HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation

Quantum Physics [Submitted on 13 Apr 2026] HHL with a Coherent Fourier Oracle: A Proof-of-Concept Quantum Architecture for Joint Melody-Harmony Generation Alexis Kirke 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 Submission history From: Alexis Kirke [view email] [v1] Mon, 13 Apr 2026 15:27:48 UTC (559 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.AI cs.SD References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)