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

Alexis Kirke

quant-phApr 13, 2026

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

Authors: Alexis Kirke

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.

Figures & tables

Figure 1: Coherent HHL $\to$ Fourier harmonic oracle pipeline: no intermediate measurement. A single joint collapse selects melody notes and two-chord progression simultaneously.

Figure 2: Fourier oracle affinity for C major across all 12 pitch classes. Left : binary hard-gate (red/blue bars) versus $K=4$ smooth weighting (orange/green bars); chord tones C, E, G are labelled. Right : effect of truncation order $K$ – lower $K$ broadens the affinity envelope, higher $K$ recovers the binary boundary.

Panel of Figure 3 — Figure 3: Four sampled outputs from the Fourier oracle ( $K=10$ ), 2-chord block.

Figure 4: Four-block 2/2 chain, $K=4$ (representative trial, 4/8 chord-tone compliance = 50%). With the broadest spectral smoothing, several melody notes fall outside the strict chord-tone set; all grammar junctions remain valid. Mean over 10 unseeded trials: 46.3% $\pm$ 16.7%.

Figure 5: Four-block 2/2 chain, $K=6$ (representative trial, 5/8 chord-tone compliance = 62.5%). Intermediate spectral precision produces a mix of chord-tone and passing notes. Mean over 10 unseeded trials: 56.3% $\pm$ 12.1%. All grammar junctions valid.

Figure 6: Four-block 2/2 chain, $K=8$ (representative trial, 5/8 chord-tone compliance = 62.5%). $K=8$ achieves the highest mean compliance across the sweep and lower variance than $K=10$ , making it the best mean-variance trade-off. Mean over 10 unseeded trials: 67.5% $\pm$ 14.7%. All grammar junctions valid.

Figure 7: Four-block 2/2 chain, $K=10$ (representative trial, 5/8 chord-tone compliance = 62.5%). Sharpest spectral truncation yields a mean comparable to $K=8$ but with slightly higher trial-to-trial spread (std $\pm$ 15.6%; range 37.5-87.5%). Mean over 10 unseeded trials: 66.3% $\pm$ 15.6%. All grammar junctions valid.

Figure 8: Four-block 2/2 chain (Method A, $K=10$ ): A4/iii, A4/vi, A4/IV, G4/I, G4/V, C4/I, C4/IV, G4/V. All three block junctions produce grammatically valid continuations: vi $\to$ IV, I $\to$ V, I $\to$ IV. Chord-tone compliance: 7/8 (87.5%).

Figure 9: HHL note pair probabilities (blue) vs. uniform (grey), top 10 pairs. HHL strongly favours stepwise motion to tonally stable notes (Narmour+KK encoding).

Interval category	Pairs	HHL %	Uniform %	Ratio	Bach sop.%
Unison (0 st)	7	10.2	14.3	$0.71\times$	15.0
Step (1-2 st)	12	39.0	24.5	$1.59\times$	67.9
Skip (3-5 st)	18	24.0	36.7	$0.65\times$	13.2
Leap (6+ st)	12	26.8	24.5	$1.09\times$	4.0

Table 1: Melodic interval distribution: HHL vs. uniform baseline vs. Bach soprano (433 chorales, 22,428 pairs, computed via music21). HHL enriches steps 1.6

\times

over uniform, consistent with Narmour Proximity, and moves in the direction of Bach’s 67.9% – a statistical tendency, not a claim of comparable musical quality. The residual gap is a range artefact: the 7-note set (B3-A4) has 24.5% leap pairs at uniform; unrestricted soprano writing has

\approx

4%, so absolute rates are not commensurable. Within-pair intervals; cross-pair stepwise rate 49.8% (Table 2 ).

Metric	Fourier ( $K=4$ )	Classical MC a
$\text{KL}(\text{pair}\\|\text{uniform})$	0.166	0.155
$\chi^{2}$ (500k samples)	189,363	170,319
V $\to$ I resolutions	8.0%	8.0%
Tonic (I) endings	28.1%	27.9%
Stepwise melody	49.8%	49.4%
Nonzero joint states	784	784

Table 2: Pipeline verification : both columns encode the same

T

, so agreement confirms implementation correctness. 500,000 samples, 2-chord blocks, 19 qubits.

Variant	Conditioning	Top-1 prob.	Conc. factor
A. Full Method A	Melody bias + hard-T	3.04%	$8.3\times$
B. Melody only	Melody bias, no hard-T	1.19%	$3.3\times$
C. Harmony only	Uniform b, hard-T	0.77%	$2.1\times$
D. Unconditioned	Uniform b, no hard-T	0.37%	$1.0\times$

Table 3: Method A ablation (100,000 samples, context C4/V): concentration factors relative to unconditioned baseline. Top-1 prob. = modal (note-pair, chord-sequence) frequency.

$c_{2}$ (cond.)	Valid $c_{1}$ successors	Top $c_{1}$	Valid?	% grammatically valid
I	IV, V, vi	vi	✓	100.0
ii	I, IV, V	I	✓	100.0
iii	IV, vi	vi	✓	100.0
IV	I, V	I	✓	100.0
V	I, vi	I	✓	100.0
vi	ii, IV, V	ii	✓	100.0

Table 4: Grammar coverage (Fourier oracle, hard-T,

n=50{,}000

per context): all 7

c_{2}

conditioning contexts produce 100% grammatically valid next-block openings.

$K$	Mean	Std	Range	Notes
4	46.3%	$\pm$ 16.7%	25-75%	Lowest mean, high variance
6	56.3%	$\pm$ 12.1%	37.5-75%	Intermediate mean, lowest variance
8	67.5%	$\pm$ 14.7%	50-100%	Highest mean, moderate variance
10	66.3%	$\pm$ 15.6%	37.5-87.5%	Similar to $K=8$ , slightly higher variance

Table 5: Melody note-chord-tone compliance across

K

values in the 4-block 2/2 chain,

n=10

independent unseeded trials per

K

. Unseeded runs simulate real quantum hardware, where each measurement is genuinely random. Grammar junctions are valid at 100% across all

K

values and all 40 trials. With

n=10

the

K=8

and

K=10

means (67.5% vs 66.3%) are not statistically distinguishable at

n=10

;

K=8

offers the better mean-variance trade-off.

Figure 10: Fourier oracle gate count (Qiskit-verified): 375 gates vs. $\approx$ 12,742 lookup-based ( $34\times$ fewer). Crossover where lookup becomes cheaper occurs at $C_{\text{vocab}}\approx 192$ .

Level	$F$	$\alpha$	PS	$\mathrm{KL}_{\mathrm{pair}}$	$\chi^{2}(48)$ / $p$ vs $p_{\mathrm{HHL}}$	$\chi^{2}(48)$ / $p$ vs unif.	$\mathrm{KL}_{\mathrm{chord}}$
Noiseless	1.00	0.00	0.022 ‡	0.049 ‡	0.03 / 1.00	51.8 / $<$ .001	0.0000
$F=80\%$	0.80	0.20	0.149	0.041	34.0 / 0.94	768 / $<$ .001	0.0000
$F=50\%$	0.50	0.50	0.281	0.017	211 / $<$ .001	301 / $<$ .001	0.0000
$F=10\%$	0.10	0.90	0.456	0.001	686 / $<$ .001	12 / 1.00	0.0001
Eff. depol. †	$\approx$ 0	$\approx$ 1	0.500	0.000	842 / $<$ .001	0.01 / 1.00	0.0002

Table 6: HHL-stage depolarising noise (oracles ideal). PS = post-selection rate;

\mathrm{KL}_{\mathrm{pair}}

= KL divergence of note-pair marginal from uniform.

$C_{\text{vocab}}$	$n_{\text{chord}}$	Fourier (unitary synth.)	Lookup
7 (current)	3	$375^{\dagger}$	12,742
64	6	8,064	101,936
256	8	129,024	407,744

Table 7: Gate counts per note pair vs. chord vocabulary size ( † exact Qiskit transpilation at

C_{\text{vocab}}=7

). Fourier advantage holds only at small vocabularies under default synthesis; crossover at

C_{\text{vocab}}\approx 192

Penalty scheme	$\kappa$	KL(pair $\\|$ unif.)	Top 5 pairs
Baseline (as reported)	11.23	0.107	B3,C4; C4,B3; E4,F4; F4,E4; A4,G4
Penalties $\times 0.5$	14.96	0.096	B3,C4; B3,A4; C4,B3; A4,B3; E4,F4
Unison+tritone only	11.85	0.096	B3,A4; A4,B3; B3,C4; C4,B3; C4,A4

Table 8: Sensitivity of HHL note-pair distribution to penalty scheme (theoretical KL from solved distribution). All schemes favour stepwise motion;

\kappa

remains bounded near 11-15.

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