Long Range Frequency Tuning for QML

Michael Poppel; Markus Baumann; Sebastian Wölckert; Claudia Linnhoff-Popien; Jonas Stein

cs.LGFeb 26, 2026

Long Range Frequency Tuning for QML

Authors: Michael Poppel, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Jonas Stein

Organizations: LMU Munich & Aqarios GmbH, Munich, Germany · LMU Munich, Munich, Germany

Abstract

Angle-encoded variational quantum circuits admit a truncated Fourier series representation of their output, but approximating functions with maximum frequency $ω_{\max}$ using fixed unary encoding requires $\mathcal{O}(ω_{\max})$ encoding gates. Trainable-frequency (TF) circuits promise a reduction by learning the data-encoding prefactors alongside the ansatz parameters, adapting the accessible frequency spectrum to the target during training. We identify a practical barrier that prevents this promise from being realized: the prefactor gradient is suppressed by the spectral gap between the circuit's accessible frequencies and the target spectrum, independently of the ansatz parameters, confining gradient-driven prefactor movement to a narrow neighborhood of initialization. We propose ternary grid initialization -- setting prefactors to $\{1, 3, 9, \ldots, 3^{k-1}\}$ -- which ensures every target frequency within $[-ω_{\max}, ω_{\max}]$ lies within $\tfrac{1}{2}$ unit of the accessible spectrum at initialization, so that the spectral-gap bound no longer constrains the target-driven gradient to be small. This is a necessary condition for reliable convergence, whose sufficiency we establish empirically. On a synthetic benchmark with target frequencies shifted well beyond the standard initialization range, ternary initialization achieves median $R^2 = 0.997$ versus $0.18$ for unary initialization, with $100\%$ of runs achieving $R^2 > 0.95$ against $0\%$ . CMA-ES with $20\times$ the evaluation budget reaches only $25\%$ success, confirming the limitation is a property of the optimization landscape rather than of gradient-based optimization specifically. Real-world validation on two benchmark datasets demonstrates consistent advantages over both fixed and trainable unary baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Trained predictions against target for the original Jaderberg et al. (2024) frequency spectrum with unary prefactors (a), the frequency-shifted variant with unary prefactors (b), and the same shifted spectrum with ternary grid initialization (c). Ternary initialization recovers accurate fits where unary fails entirely.

Panel of Figure 2 — Figure 2: Prefactor displacement and gradient analysis across 100 runs with $\alpha^{(0)}=\{1.01,1.02,1.03\}$ . Displacement remains far below what is required to reach $\Omega_{2}$ under all learning rates tested (a). Gradient magnitude is elevated near the target frequency range, consistent with the inverse spectral-gap scaling predicted by Proposition 1 (b).

Approach	Enc. gates	Enc. params	Ansatz params	2-qubit gates	Depth	Reach $\omega_{\max}$
Trainable Unary	3	3	63	38	110	3
Trainable Ternary (ours)	3	3	63	38	110	13

Table 1: Circuit resources for the synthetic benchmark (3-qubit, target

\Omega_{2}

). Encoding and ansatz parameters are listed separately; two-qubit gate count and depth are for the ansatz block transpiled to

\{R_{x},R_{y},R_{z},\mathrm{CNOT}\}

. All resource columns are identical across the two approaches; only the accessible reach

\omega_{\max}

obtained from the three encoding gates differs. The fixed-prefactor variants coincide with these except for carrying no trainable encoding parameters.

Figure 4: Ablation studies on

\Omega_{2}=\{11,11.2,13\}

, 10 randomly chosen target coefficients with 10 different weight initialization seeds each. (a)

R^{2}

distributions by encoding base, 3-qubit circuit with 1 feature map per qubit. Base-3 (ternary) and base-4 achieve

100\%

success; bases 1, 2, and 5 fail completely. (b)

R^{2}

distributions by circuit architecture, ternary initialization throughout. The 3-qubit (1 FM) and 2-qubit (2 FM) circuits achieve

100\%

success; the 4-qubit circuit is partially successful; the 2-qubit (1 FM) circuit fails. Dashed line marks

R^{2}=0.95

Figure 5:

R^{2}

distributions across 100 runs for each optimization strategy on target

\Omega_{2}

. (a) Overall distributions; percentages indicate success rates (

R^{2}\geq 0.95

). (b) Per-function breakdown for CMA-ES and ternary across all 10 target functions: CMA-ES performance varies widely and unpredictably; ternary achieves tight, consistent

R^{2}\approx 0.987

–

0.993

for every function and seed.

Approach	Flight Passengers	California Housing
Fixed Unary	$0.707$	$0.662$
Trainable Unary	$0.788$	$0.680$
Fixed Ternary	$0.906$	$0.730$
Trainable Ternary (ours)	$\mathbf{0.967}$	$\mathbf{0.744}$

Table 3: Median

R^{2}

across real-world benchmarks. Flight Passengers uses a 4-qubit parallel circuit (255 parameters); California Housing uses an 8-qubit serial circuit with 6 layers of nearest-neighbor

\mathrm{SU}(4)

gates and measurement on the central qubit (630 parameters).

Panel of Figure 6 — Figure 6: $R^{2}$ distributions across 10 initializations on the two real-world benchmarks. The same performance staircase (Fixed Unary $<$ Trainable Unary $<$ Fixed Ternary $<$ Trainable Ternary) appears on both datasets.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Prefactor evolution under identical initialization $\{1.0,1.0,1.0\}$ for three learning rates. At $\eta\leq 0.01$ prefactors evolve lock-step; at $\eta=0.1$ symmetry breaks but most prefactors still fail to reach $\Omega_{2}$ .

Panel of Figure 8 — Figure 8: $R^{2}$ distributions across 100 runs for target $\Omega_{2}=\{11,11.2,13\}$ , comparing identical and perturbed unary initializations against trainable ternary ( $\eta=0.001$ ). Both unary strategies produce similar failure distributions despite different prefactor evolution patterns.

Panel of Figure 9 — Figure 9: Prefactor evolution for unary initialization on $\Omega_{2}=\{11,11.2,13\}$ , showing the best-performing run at each learning rate. Only $\eta=0.1$ enables substantial movement, but convergence remains unreliable at that rate (Figure 8 ).

Panel of Figure 10 — Figure 10: 3-qubit circuit with parallel $R_{x}(\alpha_{i}x)$ encoding and SpecialUnitary ansatz blocks (63 trainable parameters each).

Figure 11: Held-out

R^{2}

on randomly drawn targets across the favorability grid, with full

\mathrm{SU}(2^{k})

ansatz blocks. Columns vary the frequency prior

\beta

(target frequencies scattered across the accessible range at

\beta=0

, concentrated toward the low end at

\beta=2

); rows vary the amplitude prior

\gamma

(uniform Fourier energy at

\gamma=0

, concentrated onto dominant components at

\gamma=2

). Within each panel the number of encoding gates

k

increases left to right; the top axis of the first row gives the corresponding ternary accessible reach

\omega_{\max}=(3^{k}-1)/2

. Boxes span the interquartile range, whiskers the minimum and maximum over the

10

runs, and the centre line the median. Both encodings use

k

trainable prefactors (matched cost) and share identical targets, splits and ansatz initializations at each seed; unary’s reach is

\omega_{\max}=k

against ternary’s

(3^{k}-1)/2

. The dashed line marks

R^{2}=0.95

and the grey line

R^{2}=0

, the value of a constant-mean predictor. Trainable ternary (blue) is flat across every prior and scale, with no run below

R^{2}=0.945

; trainable unary (red) tracks target favorability, rising from unusable in the scattered column to competitive only in the concentrated column, and reliably so only in the doubly-concentrated cell

(\beta=2,\gamma=2)

(outlined), where the two methods are statistically indistinguishable and ternary holds the higher worst-case run.

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