Alexis’ new quantum computer music paper on the use of the HHL algorithm for quantum music is now available on arXiv: https://arxiv.org/abs/2604.20882
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.