Task-aligned spectral inductive biases in parameterized quantum circuits

Felix Frohnert; Adrián Pérez-Salinas; Sebastian Lange; Marcus Schwaebisch; Evert van Nieuwenburg; Vedran Dunjko

quant-phOct 16, 2026

Task-aligned spectral inductive biases in parameterized quantum circuits

Authors: Felix Frohnert, Adrián Pérez-Salinas, Sebastian Lange, Marcus Schwaebisch, Evert van Nieuwenburg, Vedran Dunjko

Organizations: ⟨aQaL⟩Applied Quantum Algorithms, Universiteit Leiden, Leiden, The Netherlands · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany · Institute for Theoretical Physics, ETH Zürich, Switzerland · Instituut-Lorentz, Universiteit Leiden, Leiden, The Netherlands ·

Abstract

Parameterized quantum circuits are widely investigated in quantum machine learning, yet how their architecture can provide task-aligned inductive biases remains poorly understood. We study data reuploading quantum circuits with two distinct inputs: a signal that varies between samples and a fixed, task-defined spectral query that specifies how the signal should be probed. We show that encoding the query can change the circuit's hypothesis class over the signal by modulating how its existing signal frequencies are combined, without introducing new sample-dependent information. This mechanism matches tasks built on nonuniform Fourier transforms, whose evaluation frequencies specify the query and whose sampling positions guide its encoding. We examine this connection in a classification task inspired by synthetic aperture radar, where acquisition geometry determines a query for classifying unfocused spectral patches before image formation. Across small-scale spectral and radar-like tasks, aligned queries improve learning from limited data over parameter-matched circuits with mismatched or absent queries. These results provide a route from known spectral task structure to the design and validation of inductive biases in parameterized quantum circuits.

Figures & tables

Figure 1: From query-controlled circuits to geometry-aligned classification. (II) In a DDR circuit, a fixed query controls how signal-dependent Fourier components are combined, modifying the represented function without providing sample-specific information (Sec. II ). (III) A spectral classification task supplies a fixed query $\bm{f}^{\star}$ and, through its sampling positions and score structure, a target spectrum $\Omega_{\star}^{(f)}$ that guides the query encoding (Sec. III ). (IV) In SAR, the acquisition geometry supplies this structure, enabling geometry-aligned classification directly from raw measurements without first reconstructing a focused image (Sec. IV ).

Panel of Figure 2 — Figure 2: Parameterized quantum circuits with data reuploading architectures: (a) The $n_{q}$ -qubit single data reuploading scheme consisting of $L$ layers. Classical input features $\bm{x}$ are re-encoded at each layer via $E_{\ell}(\bm{x})$ alongside parameterized unitaries $V_{\ell}(\bm{\theta})$ . (b) The $n_{q}$ -qubit double data reuploading scheme extended to two input streams, $\bm{x}$ and $\bm{f}$ , structured across $L/2$ blocks. Each block alternates between encoding unitaries $E_{\ell}^{(f)}(\bm{f})$ and $E_{\ell}^{(x)}(\bm{x})$ and parameterized unitaries $V^{(f)}_{\ell}(\bm{\theta})$ and $V^{(x)}_{\ell}(\bm{\theta})$ .

Figure 3: Changing the query encoding or the query value can alter the hypothesis class and the function prior. The top row varies the encoding at fixed

\bm{f}=(\pi/4,\pi/3)

; the bottom row varies

\bm{f}(t)=t\bm{1}

at fixed encoding. In each row, the three series represent (I) an absorbable local encoding with Haar-distributed parameters, (II) the same encoding with a Gaussian parameter prior of width

0.35

, and (III) a nonlocal encoding with Haar-distributed parameters. The comparisons preserve the accessible query-frequency spectrum

\Omega^{(f)}

, isolating changes beyond frequency support. (a–c) Encoding interpolation. The generators

H_{\ell,n}^{(f)}(\alpha)=\frac{s_{\ell}}{2}[\cos(\alpha)A_{n}+\sin(\alpha)B_{n}]

start from

(A_{0},A_{1})=(X_{0},X_{1})

. The absorbable path ends at

(B_{0},B_{1})=(Y_{0},Y_{1})

; the nonlocal path ends at

(Y_{0}Z_{1},Z_{0}Y_{1})

. Anticommutation preserves the generator eigenvalues along both paths. (d–f) Query variation. We compare the local generators

(X_{0},X_{1})

with the nonlocal pair

(Y_{0}Z_{1},Z_{0}Y_{1})

, using layer weights

(s_{1},s_{2})=(1,2)

and the same product trainable blocks. The columns report complementary diagnostics: (a,d) the normalized change

\Delta

of the Fourier coefficient vector for a fixed parameter realization; (b,e) the class witness

S

, measuring the largest nonseparable coefficient weight across

x_{0}|x_{1}

; and (c,f) the normalized distance

D_{\Sigma}

between spectral second moments and their value at

\alpha=0

t=0

. Individual functions change in all three cases, so

\Delta

alone does not identify a change in inductive bias. The remaining diagnostics distinguish the cases:

S

and

D_{\Sigma}

remain at zero or within the Haar sampling null in Case (I); only

D_{\Sigma}

changes in Case (II); and both change in Case (III). In (e),

S(t)

vanishes at

t\in\{0,\pi,2\pi\}

, where the nonlocal query encoding gates reduce to product unitaries. The

D_{\Sigma}

axes are linear within the shaded sampling-null region and logarithmic above it. Definitions, circuit details, and sampling procedures are given in Appx. A.5.1 and Appx. A.5.2 .

Figure 4: A task-aligned fixed query improves learning in a teacher–student problem. Teachers are DDR circuits evaluated at

\bm{f}^{\star}=(\pi/4,\pi/3)

; students use the correct query, the mismatched query

2\bm{f}^{\star}

, or no query. All students share the same architecture, trainable parameters, initializations, and optimization budget. (a) Test balanced accuracy versus training-set size. Lines show averages over ten independently sampled teachers after averaging folds within each teacher; shaded regions are paired

95\%

bootstrap intervals over teachers. (b) Median best-found relative mean-square error when fitting the continuous teacher score on an independent grid. The correct-query model reaches numerical zero by construction, while the other models quantify the residual under the fixed optimization budget; the no-query class is separable by definition.

Figure 5: Classification sensitivity to the fixed query value. (a) Test balanced accuracy for students trained with

\bm{f}(\lambda)=\lambda\bm{f}^{\star}

, where

\bm{f}^{\star}=(\pi/4,\pi/3)

is the query used to generate the teacher labels. The values

\lambda=0

1

, and

2

correspond to the no-query, correct-query, and mismatched-query models, respectively. (b) Local sensitivity to perturbations

\bm{f}=\bm{f}^{\star}+\delta(\cos\phi,\sin\phi)

over eight sampled directions. The solid curve denotes the directional mean, the light band the directional

10

–

90\%

range, and the dashed curve the minimum accuracy over directions, averaged over teachers. The marked tolerance

\delta_{1\%}

estimates the largest query mismatch for which the worst-direction accuracy remains within one percentage point of its matched-query value. Parameters are retrained independently for every query; shaded confidence intervals are obtained by paired bootstrap resampling over teacher instances.

Figure 6: Geometry-derived phase correction of an unfocused SAR-like measurement. (a) Isolated target phase, delivered patch with clutter, noise, and channel phase, and the correction calculated from the acquisition. Hue shows phase; brightness shows amplitude (

|x_{ad}|

for the measured patch). (b) Intensity by azimuth cell

j

and range index

d

after the same Fourier readout with zero, wrong-sign, or geometry-derived query. The marker indicates the true cell

\ell

. The value

p(\ell)

is the readout probability of the true cell for the scene shown. (c) Conditional mean output probabilities

\mathbb{E}[p(j)\mid\ell]

over independent scenes. The displayed balanced accuracies use the most probable output cell as the predicted label. This diagnostic uses the known channel calibration and geometry, with no fitted parameters.

Figure 7: Geometry-derived queries improve SAR-like spectral classification. (a) Test balanced accuracy as the fixed query is varied relative to

f^{\star}_{\mathrm{geom}}

. The variational parameters are retrained independently at every query value using paired datasets and initializations; the vertical dashed line marks the prediction from the acquisition geometry. (b) Profile-optimal fixed query

\widehat{f}

versus

f^{\star}_{\mathrm{geom}}

for acquisitions with different carrier frequencies, squint angles, bandwidths, and reference ranges. For each acquisition, the variational parameters are retrained over a grid of fixed queries and

\widehat{f}

minimizes the mean held-out validation cross-entropy. Error bars are

95\%

intervals obtained by bootstrapping the complete validation-loss profiles and re-estimating the preferred query. They quantify variability across scene draws conditional on the local quadratic interpolation and do not include interpolation bias. The dashed line denotes

\widehat{f}=f^{\star}_{\mathrm{geom}}

. (c) Test balanced accuracy versus the number

N_{\rm train}

of labeled training scenes for the geometry-aligned query

f=f^{\star}_{\mathrm{geom}}

, a trainable global query, the equally entangling wrong-sign query

f=-f^{\star}_{\mathrm{geom}}

, and the zero-query circuit with

f=0

. The fixed-query models share the same phase upload, 48-parameter variational circuit, optimization budget, and paired initializations; the trainable-query model contains one additional global scalar. Lines and shaded regions in (a) and (c) show means and

95\%

bootstrap intervals over independent paired scene draws. The horizontal dotted line denotes chance accuracy,

1/8

. (a,b) use

N_{\rm train}=2^{9}

Symbol	Meaning
$\bm{x}$	Sample-dependent signal; represented by real features when supplied to the circuit.
$\bm{f}$ , $\bm{f}^{\star}$	Query input and its task-prescribed value, shared across samples.
$\bm{\theta}$	Trainable circuit parameters.
$M$ , $N$ , $n_{q}$	Signal-feature dimension, query dimension, and number of qubits.
$L$	Total encoding layers; alternating DDR uses $L/2$ paired signal–query blocks.
$v(\bm{x},\bm{f};\bm{\theta})$	Circuit output: the expectation value of the observable $O$ .

Table A1: Key notation. Superscripts

(x)

and

(f)

distinguish signal and query quantities; a star identifies task-prescribed or target quantities.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure A1: Circuit used in the query-encoding interpolation experiment. Each of the two DDR blocks encodes $\bm{x}=(x_{0},x_{1})$ , applies product trainable blocks $V^{(x)}_{\ell}$ and $V^{(f)}_{\ell}$ , and encodes both components of the fixed query $\bm{f}=(\pi/4,\pi/3)$ . The composite query layer $E_{\ell}^{(f)}$ contains the two commuting rotations defined in Eq. ( 62 ). The local path interpolates from $(X_{0},X_{1})$ to $(Y_{0},Y_{1})$ , whereas the nonlocal path interpolates from $(X_{0},X_{1})$ to $(Y_{0}Z_{1},Z_{0}Y_{1})$ , as defined in Eqs. ( 64 ) and ( 65 ). The output is the correlator $\langle Z_{0}Z_{1}\rangle$ .

Panel of Figure 10 — Figure A2: Circuit used in the query-value experiment. Each of the two DDR blocks encodes $\bm{x}=(x_{0},x_{1})$ , applies product trainable blocks $V^{(x)}_{\ell}$ and $V^{(f)}_{\ell}$ , and encodes the query along the diagonal path $\bm{f}(t)=t(1,1)$ , with layer weights $(s_{1},s_{2})=(1,2)$ . The composite query layer $E_{\ell}^{(f)}(t)\equiv E_{\ell}^{(f)}[\bm{f}(t)]$ contains the two commuting rotations defined in Eq. ( 69 ). The local encoding uses generators $(X_{0},X_{1})$ and is absorbable into the trainable blocks, whereas the nonlocal encoding uses $(Y_{0}Z_{1},Z_{0}Y_{1})$ and is generically nonproduct. Series (I) and (II) use the same local circuit with different parameter priors; series (III) uses the nonlocal encoding. The output is the correlator $\langle Z_{0}Z_{1}\rangle$ .

Panel of Figure 11 — Figure A3: Four-qubit DDR circuit used in the spectral-alignment experiment. The signal is uploaded twice through local $R_{Y}(x_{n})$ rotations. Each trainable block $V_{\ell}^{(x)}$ or $V_{\ell}^{(f)}$ consists of independent $R_{Y}$ rotations followed by a CNOT ring $0\to 1\to 2\to 3\to 0$ . The query rotations implement $H_{\ell,n}^{(f)}=s_{\ell,n}Z_{n}$ , with dyadic scales $s_{\ell,n}=R_{n}2^{-(\ell+1)}$ . The isospectral control replaces each query $R_{Z}$ rotation by an $R_{Y}$ rotation with the same angle. The output is $\langle Z_{0}\rangle$ .

Figure A4: A query encoding can match the task spectrum while changing the function prior. We test these two requirements in a four-qubit, two-layer DDR circuit for the MNUDFT intensity task described in the text. (a) Worst-case distance from the task modes

\Omega_{\star}^{(f)}

to the accessible encoding spectrum

\Omega^{(f)}

as the number

L_{f}

of query layers increases. The dyadic generator scales of Proposition 6 halve the distance with every layer and satisfy the bound

\sqrt{N}R\,2^{-L_{f}}

, whereas uniformly spaced scales converge only polynomially. (b) Normalized distance

D_{\Sigma}

between the spectral second moments at

\bm{f}=t\bm{f}^{\star}

and

\bm{f}=\bm{0}

for

L_{f}=2

. The task-aligned

Z_{n}

encoding is compared with a

Y_{n}

control having the same generator eigenvalues and therefore the same

\Omega^{(f)}

. The control is absorbed by the adjacent

R_{Y}

trainable blocks and remains at the sampling floor, whereas the aligned encoding produces a clear change in the parameter-induced function prior. Thus, matching the task spectrum and changing the function prior are compatible, while identical frequency support alone does not guarantee the latter. Shaded regions denote jackknife

95\%

confidence intervals. Circuit, task, and estimator details are given in Appx. B.3 .

Figure A5: Decision boundaries for one teacher: (a) Teacher score

v_{T}

and its zero level set. (b–d) Students trained on

N_{\rm train}

labeled points (dots) with the correct query

\bm{f}^{\star}

(b,c) or without a query,

\bm{f}=\bm{0}

(d). Solid line: teacher boundary; dashed line: student boundary; shading: misclassified region. In (c),

56\%

of misclassified test points satisfy

|v_{T}|<0.05

, a region that contains

8.7\%

of all test inputs. The separable student in (d) is restricted to axis-aligned boundaries.

Panel of Figure 14 — Figure A6: Circuit used in the SAR-like classification experiment. The three-qubit registers $A$ and $D$ carry the binary digits of the azimuth index $a$ and the range index $d$ of the $8\times 8$ patch. Each of the two blocks applies product trainable blocks $W_{\ell}^{A}\otimes W_{\ell}^{D}$ and $V_{\ell}^{A}\otimes V_{\ell}^{D}$ , each consisting of $R_{Z}R_{Y}$ rotations on every qubit followed by a ring of CZ gates within the register, which gives $48$ trainable parameters. The signal is reuploaded as the phase-only diagonal $E^{(x)}(\bm{x})=\mathrm{diag}_{ad}\,\exp(i\arg(x_{ad})/L_x)$ , and the fixed query as $E^{(f)}(f)=\exp[-i(f/L_{f})\widetilde{A}\widetilde{D}]$ , which compiles into nine cross-register $R_{ZZ}$ rotations. The readout applies the quantum Fourier transform $F_{8}$ to $A$ and measures it, giving the class distribution $p(j)$ ; $D$ is traced out.

Acquisition	$f_{\rm c}$ (GHz)	$\delta f$ (MHz)	$k_{x0}$ (rad m -1 )	$R_{0}$ (km)	$f^{\star}_{\rm geom}$
High carrier	14	5	125	3.7	0.377
Low squint	10	5	80	3.7	0.467
Carrier + bandwidth	14	9	125	3.7	0.679
Baseline	10	5	125	3.7	0.793
Far range	10	5	125	4.8	1.029
High squint	10	5	180	3.7	1.348

Appendix

Table A2: Acquisitions used in Fig. 7 b. The query is calculated from Eq. ( 33 ).

Panel	Draws	Training	Validation	Test
a	12	512	–	4096
b	4	512	1024	–
c	12	32–512	–	4096

Appendix

Table A3: Dataset sizes for Fig. 7 . For panel b, draws are performed separately for each acquisition.

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Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.

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