HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation
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Quantum Physics arXiv:2604.20882 (quant-ph) [Submitted on 13 Apr 2026] Title:HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation Authors:Alexis Kirke View a PDF of the paper titled HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation, by Alexis Kirke View PDF HTML (experimental) Abstract:Quantum algorithms with a proven theoretical speedup over classical computation are rare. Among the most prominent is the Harrow-Hassidim-Lloyd (HHL) algorithm for solving sparse linear systems. Here, HHL is applied to encode melodic preference: the system matrix encodes Narmour implication-realisation and Krumhansl-Kessler tonal stability, so its solution vector is a music-cognition-weighted note-pair distribution. The key constraint of HHL is that reading its output classically cancels the quantum speedup; the solution must be consumed coherently. This motivates a coherent Fourier harmonic oracle: a unitary that applies chord-transition weights directly to the HHL amplitude vector, so that a single measurement jointly selects both melody notes and a two-chord progression. A two-note/two-chord (2/2) block is used to contain the exponential growth of the joint state space that would otherwise make classical simulation of larger blocks infeasible. For demonstrations of longer passages, blocks are chained classically - each block's collapsed output conditions the next -- as a temporary workaround until fault-tolerant hardware permits larger monolithic circuits. A four-block chain produces 8 notes over 8 chords with grammatically valid transitions at every block boundary. Independent rule-based harmony validation confirms that 97% of generated chord progressions are rated strong or acceptable. The primary motivation is that HHL carries a proven exponential speedup over classical linear solvers; this work demonstrates that a coherent HHL+oracle pipeline - the prerequisite for that speedup to be realised in a musical setting - is mechanically achievable. Audio realisations of representative outputs are made available for listening online. Subjects: Quantum Physics (quant-ph); Artificial Intelligence (cs.AI); Sound (cs.SD) Cite as: arXiv:2604.20882 [quant-ph] (or arXiv:2604.20882v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.20882 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Alexis Kirke [view email] [v1] Mon, 13 Apr 2026 15:27:48 UTC (559 KB) Full-text links: Access Paper: View a PDF of the paper titled HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation, by Alexis KirkeView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 Change to browse by: cs cs.AI cs.SD References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)