Data re-uploading depth vs Fourier expressivity for variational quantum regression
Domain: quantum (quantum-machine-learning, data-reuploading, expressivity)
Topic: Data re-uploading depth vs Fourier expressivity for variational quantum regression: characterizing how the trainable bandwidth of a parameterized re-uploading circuit scales with re-uploading layers L on synthetic 1D regression targets
Venue: ARC-Bench 2026
Headline metric: test_mse (minimize)

Research briefing:
Data re-uploading is a variational quantum circuit pattern in which the
classical input x is loaded into the same qubits multiple times,
interleaved with trainable rotation layers: the circuit
W(theta_L) U(x) W(theta_{L-1}) U(x) ... W(theta_1) U(x) |0> applies L
encoding blocks U(x) and L trainable blocks W(theta). Perez-Salinas et al.
proved in 2020 that a single qubit with sufficient re-uploading depth is
a universal function approximator on bounded intervals. Schuld, Sweke,
and Meyer (PRA 2021) then proved the structural reason: a re-uploading
circuit's output function is a truncated Fourier series in the input x,
with the set of representable frequencies determined by the eigenvalue
spectrum of the encoding generators and growing with L.

The practical question that follows is whether the theoretical
expressivity claim translates into empirical fit quality. As L grows we
expect (i) more representable frequencies (better fit on high-bandwidth
targets), and (ii) more parameters and more risk of overfitting on
finite-sample regression with discontinuous targets. Classical kernel
regression (RBF) and polynomial regression provide the natural baselines
because they too can be interpreted as choosing a basis of functions
(Gaussian or polynomial) of fixed expressivity; comparing quantum
re-uploading against these classical baselines is the cleanest way to
ask "does the Fourier structure imposed by re-uploading give a useful
inductive bias on these targets, or do classical models with comparable
capacity match it?"

Implementation guidance is provided by the quantum-qiskit skill, which
is automatically injected at stage 10 and contains canonical code for
building re-uploading circuits using qiskit.circuit.ParameterVector and
rotation gates. Training uses Adam through parameter-shift gradients
(do NOT roll your own gradient finite-difference loop). The simulator
is fixed to AerSimulator(method='statevector') for noiseless evaluation
so the Fourier-spectrum probe is clean.

Experimental protocol. Four quantum re-uploading depths (L=1, L=3, L=5,
L=7) are compared against two classical baselines (RBF kernel ridge
regression with bandwidth chosen by cross-validation; polynomial
regression of degree 7). All six conditions are evaluated on two 1D
regression targets: a sum-of-sinusoids signal with known frequency
content, and a step function (tests handling of discontinuities).
Each (condition, dataset) cell is averaged over 2 random seeds, giving
6 conditions x 2 datasets x 2 seeds = 24 total cells.

Research question: *On 1D regression with known target bandwidth, does
increasing data re-uploading depth L produce monotonically lower test
MSE on bandwidth-rich targets (sinusoid mixture), and does the
recovered Fourier spectrum overlap with the target spectrum scale
predictably with L?*

Hypotheses:
statement: On the sinusoid-mixture target, the L=5 re-uploading model achieves test MSE at least 4 times lower than the L=1 re-uploading model (2-seed mean), demonstrating that increasing re-uploading depth improves regression accuracy on bandwidth-rich targets., statement: On the step-function target, the L=7 re-uploading model exhibits overfitting compared to L=5: L=7 test MSE is at least 15 percent higher than L=5 test MSE (2-seed mean), even though L=7 has more parameters and lower training MSE., statement: At L=7 on the sinusoid-mixture target, the recovered Fourier spectrum (FFT of the trained circuit's output sampled on a uniform 256-point grid) has cosine similarity at least 0.7 with the target Fourier spectrum, restricted to the lowest 5 nonzero frequencies.

Experiment design:
research_question: On 1D regression with known target bandwidth, does increasing data re-uploading depth L produce monotonically lower test MSE on bandwidth-rich targets, and does the recovered Fourier spectrum overlap with the target spectrum scale predictably with L?
conditions: name: reuploading_L1
description: Single-qubit re-uploading circuit with L=1 encoding block. Encoding: U(x) = RZ(x). Trainable block: W(theta) = RY(theta_0) RZ(theta_1). Output: <Z>. 2 trainable parameters total., name: reuploading_L3
description: Same single-qubit re-uploading template with L=3 encoding blocks. 6 trainable parameters., name: reuploading_L5
description: Same template with L=5 encoding blocks. 10 trainable parameters., name: reuploading_L7
description: Same template with L=7 encoding blocks. 14 trainable parameters.
baselines: rbf_kernel_ridge: sklearn.kernel_ridge.KernelRidge(kernel='rbf', alpha=1e-3, gamma=tuned_via_grid_search). Strong nonparametric regression baseline., polynomial_regression: sklearn.pipeline.make_pipeline(PolynomialFeatures(degree=7), Ridge(alpha=1e-3)). Matched-capacity polynomial basis baseline.
metrics: name: test_mse
direction: minimize
description: Mean squared error on a 200-point held-out test set, averaged over 2 random seeds per (condition, dataset) cell. Primary metric., name: train_mse
direction: minimize
description: Mean squared error on the 200-point training set after training completes., name: fourier_spectrum_cosine
direction: maximize
description: Cosine similarity between the FFT of the trained model's output sampled on a uniform 256-point grid over [0, 1] and the FFT of the target function on the same grid, restricted to the lowest 5 nonzero frequency bins. Higher means the model has learned the target's Fourier content.
datasets: name: sinusoid_mixture_1d
source: Synthetic generator
description: Target function f(x) = sum_k a_k * sin(2*pi*k*x) for k in {1, 2, 3, 5, 7} with random amplitudes a_k drawn from Uniform[0.5, 1.0] using a fixed dataset-generation seed (independent of per-cell training seeds). 200 training x-values + 200 test x-values uniformly sampled on [0, 1]. Targets are observed with Gaussian noise N(0, 0.05)., name: step_function_1d
source: Synthetic generator
description: Target function f(x) = 1.0 if x > 0.5 else 0.0, plus Gaussian noise N(0, 0.05). Tests how the re-uploading model handles a discontinuity (its Fourier series will exhibit Gibbs ringing). 200 training + 200 test samples.
compute_requirements: gpu_required: no
estimated_wall_clock_sec: 900

Requirements:

Scoring rubric:
requirements: A credible empirical study of data re-uploading depth vs Fourier expressivity in variational quantum regression. The agent must (a) implement 4 re-uploading circuits at L in {1, 3, 5, 7} using qiskit ParameterVector + RY/RZ rotation gates, (b) train each with parameter-shift Adam on 200 samples of two 1D regression targets (sinusoid mixture and step function), (c) compare against two classical baselines (RBF kernel ridge regression, polynomial regression degree 7) on the same data, (d) measure test MSE per cell and the recovered Fourier spectrum of the trained quantum model, and (e) score H1/H2/H3 with numerical evidence.
judging_note: Re-uploading topics are scored on (i) whether the 4 quantum circuits actually differ by depth (sanity check: parameter counts must be 2/6/10/14), (ii) whether training was real and produced different outputs across L levels (3 quantum cells producing bit-identical MSE would indicate dispatch failure, same anti-pattern as Q01 ablation failure), (iii) whether classical baselines are computed on the exact same data as quantum cells, and (iv) whether H1/H2/H3 are supported by reported MSE and Fourier-spectrum numbers. Quantitative result leaves use a graded scale: 100% if hypothesis threshold cleanly met, 67% if trend in predicted direction without clearing threshold, 33% if ambiguous, 0% if contradicted or evidence missing.
sub_tasks: requirements: Code-development bucket: 4 re-uploading circuits + 2 classical baselines are implemented correctly with shared training pipeline and reproducible per-cell seed control.
sub_tasks: requirements: All four re-uploading circuits use qiskit.QuantumCircuit + qiskit.circuit.ParameterVector. The encoding gate is RZ(x) and the trainable block is RY(theta_2k) RZ(theta_2k+1). Output is the expectation <Z> on the single qubit (via qiskit.quantum_info.Statevector or the reference Estimator primitive). Parameter counts match: L=1 has 2 params, L=3 has 6, L=5 has 10, L=7 has 14. A sanity check asserts that the four circuits produce different output values for a fixed nonzero input x=0.3 with a fixed random parameter vector.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: Training uses Adam with parameter-shift gradients computed via the qiskit parameter-shift rule (NOT scipy finite differences). Loss is mean squared error on the 200 training samples. Same optimizer hyperparameters across all four L levels (Adam lr=0.05, 300 steps). Both classical baselines use sklearn (KernelRidge with grid-searched gamma, and a Pipeline of PolynomialFeatures+Ridge).
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: Each per-(condition, dataset, seed) result is emitted to stdout as a single line starting with METRIC_RESULT followed by JSON containing condition, dataset, seed, test_mse, train_mse, and fourier_spectrum_cosine. The FFT is computed on the trained model's output sampled on a uniform 256-point grid over [0, 1].
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: Execution-validity bucket: all 24 cells ran and produced numerically valid outputs.
sub_tasks: requirements: At least 22 cells out of 24 expected (6 conditions x 2 datasets x 2 seeds) completed without unhandled errors and produced a test_mse value. Missing more than 2 cells (10 percent) without documented justification fails this leaf.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Numerical validity: no NaN or Inf in test_mse; train_mse <= test_mse * 1.5 across conditions (otherwise model is broken); the 4 quantum L levels produce different test MSEs (max - min across the 4 quantum conditions on the sinusoid-mixture dataset is at least 0.005). Fourier spectrum values are real, finite, and the cosine similarity is in [-1, 1].
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Results bucket: quantitative tests of H1/H2/H3 plus a per-hypothesis writeup.
sub_tasks: requirements: Quantitative test of H1. On the sinusoid-mixture dataset, is test_mse(L=5) <= test_mse(L=1) / 4 (2-seed mean)? 100% if ratio >= 4, 67% if 2 <= ratio < 4, 33% if 1 < ratio < 2, 0% if L=5 is not better than L=1.
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H2. On the step function dataset, is test_mse(L=7) >= test_mse(L=5) * 1.15 (2-seed mean)? 100% if overshoot is >= 15 percent, 67% if 5-15 percent, 33% if any small overshoot (0-5 percent), 0% if L=7 still has lower MSE.
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H3. On the sinusoid-mixture dataset at L=7, does the recovered Fourier spectrum cosine similarity (restricted to lowest 5 nonzero frequencies) reach 0.7? 100% if >= 0.7, 67% if >= 0.5, 33% if >= 0.3, 0% otherwise.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Writeup of at least 200 words (submission/README.md ## Agent-produced writeup section) with explicit supported / refuted / inconclusive verdict for each of H1, H2, H3 backed by specific test MSE values per (L, dataset) and Fourier spectrum cosine values. Discusses how the Fourier-series view (Schuld, Sweke, Meyer 2021) predicts the observed trend, and identifies a dominant systematic uncertainty (seed variance, optimizer convergence, parameter-shift gradient noise).
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation