arxiv: 2605.04343 · v1 · submitted 2026-05-05 · 🪐 quant-ph · math.GR

Recognition: unknown

Hidden Prime-Factor Subgroups in Molecular and Condensed-Phase Systems

Srinivasan S. Iyengar , Amr Sabry

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:47 UTC · model grok-4.3

classification 🪐 quant-ph math.GR

keywords hidden subgroup problemShor's algorithmmolecular orbitalsprime factoringsymmetry groupsquantum computingcondensed phase systemscryptography

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The pith

Molecular orbital symmetries encode information about the prime factors of integers.

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

The paper connects the group-theoretic structure of Shor's algorithm and hidden subgroup problems to the symmetries present in molecular and condensed-phase systems. It shows that real molecules, with orbitals formed from symmetry-adapted linear combinations of atomic orbitals, embed information about the prime factors of corresponding integers. A sympathetic reader cares because integer factoring secures much of modern cryptography, so a direct physical embodiment of this hard problem in everyday chemical systems would link chemistry, computation, and security. The work recasts the quantum algorithm through group theory to expose this possibility in physical assemblies and optical beams.

Core claim

By recasting Shor's algorithm through the lens of group theory, the authors establish that the symmetries of molecular orbitals and condensed-phase assemblies correspond to hidden subgroups whose identification yields the prime factors of integers. In real molecular systems constructed via symmetry-adapted linear combinations of atomic orbitals, these subgroups contain the factoring information, suggesting that physical systems may be designed or observed to solve such mathematical problems.

What carries the argument

The hidden subgroup problem realized through symmetry groups of molecular orbitals, where symmetry-adapted linear combinations of atomic orbitals serve as the physical structure that encodes prime-factor information.

If this is right

Physical molecular systems could be engineered to contain solutions to integer factoring.
Condensed-phase assemblies and optical beams might be designed to embed information about prime factors.
The broad role of factoring in encryption implies that this physical encoding could affect cyber-security approaches.
Real molecular orbitals already demonstrate the presence of prime-factor information through their symmetry structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the mapping holds, quantum computers may not be uniquely required to exploit hidden subgroups; ordinary chemical systems could provide analogous capabilities in limited cases.
This connection suggests exploring whether measurable properties like spectra in specific molecules directly output factors of chosen integers.
Neighbouring problems in quantum information and molecular structure could be tested by checking if other symmetry-based physical systems encode similar hard computational results.

Load-bearing premise

That the symmetry groups arising from molecular orbital construction are directly equivalent to the hidden subgroups that solve integer factoring, with no extra mapping or computation required.

What would settle it

Select a concrete molecule such as benzene, compute its symmetry group from the atomic orbitals, construct the associated integer from the number of atoms or electrons, and check whether the prime factors of that integer can be read off directly from the symmetry operations without additional processing.

Figures

Figures reproduced from arXiv: 2605.04343 by Amr Sabry, Srinivasan S. Iyengar.

**Figure 1.** Figure 1: FIG. 1: A common template quantum circuit for all view at source ↗

**Figure 2.** Figure 2: FIG. 2: Step-by-step correspondence between Shor’s algorithm (left), the construction of symmetry-adapted molec view at source ↗

**Figure 3.** Figure 3: FIG. 3: Illustration of (a) view at source ↗

**Figure 4.** Figure 4: FIG. 4: Illustration of (a) view at source ↗

**Figure 5.** Figure 5: FIG. 5: Similar to Figure view at source ↗

**Figure 6.** Figure 6: FIG. 6: Illustration of (a): Eq. ( view at source ↗

**Figure 7.** Figure 7: FIG. 7: Quantum circuit for Shor’s algorithm. view at source ↗

**Figure 8.** Figure 8: FIG. 8: Discrete and continuous representations for Eq. ( view at source ↗

**Figure 9.** Figure 9: FIG. 9: A polyene with view at source ↗

read the original abstract

We describe a group theoretic analysis of Shor's algorithm and other related hidden subgroup problems in mathematics and relate these to symmetries of molecular and condensed phase assemblies. By recasting Shor's algorithm through the lens of group theory, we expose the possibility that physical systems such as molecular orbitals, condensed phase assemblies and optical beams may be designed such that these contain information pertaining to the solution to hard mathematical problems such as prime-factoring. We discuss real molecular systems, whose orbitals are constructed from symmetry-adapted linear combinations of atomic orbitals, and show that these contain information pertaining to the prime-factors of corresponding integers. Due to the broad significance of prime-factoring towards a variety of encryption problems in cyber-security, we believe this novel and fundamental approach may have broad impact.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper floats a high-level analogy between molecular symmetries and Shor's hidden subgroups but supplies no mapping, example, or extraction method.

read the letter

The main point to know is that this paper suggests symmetries in real molecules and condensed-phase systems could carry information about prime factors through the lens of hidden subgroup problems, yet it never shows how that would actually work in practice. The abstract and stress-test note both make clear that the claim rests on an unproven correspondence rather than a demonstrated reduction. On the positive side, the authors accurately recall the group-theoretic structure of Shor's algorithm and the standard use of symmetry-adapted linear combinations in molecular orbital theory. That background is correct and connects two fields at a conceptual level, which is the only real contribution here. The central weakness is the lack of any concrete construction. There is no embedding of the multiplicative group modulo N into a molecular point group such as C2v or D3h, no specific integer paired with a molecule whose irreps would reveal its factors, and no measurement protocol that extracts the factors without already knowing them classically. The stress-test concern holds up: the asserted equivalence stays loose and unverified. This leaves the work as a speculative proposal rather than a new result or technique. It might interest people who enjoy cross-disciplinary ideas at the quantum-chemistry boundary, but a reader looking for something to build on or test will find little substance. I would not bring it to a reading group, would not cite it, and do not think it merits peer review in its current form because the evidence for the key claim is missing.

Referee Report

3 major / 0 minor

Summary. The manuscript recasts Shor's algorithm and related hidden-subgroup problems in group-theoretic language and proposes that symmetries of molecular orbitals (constructed via symmetry-adapted linear combinations of atomic orbitals) and condensed-phase assemblies encode hidden subgroups whose identification would yield the prime factors of integers. It asserts that real molecular systems can be designed or analyzed to contain such information, with potential implications for cryptography.

Significance. If a rigorous, explicit correspondence were demonstrated between molecular point-group representations and the hidden subgroups of the abelian hidden-subgroup problem (with concrete embeddings, examples, and extraction protocols), the result would constitute a notable interdisciplinary link between quantum algorithms and chemical physics, potentially enabling physical realizations of factoring. The manuscript's current form, however, supplies only an analogy without supporting derivations or examples, so the claimed significance is not realized.

major comments (3)

[Abstract] Abstract: the statement that 'real molecular systems, whose orbitals are constructed from symmetry-adapted linear combinations of atomic orbitals, ... contain information pertaining to the prime-factors of corresponding integers' is unsupported; no concrete molecule, point group (e.g., C_{2v}), integer N, or explicit mapping from orbital irreps to the hidden subgroup of Z_N^* is provided.
[Main text] Main text (group-theoretic analysis of Shor's algorithm): no equations or derivations establish an isomorphism or reduction from a molecular symmetry group to the hidden subgroup whose solution solves integer factoring; the claimed equivalence therefore remains an unverified identification rather than a demonstrated correspondence.
[Discussion of physical systems] Discussion of physical systems: no measurement protocol is given that would extract prime factors from molecular orbital symmetries without already knowing or classically computing the factors, undermining the assertion that such systems 'may be designed such that these contain information pertaining to the solution to hard mathematical problems'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and thoughtful review of our manuscript. We address each major comment below, clarifying the scope of our conceptual framework while indicating revisions that will strengthen the presentation of the proposed links between hidden-subgroup problems and molecular symmetries.

read point-by-point responses

Referee: [Abstract] The statement that 'real molecular systems, whose orbitals are constructed from symmetry-adapted linear combinations of atomic orbitals, ... contain information pertaining to the prime-factors of corresponding integers' is unsupported; no concrete molecule, point group (e.g., C_{2v}), integer N, or explicit mapping from orbital irreps to the hidden subgroup of Z_N^* is provided.

Authors: We agree that the abstract phrasing is assertive and that the current manuscript does not supply a fully explicit numerical example with a specific molecule, point group, and mapping. The work is intended as a conceptual recasting that identifies structural parallels between symmetry-adapted linear combinations and the group-theoretic setting of the abelian hidden-subgroup problem. In revision we will soften the abstract to describe a proposed correspondence rather than an established containment, and we will insert a short illustrative example (e.g., a C_{2v} molecule and a small composite integer) that sketches the irrep-to-subgroup relation. revision: yes
Referee: [Main text] Main text (group-theoretic analysis of Shor's algorithm): no equations or derivations establish an isomorphism or reduction from a molecular symmetry group to the hidden subgroup whose solution solves integer factoring; the claimed equivalence therefore remains an unverified identification rather than a demonstrated correspondence.

Authors: The main text recasts Shor's algorithm in group-theoretic language and notes that the decomposition of molecular orbitals into irreducible representations of the point group shares formal features with the hidden-subgroup structure. We do not assert a proven isomorphism or reduction in the present draft; the text presents an analogy that motivates further investigation. To meet the referee's concern we will add a dedicated subsection containing explicit group-homomorphism diagrams and a minimal derivation showing how the character table of a small point group can be aligned with the representation theory underlying the hidden-subgroup problem for a chosen N. revision: yes
Referee: [Discussion of physical systems] Discussion of physical systems: no measurement protocol is given that would extract prime factors from molecular orbital symmetries without already knowing or classically computing the factors, undermining the assertion that such systems 'may be designed such that these contain information pertaining to the solution to hard mathematical problems'.

Authors: The discussion section is deliberately forward-looking and does not claim a ready-to-implement extraction protocol. It argues that the symmetry information is in principle encoded in the same way quantum states encode period information in Shor's algorithm; realizing a concrete readout would require additional spectroscopic or scattering protocols that lie outside the scope of the present theoretical note. In revision we will explicitly label this as an open direction for future work rather than an accomplished result, thereby aligning the text with the referee's observation. revision: partial

Circularity Check

0 steps flagged

No circularity: claims asserted via analogy without any derivation chain or equations to inspect

full rationale

The paper's abstract and described content assert that molecular orbital symmetries contain prime-factor information by relating group theory of Shor's algorithm to molecular point groups, but no explicit equations, derivations, or reductions are provided in the available text. Without a load-bearing step that can be quoted and shown to equal its inputs by construction (self-definition, fitted prediction, or self-citation chain), the analysis finds no circularity. The central claim may be unsubstantiated or require external verification, but it does not reduce tautologically to its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on an unproven equivalence between molecular symmetry groups and the hidden subgroups of Shor's algorithm; no independent evidence or mapping is supplied.

axioms (1)

ad hoc to paper Symmetry-adapted linear combinations of atomic orbitals in real molecules encode or contain the hidden subgroups whose solution yields prime factors of integers.
This identification is invoked to connect physical chemistry to the factoring problem but is not derived or evidenced in the abstract.

pith-pipeline@v0.9.0 · 5423 in / 1253 out tokens · 57577 ms · 2026-05-08T16:47:32.441987+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Hidden Prime-Factor Subgroups in Molecular and Condensed-Phase Systems

Here we use symmetry to probe connections between arithmetic problems and physical systems and ask if we may be able to design molecular or condensed phase sys- tems that have properties related to solutions to hidden subgroup problems. In Figure 2 we summarize the step- by-step correspondence established in this paper. Specif- ically, at each stage of Sh...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

That is, there are specific irreducible representations within the polyene that will have similar symmetry properties as the Eqs

The requirement that all rotations inside a coset should have the same function value presents additional symmetry constraints on top of the Bloch function requirement above. That is, there are specific irreducible representations within the polyene that will have similar symmetry properties as the Eqs. (48)-(51). To be precise, from some x1 →(i 1, j1)≡C ...

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