Coutinho, Guo and Schmeits (arXiv:2505.11986v4) introduced peak state transfer for continuous quantum walks and remarked in Section 7 that, for adjacency-based dynamics, their computations suggest that there may be infinitely many trees that support peak state transfer. This note gives an elementary positive construction for that remark within unweighted trees; the source itself already gives an infinite family of edge-weighted trees (the weighted paths X_n of its Theorem 6.2) with adjacency peak state transfer, so the novelty claimed here, if any, is restricted to unweighted trees. Every star has adjacency peak state transfer between its centre and a leaf. The trees T(k,m), obtained by attaching k >= 2 copies of the star K_{1,m} to a common root through their centres, form an infinite family of pairwise non-isomorphic non-star trees with adjacency peak state transfer from the root. For vertices in different branches of T(k,m), peak state transfer occurs exactly when sqrt((k+m)/m) = p/q is rational in lowest terms and p, q satisfy an explicit parity or congruence condition; this yields infinitely many trees with adjacency peak state transfer at distance 3 and at distance 4. All peak times are determined. Finite numerical observations (not used in the proofs) and verification code are included. The remark in the source is an observation rather than a formal open problem, and this note makes no claim about stronger questions the source authors may have intended. Trees of the same root-hub-leaf shape were used before for the Laplacian analogue (Open Problem 7.1 of the source): in the-omega-institute/trureturing (issue #14801, pull request #14855, merged on 10 October 2026, Lean 4 formalisation; commit 71980b170a), in the author's own earlier record Zenodo 23216901 (DOI 10.5281/zenodo.23216901, 10 October 2026), and, for the ten-vertex member T(3,2), in Figure 10 of the source. Those results concern the Laplacian; the present note concerns the adjacency matrix and does not use them. The recent arXiv:2610.20855 (Awuah and Guo) reports experiments on perfect and peak state transfer on IBM quantum hardware; its full text contains no infinite family of trees, and its family X_n is the weighted path family of Section 6.3 of the source. Versions 0.1.4 and 0.1.5 change attribution and notation in the prior-work section only. Disclosure: This is a record of personal, independent (hobby) research in which AI was used for exploration, drafting of proof candidates, verification computations and writing. The claims were checked only by an internal AI reviewer; the work has not been reviewed by human experts. Files: main.pdf, main.tex, code.zip (folder code/: Python verification scripts and outputs; unchanged since v0.1.2), README.md, CHANGELOG.md.
Coutinho, Guo and Schmeits (arXiv:2505.11986v4) introduced peak state transfer for continuous quantum walks and remarked in Section 7 that, for adjacency-based dynamics, their computations suggest that there may be infinitely many trees that support peak state transfer. This note gives an elementary positive construction for that remark within unweighted trees; the source itself already gives an infinite family of edge-weighted trees (the weighted paths X_n of its Theorem 6.2) with adjacency peak state transfer, so the novelty claimed here, if any, is restricted to unweighted trees. Every star has adjacency peak state transfer between its centre and a leaf. The trees T(k,m), obtained by attaching k >= 2 copies of the star K_{1,m} to a common root through their centres, form an infinite family of pairwise non-isomorphic non-star trees with adjacency peak state transfer from the root. For vertices in different branches of T(k,m), peak state transfer occurs exactly when sqrt((k+m)/m) = p/q is rational in lowest terms and p, q satisfy an explicit parity or congruence condition; this yields infinitely many trees with adjacency peak state transfer at distance 3 and at distance 4. All peak times are determined. Finite numerical observations (not used in the proofs) and verification code are included. The remark in the source is an observation rather than a formal open problem, and this note makes no claim about stronger questions the source authors may have intended. Trees of the same root-hub-leaf shape were used before for the Laplacian analogue (Open Problem 7.1 of the source): in the-omega-institute/trureturing (issue #14801, pull request #14855, merged on 10 October 2026, Lean 4 formalisation; commit 71980b170a), in the author's own earlier record Zenodo 23216901 (DOI 10.5281/zenodo.23216901, 10 October 2026), and, for the ten-vertex member T(3,2), in Figure 10 of the source. Those results concern the Laplacian; the present note concerns the adjacency matrix and does not use them. The recent arXiv:2610.20855 (Awuah and Guo) reports experiments on perfect and peak state transfer on IBM quantum hardware; its full text contains no infinite family of trees, and its family X_n is the weighted path family of Section 6.3 of the source. Versions 0.1.4 and 0.1.5 change attribution and notation in the prior-work section only. Disclosure: This is a record of personal, independent (hobby) research in which AI was used for exploration, drafting of proof candidates, verification computations and writing. The claims were checked only by an internal AI reviewer; the work has not been reviewed by human experts. Files: main.pdf, main.tex, code.zip (folder code/: Python verification scripts and outputs; unchanged since v0.1.2), README.md, CHANGELOG.md.
Unrefereed AI-assisted preprint, version 0.2.2, 9 October 2026. No author affiliation is given. For each m >= 1, set q = m^2 + m + 1. Attach q intermediate vertices to a root and q - 1 leaves to each intermediate vertex. This produces infinitely many pairwise nonisomorphic non-star trees of order q^2 + 1, with root Laplacian spectral support {0, m^2 + 1, (m + 1)^2 + 1}. The symbolic argument establishes Laplacian peak state transfer at time pi from the root to an intermediate vertex for odd m and to a leaf for even m, addressing Open Problem 7.1 of Coutinho, Guo and Schmeits, arXiv:2505.11986v4. The ten-vertex m = 1 member is their known Figure 10 example. Phase alignment is an application of their Lemma 5.2 and Corollary 5.3; the general peak-time classification follows their Theorem 5.4 and Corollary 5.5. The contribution presented here is the explicit infinite family and its spectral coefficients, targets and gcd calculation. For the specified pairs, all positive peak times are odd multiples of pi/g, where g = 5 when m = 2 mod 5 and otherwise g = 1. Novelty and priority are not certified. Peak transfer means attaining the mutual-projector absolute-sum bound, not unit amplitude. The probabilities tend to zero along both parity subsequences. This does not establish perfect transfer, efficient long-distance quantum communication or a classification of all trees. Exact standard-library reruns check actual full-tree Laplacians for m = 1..5, quotients for m = 1..1000, and gcd/phase regressions for m = 1..10000. Those finite checks support, not replace, the symbolic all-parameter proof and the analytic all-real-time Bezout argument. No complete Lean or other proof-assistant verification is supplied. 本研究は、清水竣平が個人の自主研究として企画・進行した、AIを活用した数学探索です。清水は探索方針の指示、検証の手配、成果の取りまとめを担当しました。証明の構成、計算による検証、原稿作成の大部分にはAIを使用しています。検証方法とその範囲は本文に記載しており、人間の専門家による査読は未実施です。 This work is an AI-assisted mathematical exploration initiated and coordinated by Shunpei Shimizu as a personal independent research project. Shimizu directed the exploration, arranged verification, and compiled the results. AI systems performed most proof construction, computational checking, and manuscript preparation. The verification methods and their scope are documented in the manuscript. The work has not received specialist human peer review. The submitter has not independently certified every mathematical step. Separate AI analyses and implementations are not human refereeing. No cited-author or institutional endorsement is implied. Read peak_trees_en_v0.2.2.tex first. The current package is Quantum_peak_trees_publication_v0.2.2_20261009.zip; QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md explains its contents. Version 0.2.2 removes author affiliation; mathematical text, equations, statements, proofs and verification programs are unchanged from version 0.2.1. The current package omits old manuscript sources and old full ZIPs. Earlier verification records retain their original dates; they are not human peer review. The built-in LaTeX compiler failed at Windows environment setup before TeX diagnostics. No PDF has been generated or visually verified; editable TeX and exact-check code are supplied. Documentation clarification, 10 October 2026. The retained README_EN.txt and README_JA.md are historical documentation; the corresponding preserved v0.1.0 copies describe the 8 October 2026 working draft. QUANTUM_TREES_README_PUBLICATION_v0.2.1_EN.md describes the superseded v0.2.1 package. Their source names, payload inventories, draft-status statements and no-Publish instructions are dated history, not the current record state or operative instructions. Independent checks in those documents mean separate AI analyses or implementations, not human refereeing. The current entrypoints are peak_trees_en_v0.2.2.tex, Quantum_peak_trees_publication_v0.2.2_20261009.zip and QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md. Refer to the current TeX for the argument where historical checker comments mention an absent review.md. This notice makes no change to the mathematical manuscript or verification programs. Publication packaging clarification, 10 October 2026. In addition to the three historical READMEs identified above, the two QUANTUM_TREES_ATTRIBUTION_AI_ROLE_ADDENDUM files and novelty_attribution_math_monitor_review_20261009.zip are dated supplementary review records. The r1 addendum and the current v0.2.2 manuscript give the operative author and AI contribution disclosure. The novelty archive also compares other manuscripts; those comparisons are context, not additional results of this Quantum record. Earlier private-save and no-publication instructions describe past workflow only. Current scientific claims are the explicitly stated tree-family construction and specified-pair peak times. No mathematical content, verification program or frozen historical attachment is changed by this clarification.
This record contains a short preliminary note (TeX source and PDF), Python code and its output, on the question posed in Section 11 of H. Monterde, H. Pal and S. Kirkland, "Laplacian quantum walks on blow-up graphs" (Linear Algebra Appl., 2025, doi:10.1016/j.laa.2025.10.001; numbering of arXiv:2504.11585v1): whether their Theorem 10 (Laplacian perfect state transfer after adding a matching inside the copies of a vertex in the blow-up of n copies of a graph, n divisible by 4) holds for the adjacency matrix. The same question is Problem 10 in H. Monterde's PhD thesis "Quantum pure state transfer" (University of Manitoba, 2025). For matchings whose edges join twin vertices of the blow-up and even n, the note proves that adjacency perfect state transfer between the ends a, b of an added edge forces the matching to cover every neighbour of a, and gives parity conditions on the uncovered vertices. Consequently a nonempty matching inside the copies of a single non-isolated vertex never gives adjacency perfect state transfer, for every graph and every even n. These results, and exact computations on small graphs supporting a conjecture that all sets of copies must be perfectly matched (finite computations only), are the parts claimed as new; a literature search in October 2026 found no earlier treatment of them, which does not establish that none exists. When every set of copies carries a perfect matching (a lexicographic product G[(n/2)K_2]), the note shows that for n divisible by 4 adjacency perfect state transfer between the ends of an added edge occurs if and only if the adjacency eigenvalue support of the vertex is integral, at time pi/2. This is not claimed as new: since G[(n/2)K_2] is isomorphic to Z[K_2], where Z is the blow-up of n/2 copies of G, it follows in a few lines from Proposition 6.3 of A. S. Árnadóttir and K. Guo (arXiv:2608.20476), and its sufficiency for integral graphs is a special case of a lemma of Ge, Greenberg, Perez and Tamon (arXiv:1009.1340). The criterion used for perfect state transfer between adjacent twins is a special case of a theorem of Kirkland, Monterde and Plosker (arXiv:2201.02720). When the matching lies inside the sets of copies, the graphs are edge-perturbed blow-ups in the sense of Monterde and Pal (arXiv:2505.07982v5, Section 9); their Theorem 20 concerns states orthogonal to the all-ones vector of a set of copies and does not cover vertex states. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.
Draft note (v0.1.3). We give a connected simple (unweighted) graph on 12 vertices in which the continuous-time quantum walk exp(itA) has perfect state transfer from the pair state e_1 - e_11 to the plus state e_0 + e_2 at time pi/sqrt(2), with a proof that can be checked by hand, and an infinite family obtained by Cartesian products with powers of the path P3. This concerns the question of Monterde and Pal (arXiv:2505.07982v5, Section 10). The product step is a known construction (Godsil, Kirkland and Monterde, SIAM J. Matrix Anal. Appl. 46 (2025) 2093-2115, Corollary 48 and Example 50 in arXiv:2502.08103v4; see also Monterde and Pal, Section 9, and Kirkland, Monterde and Plosker, arXiv:2609.10713, equation (4.1)); we did not find an example of perfect state transfer from a pair state to a plus state in these papers, and no claim of priority is made. Exhaustive computer searches, with numerical tests at stated tolerances, found no example on at most 10 vertices (adjacency and signless Laplacian matrices); these are finite computations, not proofs. Code and outputs are included. Versions 0.1.2 and 0.1.3 correct attribution and citations only. This is a personal hobby/independent research record in which AI was used for exploration, drafting proof candidates, checking and writing; it has not been peer-reviewed by human experts.
A weighted graph G with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in G. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each k≥3, (i) there are infinitely many connected unweighted graphs with maximum degree k admitting pair state transfer at τ∈{2π,2π}, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree k admitting plus state transfer at τ∈{2π,2π}. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at τ∈{2π,2π}, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at τ∈{2π,2π}. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.
Given a graph with Hermitian adjacency matrix H, perfect state transfer occurs from vertex a to vertex b if the (b,a)-entry of the unitary matrix exp(−iHt) has unit magnitude for some time t. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i), the oriented 3-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
This paper is a sequel to the work of Bhattacharjya et al. (2024) [2] on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacian matrix as the time-independent Hamiltonian associated with a blow-up graph. We characterize Laplacian strong cospectrality, periodicity, perfect state transfer (LPST) and pretty good state transfer (LPGST) on blow-up graphs. We present several constructions of blow-up graphs with LPST and produce new infinite families of regular graphs where each vertex is involved in LPST. We also determine LPST and LPGST in blow-ups of classes of trees. Finally, if n ≡ 0 (mod 4), then the blow-up of n copies of a graph has no LPST, but we show that under certain conditions, the addition of an appropriate matching to this blow-up graph results in LPST.