{"id":"218db38a-8184-4acd-92cc-48d914b75dd1","arxiv_id":"2608.05321","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Polynomial-time quantum algorithms are given for the hidden subgroup problem over scalar-action semidirect products A semidirect Z_{p^k} and over quasi-Hamiltonian groups with a structured presentation.","lead":"Two new families of non-Abelian groups are added to the list of hidden subgroup problems that quantum computers can solve in polynomial time: scalar-action semidirect products and quasi-Hamiltonian groups given a structured presentation. The second result is the first to use the modular structure of the subgroup lattice in a quantum hidden subgroup algorithm.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem C's structured-input assumption is the main load-bearing limitation; without a constructive-recognition algorithm, it does not apply to black-box quasi-Hamiltonian groups.","rationale":"I focused on what would have to be true for the central claim to hold. For Theorem C, the algorithm is only guaranteed to run when the structured quasi-Hamiltonian presentation is supplied. The paper is transparent about this, but the assumption is load-bearing because the standard HSP formulation is the black-box model, and the paper gives no method or reference for obtaining the presentation from a black-box generating set. I re-examined the internal mathematics: the scalar-action reduction in Theorem B is sound, including the use of the Hidden Multiple Shift problem and the linear-search verification; the crossed isomorphism in Theorem C satisfies the hypotheses of Baer's theorem, and the mapping lambda preserves cosets as required. Lemma 32, whose proof the authors disclose was AI-gap-fixed, appears correct after independent inspection: the digit-lifting argument and the congruence (34) check out, and the binomial coefficient bound is valid for the stated ranges of p and s. Thus the reader's weakest-assumption identification is correct: the structured input is the most load-bearing concern. The reader's CONDITIONAL verdict is appropriate, and my review does not change it.","tokens_in":32834,"tokens_out":37866,"duration_ms":309022,"concrete_test":"Test whether the structured presentation can be derived efficiently from an arbitrary black-box presentation of a small quasi-Hamiltonian group, e.g., Z_81 ⋊_4 Z_27 from Appendix A.3, by using known polynomial-time quantum algorithms for solvable groups (e.g., Watrous's algorithm [26]) to compute a polycyclic presentation and then extracting A, b, p, and s. If this succeeds in polynomial time, the input assumption is mild and Theorem C can be made unconditional for black-box inputs. If it fails or is shown to require solving the HSP itself, then Theorem C is conditional on a non-trivial recognition step that the paper leaves open.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem C is conditional on the input being a structured quasi-Hamiltonian presentation: the algorithm requires the Sylow decomposition and, for each non-Hamiltonian factor, the data P=A<b>, A normal in P, P/A cyclic, and parameters p and s for the action b^{-1}ab = a^{1+p^s}. The paper explicitly states (Section 1.1 and the input-model paragraph of Section 4) that it does not address constructive recognition of this presentation from an arbitrary generating set. Since the standard HSP input model is the black-box group model, this is a substantial restriction. If no efficient recognition algorithm exists, Theorem C does not solve the HSP for the class of quasi-Hamiltonian groups in the standard model; it only solves the problem when the group is handed to the algorithm in an already structured form. The abstract's phrase 'mild assumption' overstates how mild this is. The rest of the proof—the crossed isomorphism construction, the use of Baer's theorem, and the reduction to the Abelian HSP on B—appears internally consistent; this is not a correctness flaw in the algorithm as stated, but it is a load-bearing limitation of its scope. The reader's CONDITIONAL verdict is therefore appropriate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":33070,"tokens_out":43759,"duration_ms":373383,"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":[{"comment":"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.","section":"Section 4, Input model; Theorem C"},{"comment":"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.","section":"Sections 3.3.4 and 3.2.2"}],"minor_comments":[{"comment":"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":"Abstract and Section 1.1"},{"comment":"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}}.","section":"Section 4.2"},{"comment":"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":"Lemma 37"},{"comment":"There is a typo in the description of scalar multiplication: 'i.e., my the map' should be 'i.e., by the map'.","section":"Section 3.3.2"},{"comment":"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'.","section":"Appendix A.1"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the paper appears sound, and the reductions are carefully organized. The main issue is scope: Theorem C is not a solution to the HSP for quasi-Hamiltonian groups in the standard black-box model, and this should be made much more prominent. There is also a small but real gap in the unknown-t analysis for Theorem B concerning the quantitative success probability of the HMS subroutine. Both issues are fixable within the manuscript's scope, but they affect how the central claims should be stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a solid contribution to the non-Abelian HSP literature, and the quasi-Hamiltonian result is the real novelty. Theorem B cleanly extends prior semidirect-product algorithms to bounded-rank Abelian A with scalar action, and the reduction to Hidden Multiple Shift is well motivated. Theorem C, which transports the HSP on a non-Hamiltonian modular p-group to an Abelian p-group via a crossed isomorphism, is genuinely new and is the first HSP algorithm to exploit modularity of the subgroup lattice. The construction of B, the proof that the crossed isomorphism induces a projectivity, and the coset-preservation argument are careful and convincing.\n\nThe paper is also unusually honest. The input model for Theorem C is stated plainly: the algorithm needs a structured quasi-Hamiltonian presentation, including Sylow decomposition and explicit parameters for the action; constructive recognition is left open. The stress-test note is right that the abstract's 'mild assumption' overstates this. It is a real limitation, not a hidden flaw. Similarly, the proof of Lemma 32 includes the full digit-lifting argument, and the AI-usage statement says a gap in that lemma was found and fixed. The proof is explicit enough for a referee to check, but it is not machine-verified and the rest of the paper leans on several external theorems. None of these are disqualifying.\n\nThe soft spots are proportionate. The biggest is that Theorem C solves the HSP only for the structured-input class; if recognition is hard, the black-box quasi-Hamiltonian HSP remains open. The efficiency condition Exp(A)/p = polylog(|G|) in Theorem B is also restrictive, but it is in line with prior work. The PGM analysis in Appendix B is not the main route, but the HMS reduction depends on the algorithm of [16] which has its own parameters. I did not find a circular dependency or a fitted constant.\n\nWho is this for? Researchers in quantum algorithms for algebraic problems, especially those tracking which group families admit efficient HSP algorithms. The paper extends the map of solvable groups and gives a clean new technique. It deserves serious peer review. The referee should focus on Lemma 32, the crossed-isomorphism verification, and the precise framing of Theorem C's input model. I would engage with it.","headline":"Solid new HSP results for quasi-Hamiltonian groups, with the structured-input assumption as the key honest caveat.","tokens_in":33580,"tokens_out":2442,"would_cite":true,"duration_ms":22581,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","20D15","20D30"],"pacs":["03.67.Ac"],"model":"deepseek-v4-flash","headline":"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.","keywords":["hidden subgroup problem","quantum algorithms","semidirect products","scalar action","quasi-Hamiltonian groups","modular subgroup lattice","crossed isomorphism","hidden multiple shift"],"falsifier":"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 $\\varphi(N)p/N^2$.","tokens_in":32641,"feed_emoji":"⚛️","tokens_out":25473,"duration_ms":188598,"temperature":0.7,"pith_summary":"The paper claims polynomial-time quantum algorithms for the hidden subgroup problem (HSP) in two families that contain non-Abelian groups. The first (Theorems A and B) handles semidirect products $G = A\\rtimes \\mathbb{Z}_{p^k}$ in which the finite Abelian group $A$ is acted on by the scalar map $a\\mapsto \\mu^b a$, and is efficient when $A$ has bounded generator rank and $\\mathrm{Exp}(A)/p = \\mathrm{polylog}(|G|)$, subsuming the previously studied cyclic families $\\mathbb{Z}_N \\rtimes \\mathbb{Z}_p$ and $\\mathbb{Z}_{q^r}$. The second (Theorem C) handles finite quasi-Hamiltonian groups — nilpotent groups whose subgroup lattice is modular, equivalently groups in which every subgroup is permutable — in quantum polynomial time whenever a structured presentation is supplied, and the paper states that this is the first HSP algorithm to exploit modularity of the subgroup lattice. If correct, these results extend efficient quantum HSP algorithms to groups of unbounded nilpotency class that sit between the Abelian cases and the dihedral and symmetric cases of cryptographic and graph-isomorphism interest. The stated caveat is the input model: Theorem C requires the Sylow decomposition and the action parameters $p$ and $s$ as part of the input, and constructive recognition of this presentation from an arbitrary generating set is left open.","feed_headline":"Quantum algorithms crack two new non-Abelian group families","feed_subtitle":"They extend quantum algorithms beyond Abelian groups, toward the dihedral and symmetric cases tied to crypto and graph isomorphism.","key_machinery":"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 $\\varphi(N)p/N^2$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the pretty good measurement approach for $\\mathbb{Z}_N \\rtimes \\mathbb{Z}_p$ that Theorem A extends from a prime-order complement to $\\mathbb{Z}_{p^k}$, including the coset-state construction and block measurement adapted here.","marker":"[1]"},{"why":"Gives the Hidden Multiple Shift algorithm that Theorem B invokes to recover the cyclic parameter $d$ for the general Abelian normal factor.","marker":"[16]"},{"why":"Source of the modular $p$-group classification (Proposition 14), Corollary 30, the crossed-isomorphism material, and Baer's theorem as quoted; the structural backbone of Theorem C.","marker":"[17]"},{"why":"Provides the HSP algorithm for Dedekind groups, which handles the Hamiltonian factors $Q_8\\times E$ and serves as the baseline that Theorem C extends.","marker":"[3]"},{"why":"Supplies the finite-Abelian-group decomposition and Smith normal form routines used for the invariant-factor input, the quotient $Q$, and the QFT basis of the Abelian group $B$.","marker":"[6]"},{"why":"The original Ettinger–Høyer reduction that the paper generalizes to obtain the cyclic form $\\langle (d, p^t)\\rangle$ of the residual hidden subgroup.","marker":"[25]"},{"why":"Baer's theorem on crossed isomorphisms, used to prove that $\\sigma$ induces a projectivity preserving right cosets — the key step transporting the HSP to an Abelian group.","marker":"[21]"}],"fun_headline_variants":["Quantum algorithms crack HSP in two new non-Abelian group families","HSP solved for semidirect products and quasi-Hamiltonian groups","Polynomial-time quantum HSP for semidirect and quasi-Hamiltonian groups","First quantum HSP algorithm exploiting modular subgroup lattices","Quantum algorithms now handle semidirect products and quasi-Hamiltonian HSP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum algorithms crack HSP in two new non-Abelian group families","HSP solved for semidirect products and quasi-Hamiltonian groups","Polynomial-time quantum HSP for semidirect and quasi-Hamiltonian groups","First quantum HSP algorithm exploiting modular subgroup lattices","Quantum algorithms now handle semidirect products and quasi-Hamiltonian HSP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001611,"raw_usage":{"total_tokens":6597,"prompt_tokens":1311,"completion_tokens":5286,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":927,"completion_tokens_details":{"reasoning_tokens":5195}},"tokens_in":927,"tokens_out":5286,"duration_ms":36592,"temperature":1.0,"reasoning_tokens":5195,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T15:23:42.418208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\varphi(N)p/N^2$.","supporting_citations":[{"cited_title":"From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups","cited_arxiv_id":null,"evidence_quote":"Supplies the pretty good measurement approach for $\\mathbb{Z}_N \\rtimes \\mathbb{Z}_p$ that Theorem A extends from a prime-order complement to $\\mathbb{Z}_{p^k}$, including the coset-state construction and block measurement adapted here."},{"cited_title":"On learning linear functions from subset and its applications in quantum computing","cited_arxiv_id":"1806.09660","evidence_quote":"Gives the Hidden Multiple Shift algorithm that Theorem B invokes to recover the cyclic parameter $d$ for the general Abelian normal factor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the modular $p$-group classification (Proposition 14), Corollary 30, the crossed-isomorphism material, and Baer's theorem as quoted; the structural backbone of Theorem C."},{"cited_title":"The Hidden Subgroup Problem and Quantum Computation Using Group Representations","cited_arxiv_id":null,"evidence_quote":"Provides the HSP algorithm for Dedekind groups, which handles the Hamiltonian factors $Q_8\\times E$ and serves as the baseline that Theorem C extends."},{"cited_title":"Decomposing finite Abelian groups","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-Abelian-group decomposition and Smith normal form routines used for the invariant-factor input, the quotient $Q$, and the QFT basis of the Abelian group $B$."},{"cited_title":"On Quantum Algorithms for Noncommutative Hidden Subgroups","cited_arxiv_id":null,"evidence_quote":"The original Ettinger–Høyer reduction that the paper generalizes to obtain the cyclic form $\\langle (d, p^t)\\rangle$ of the residual hidden subgroup."},{"cited_title":"Crossed Isomorphisms","cited_arxiv_id":null,"evidence_quote":"Baer's theorem on crossed isomorphisms, used to prove that $\\sigma$ induces a projectivity preserving right cosets — the key step transporting the HSP to an Abelian group."}],"review_version":1}