REVIEW 2 major objections 5 minor 28 references
The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups
T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read The paper proves polynomial-time quantum algorithms for the hidden subgroup problem in scalar semidirect products and in structured quasi-Hamiltonian groups, the first to exploit modular subgroup lattices.
desk verdict Solid new HSP results for quasi-Hamiltonian groups, with the structured-input assumption as the key honest caveat. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Three mechanisms carry the argument. (1) Cyclic reduction: in $G = A\rtimes \mathbb{Z}_{p^k}$, the hidden subgroup, after quotienting by $H\cap (A\times\{0\})$, is either trivial or $\langle (d,p^t)\rangle$, and the verification test $V(s,c) = [f(0,0) = f(c,p^s)]$ certifies the correct $t$, so a low-probability subroutine can be repeated and verified into a correct algorithm. (2) Hidden Multiple Shift: an oracle $f_s(x,h) = f(x - hs)$ over $\mathbb{Z}_q^n \times R$; the scalar action makes the coset function satisfy $f(x, bp^t) = f_0(x - M_t^{(b)}d)$, matching this format, and the injective embedding of $Q$ into $\mathbb{Z}_{E_Q}^m$ with coset labels preserves it. (3) Crossed isomorphism: the bijection $\sigma(ac^{\mu(j)}) = b^j a$ from the Abelian $p$-group $B = \langle A, c \mid ac = ca,\ c^q = a_0\rangle$ to the modular Sylow factor $P = A\langle b\rangle$, whose failure to be a homomorphism is governed by twisting automorphisms $x\mapsto x^{r^j}$; since these are power automorphisms, Baer's theorem (condition 2 of Theorem 36) implies $\sigma$ maps subgroups to subgroups and preserves right cosets — exactly the property that transports the HSP from $P$ to the Abelian group $B$. For the cyclic instance $A = \mathbb{Z}_N$, the fixed-$t$ subroutine is a pretty good measurement with one-copy success at least $\varphi(N)p/N^2$.
What would settle it
Take a small non-Abelian quasi-Hamiltonian group such as the paper's own example $P = \mathbb{Z}_{81}\rtimes_4 \mathbb{Z}_{27}$, build $B$ and $\sigma$ by the Section 4 construction, and exhaustively verify the identity $\sigma(Kx) = \sigma(K)\sigma(x)$ for every subgroup $K \leq B$ and every element $x \in B$; then simulate the full algorithm for every subgroup $H \leq P$, replacing the oracle by the true coset function, and check that the output generates exactly $H$. Any single violation of the coset identity, or any $H$ whose preimage $\lambda^{-1}(H)$ is not a subgroup, would refute the transport step behind Theorem C. For the semidirect-product claim, evaluate the one-copy pretty-good-measurement formula of Theorem 40 numerically for small $N$ and compare with the claimed bound $\varphi(N)p/N^2$.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is a pair of reductions. For scalar semidirect products, every hidden subgroup reduces to the cyclic case $\langle (d, p^t)\rangle$: the intersection $H \cap (A\times\{0\})$ is recovered by an Abelian HSP and is automatically normal because the scalar action fixes every subgroup of $A$, so after the quotient the hidden subgroup is either trivial or cyclic, with $t$ located by a linear search whose verification test compares $f(0,0)$ with $f(c, p^s)$, and $d$ recovered either by a pretty good measurement (Theorem A, $A = \mathbb{Z}_N$) or by embedding the quotient $Q = A/A_1$ into $\mathbb{Z}_{E_Q}^m$ and solving a Hidden Multiple Shift instance (Theorem B). For quasi-Hamiltonian groups, the discovery is that a non-Hamiltonian modular Sylow factor $P = A\langle b\rangle$ with $b^{-1}ab = a^{1+p^s}$ is carried by a crossed isomorphism $\sigma(ac^{\mu(j)}) = b^j a$ to an Abelian $p$-group $B$, and because every twisting map is a power automorphism, Baer's theorem on crossed isomorphisms implies that $\sigma$ maps subgroups to subgroups and preserves right cosets; the HSP oracle on $P$ therefore becomes an Abelian HSP oracle on $B$, and the recovered generators are mapped back by $\lambda(x) = \sigma(x^{-1})^{-1}$. The paper claims both reductions run in quantum polynomial time under the stated parameter conditions and structured input assumptions.
Load-bearing premise
If the structured presentation is not actually available, the quasi-Hamiltonian algorithm cannot start: Theorem C assumes the input already contains the Sylow decomposition and, for each non-Hamiltonian factor, the presentation $P = A\langle b\rangle$ together with the parameters $p$ and $s$ of the action $b^{-1}ab = a^{1+p^s}$, and the paper does not solve the constructive-recognition problem of extracting this presentation from an arbitrary generating set; Theorem B similarly assumes $A$ is given in invariant-factor form with coordinate maps.
Editorial extensions
If this is right
- The scalar semidirect-product algorithm unifies and extends the earlier cyclic results: the families $\mathbb{Z}_N \rtimes \mathbb{Z}_p$ and the prime-power cyclic cases studied previously are subsumed by Theorems A and B.
- The quasi-Hamiltonian algorithm applies to non-Abelian modular $p$-groups of the form $A\langle b\rangle$ with power action, including groups such as $\mathbb{Z}_{p^k} \rtimes_{1+p^s} \mathbb{Z}_{p^{k-s}}$ whose nilpotency class is unbounded; together with the known nilpotency-class-2 algorithm, the HSP is now solved in polynomial time for a wider and strictly incomparable set of nilpotent groups.
- Because every quasi-Hamiltonian group splits into its Sylow factors, and the Hamiltonian factors $Q_8\times E$ are already covered by the Dedekind-group algorithm, Theorem C solves the HSP on the entire group by combining the per-factor solutions.
- The paper's open question — the HSP with the hidden subgroup merely promised to be permutable — would subsume the Dedekind-group result if solved, and would not resolve the dihedral HSP, since the dihedral group contains non-permutable subgroups.
Reading between the lines
- In our reading, the essential resource in Theorem C is not modularity itself but the coset-preserving projectivity: any finite group admitting a crossed isomorphism to an Abelian group whose twisting maps are power automorphisms would inherit the same HSP reduction, so seeking such projectivities could serve as a general strategy for new group families.
- The 'guess, verify, repeat' pattern of Section 3.2 — a weak pretty-good-measurement candidate checked against two oracle queries — is portable: any HSP reduction that yields candidate subgroups together with a cheap membership test can be boosted to a correct polynomial algorithm.
- A concrete next step is constructive recognition of modular $p$-groups: Theorem C would become a black-box HSP algorithm the moment an efficient procedure extracts $P = A\langle b\rangle$, $p$, and $s$ from arbitrary generators, and the power-automorphism structure the algorithm exploits is a plausible recognition handle.
- Because the paper identifies the two roles of scalarity (normality of $H\cap A$ and the multiple-shift structure), non-scalar actions on bounded-rank Abelian factors mark the boundary of the method: the reduction's fallback descent shrinks the complement but still lands on the same cyclic recovery problem.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper gives polynomial-time quantum algorithms for two families of hidden subgroup problems. The first family (Theorems A and B) consists of semidirect products G = A ⋊_φ Z_{p^k} with A finite abelian, a scalar action, bounded generator rank of A, and Exp(A)/p = polylog(|G|); Theorem A treats A = Z_N via a pretty-good-measurement construction, while Theorem B reduces the general case to the Hidden Multiple Shift problem of Ivanyos–Prakash–Santha. The second family (Theorem C) is the class of finite quasi-Hamiltonian groups; the algorithm uses a crossed isomorphism to transport the hidden subgroup to an auxiliary abelian p-group B and then applies the standard abelian HSP algorithm. The paper explicitly restricts Theorem C to a structured quasi-Hamiltonian presentation in which the Sylow decomposition and the action parameters are part of the input, and it disclaims any constructive-recognition procedure.
Significance. If the results are correct, Theorem B unifies and extends several earlier semidirect-product HSP algorithms, and Theorem C is, to my knowledge, the first HSP algorithm that exploits modularity of the subgroup lattice rather than nilpotency class or near-Hamiltonian structure. The proofs are built from external, established subroutines — abelian HSP, the Hidden Multiple Shift algorithm of [16], Smith normal form decomposition of [6], and Baer's theorem on crossed isomorphisms — with no fitted constants or self-referential assumptions. The main strength is the explicit reduction structure; the main weakness is the input model, particularly for Theorem C, where the algorithm solves a promise version of the HSP rather than the standard black-box version.
major comments (2)
- [Section 4, Input model; Theorem C] The algorithm of Theorem C does not solve the hidden subgroup problem for quasi-Hamiltonian groups in the standard black-box group model. It requires, as input, the Sylow decomposition and, for each non-Hamiltonian factor, a structured presentation P = A⟨b⟩ together with the parameters p and s of the action b^{-1}ab = a^{1+p^s}. The manuscript explicitly states that it does not address the constructive-recognition problem of obtaining this presentation from an arbitrary generating set. This is a load-bearing limitation: without an efficient recognition procedure, Theorem C applies only to groups that are already handed to the algorithm in a decomposed, structured form. The abstract's phrase 'mild assumption' understates this restriction. The theorem should be framed as a promise-model result, and the abstract and introduction should be adjusted accordingly.
- [Sections 3.3.4 and 3.2.2] The unknown-t search for Theorem B is said to 'simply follow the method in Section 3.2.2', but that method's correctness analysis in Proposition 22 depends on a quantitative lower bound q0 on the success probability of each fixed-t subroutine. For the PGM-based subroutine of Theorem A, such a bound is proved in Appendix B. For the HMS-based subroutine of Theorem B, only 'with high probability' is cited from Theorem 23 of [16]; no q0 is stated. Without an explicit lower bound and a corresponding amplification argument, the repetition count R and the proof of Proposition 22 do not automatically carry over to the HMS setting. This is fixable by amplifying the HMS algorithm and stating the resulting success-probability bound, but as written it is a gap in the correctness proof of Theorem B.
minor comments (5)
- [Abstract and Section 1.1] The phrase 'mild assumption on the input structure' is too weak for the structured quasi-Hamiltonian presentation required by Theorem C; the input model described in Section 4 is a substantial promise and should be described as such in the abstract and introduction.
- [Section 4.2] The definition of the generators \tilde g_i after the Smith normal form step is garbled in the text ('gU1,i1 · · ·gUk,i d+1') and should be written as g_1^{U_{1,i}} ··· g_{d+1}^{U_{d+1,i}}.
- [Lemma 37] In the discussion of property (b), 'there are no elements in b with a power of 2 order' should read 'there are no elements in B with a power of 2 order'.
- [Section 3.3.2] There is a typo in the description of scalar multiplication: 'i.e., my the map' should be 'i.e., by the map'.
- [Appendix A.1] The sentence 'This is falls under case (2) of Proposition 14' contains a grammatical error and should read 'This falls under case (2) of Proposition 14'.
Circularity Check
No circularity found: the algorithms reduce to independently established subroutines (Abelian HSP, Hidden Multiple Shift, Smith normal form, and classical group-structure theorems), and the structured-input caveat is a scope restriction rather than a definitional recycling of the target.
full rationale
The paper's derivations are self-contained reductions to external results, not re-statements of their inputs. Theorem B reduces the semidirect-product HSP to the Hidden Multiple Shift problem of Ivanyos–Prakash–Santha (ref. [16]) after first solving an Abelian HSP over A, using the scalar-action assumption only to make H∩(A×{0}) normal and to preserve the shift structure under the embedding into Z_{E_Q}^m. No parameter is fitted to the hidden subgroup, and the efficiency condition Exp(A)/p = polylog is a constraint on the input family, not a fitted quantity. Theorem A likewise uses a proven PGM success lower bound with no fitted parameter; the repetition count R is derived from the proven bound φ(N)p/N^2, and verification against the hiding oracle is explicit. Theorem C maps the quasi-Hamiltonian group P to an explicitly constructed Abelian p-group B via a crossed isomorphism, proves the projectivity and coset-preservation properties using Baer's theorem (external classical result), and then applies the standard Abelian HSP algorithm. The construction of B and the maps σ and λ depend only on the supplied presentation of P, not on the hidden subgroup. The paper explicitly restricts the input model to a structured quasi-Hamiltonian presentation, which weakens the scope of Theorem C relative to the black-box model, but this is an input-assumption limitation, not a circular derivation: the structured data are needed to implement the reduction, and the conclusion is not assumed in the premises. All external citations (Abelian HSP, HMS, Smith normal form, Schmidt's classification, Baer's theorem) are independent mathematical or algorithmic results; none is authored by this paper's author, and none is used to forbid alternatives. The AI-usage statement and the appendix comparisons are not load-bearing in a circular way. Accordingly, no circular step is identifiable from the paper's own equations or citations.
Assumptions & free parameters
assumptions (6)
- standard math Classification of modular p-groups (Proposition 14) and related structural results from Schmidt [17].
- standard math Baer's theorem on crossed isomorphisms (Theorem 36).
- domain assumption Hidden Multiple Shift algorithm of Ivanyos, Prakash and Santha (Theorem 23).
- domain assumption Abelian HSP algorithm and finite Abelian group decomposition of Cheung and Mosca [6].
- domain assumption Black-box group access with reversible circuits and unique encodings where required.
- standard math Shor's algorithms for factoring and discrete logarithm.
invented entities (1)
-
Auxiliary Abelian p-group B with formal generator c
Cite this review
Pith. "Pith review of The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups." pith.science (2026). https://pith.science/paper/V22R2BLI
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author = {Pith},
title = {Pith review of: The Hidden Subgroup Problem in Semidirect Products and Quasi-Hamiltonian Groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/V22R2BLI}},
note = {Machine review of arXiv:2608.05321}
}
abstract
Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively. In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form $G=A\rtimes_{\varphi} \mathbb{Z}_{p^k}$, with $A$ finite Abelian, $p$ prime, $k\in \mathbb{N}$ and the action of $\mathbb Z_{p^k}$ is generated by the scalar automorphism $a\mapsto\mu a$, for some $\mu\in\mathbb Z_{\operatorname{Exp}(A)}^\times$, where $\mathrm{Exp}(A)$ is the exponent of $A$. Our algorithm is efficient when $A$ has bounded generator rank and $\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)$. This includes the case $A=\mathbb{Z}_N$ for $N\in\mathbb{N}$ and $k=1$, studied by Bacon, Childs and van Dam (FOCS 2005), and $A=\mathbb{Z}_{q^r}$ with $q$ prime and $r\in \mathbb{N}$ studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).
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