from quantum optimal control , Quantum, 6 (2022), p. 824

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V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, M. S. Alam, S. Ahmed, J. M. Arra- zola, C. Blank, A. Delgado, S. Jahangiri, K. McKiernan, J. J. Meyer, Z. Niu, A. Sz ´ava, and N. Killoran , PennyLane: Automatic differentiation of hybrid quantum- classical computations, arXiv:1811.04968, (2020)

arXiv 2020
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Fontana, D

E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia , Characterizing barren plateaus in quantum ans¨ atze with the adjoint representation , Nat. Commun., 15 (2024), p. 7171

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M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage , Lie-algebraic classical simulations for quantum computing , arXiv:2308.01432, (2023)

arXiv 2023
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K¨okc¨u, R

E. K¨okc¨u, R. Wiersema, A. F. Kemper, and B. N. Bakalov , Classification of dynamical lie algebras generated by spin interactions on undirected graphs , arXiv:2409.19797, (2024)

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M. Larocca, N. Ju, D. Garc ´ıa-Mart´ın, P. J. Coles, and M. Cerezo , Theory of over- parametrization in quantum neural networks , Nat. Comput. Sci., 3 (2023), pp. 542–551

work page 2023
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Monbroussou, E

L. Monbroussou, E. Z. Mamon, J. Landman, A. B. Grilo, R. Kukla, and E. Kashefi , Trainability and expressivity of hamming-weight preserving quantum circuits for machine learning, Quantum, 9 (2025), p. 1745

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [6]

Larocca, P

M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo , Diagnosing barren plateaus with tools from quantum optimal control , Quantum, 6 (2022), p. 824

work page 2022

[2] [1]

Aguilar, S

G. Aguilar, S. Cichy, J. Eisert, and L. Bittel , Full classification of pauli lie algebras , arXiv:2408.00081, (2024)

arXiv 2024

[3] [2]

Bergholm, J

V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, M. S. Alam, S. Ahmed, J. M. Arra- zola, C. Blank, A. Delgado, S. Jahangiri, K. McKiernan, J. J. Meyer, Z. Niu, A. Sz ´ava, and N. Killoran , PennyLane: Automatic differentiation of hybrid quantum- classical computations, arXiv:1811.04968, (2020)

arXiv 2020

[4] [3]

Fontana, D

E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia , Characterizing barren plateaus in quantum ans¨ atze with the adjoint representation , Nat. Commun., 15 (2024), p. 7171

work page 2024

[5] [4]

M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage , Lie-algebraic classical simulations for quantum computing , arXiv:2308.01432, (2023)

arXiv 2023

[6] [5]

K¨okc¨u, R

E. K¨okc¨u, R. Wiersema, A. F. Kemper, and B. N. Bakalov , Classification of dynamical lie algebras generated by spin interactions on undirected graphs , arXiv:2409.19797, (2024)

arXiv 2024

[7] [7]

Larocca, N

M. Larocca, N. Ju, D. Garc ´ıa-Mart´ın, P. J. Coles, and M. Cerezo , Theory of over- parametrization in quantum neural networks , Nat. Comput. Sci., 3 (2023), pp. 542–551

work page 2023

[8] [8]

Monbroussou, E

L. Monbroussou, E. Z. Mamon, J. Landman, A. B. Grilo, R. Kukla, and E. Kashefi , Trainability and expressivity of hamming-weight preserving quantum circuits for machine learning, Quantum, 9 (2025), p. 1745

work page 2025