Barren plateau onset under cost-function locality
Domain: quantum (quantum-machine-learning, barren-plateaus, trainability)
Topic: Barren plateau onset under cost-function locality: empirically measuring gradient variance of hardware-efficient ansatze under global vs local cost functions across depth, testing the Cerezo 2021 prediction that local costs avoid exponential gradient suppression
Venue: ARC-Bench 2026
Headline metric: log_gradient_variance (maximize)

Research briefing:
A barren plateau is the phenomenon, first identified by McClean et al.
in 2018, that the gradient of a variational quantum circuit's loss
becomes exponentially small in the number of qubits when the ansatz is
drawn from a sufficiently random circuit ensemble. Concretely, for a
loss L(theta) = <psi(theta)|H|psi(theta)> where the parameterized
state is generated by a random-ish unitary, the variance of the
gradient dL/dtheta_k scales as O(1/2^n). For n = 20 qubits this
variance is below machine epsilon, and SGD or any gradient-based
optimizer has effectively zero signal to follow. This is the central
trainability obstruction in variational quantum machine learning.

Cerezo et al. ("Cost function dependent barren plateaus in shallow
parametrized quantum neural networks", Nature Communications 2021)
showed that the locality of the cost function matters. If the cost is
a sum of single-qubit Pauli expectations (a local cost), then the
gradient variance can scale only polynomially in n even at moderate
ansatz depth. If the cost is a product or projection over all qubits
(a global cost), the variance is exponentially suppressed at any
depth. The mechanism is that local-cost observables overlap with only
a constant-size subspace of the 2^n dimensional Hilbert space, while
global-cost observables are spread thinly across the entire space and
almost always cancel.

The empirical question this topic addresses is whether the local-cost
free lunch holds at moderate depth on a CPU-feasible 6-qubit
benchmark. We fix n=6 qubits and the EfficientSU2 hardware-efficient
ansatz family (a standard choice in qiskit). We vary depth (number of
ansatz layers L) and cost-function locality, then directly probe the
gradient distribution by sampling 100 random parameter vectors per
cell, computing the parameter-shift gradient with respect to theta_0
for each, and reporting the empirical variance. A subsidiary check
trains each condition for 100 Adam iterations and reports the loss
change as a practical sanity probe of whether the gradient signal is
large enough to drive optimization at all.

Implementation guidance is provided by the quantum-qiskit skill,
which is automatically injected at stage 10. Use qiskit.circuit.library
EfficientSU2 verbatim; do NOT hand-write H/CNOT/RY gate stacks (a
prior failed run produced an "ablation failure" where three nominally
different encodings collapsed to identical circuits because they
were hand-coded). Use qiskit_algorithms.utils.algorithm_globals
random_seed for reproducibility. Gradients are computed via the
parameter-shift rule on a statevector backend.

Experimental protocol. Four conditions cover the cost x depth grid:
(local_cost_L2) shallow + local, the trainable control; (local_cost_L10)
medium + local, the regime Cerezo's theorem predicts is still
trainable; (local_cost_L20) deep + local, stressing the prediction;
(global_cost_L10) medium + global, the negative control where a barren
plateau is expected. Each condition is run with 3 different random
ansatz-structure seeds (controlling which qubits are entangled in each
EfficientSU2 layer) for variance estimation across architectures, and
100 random parameter vectors per (condition, structure-seed) cell are
used to estimate gradient variance. Total cells: 4 conditions x 3
ansatz seeds = 12 cells. No external dataset is required because this
is a measurement experiment on the ansatz family itself.

Research question: *Does the local-cost trick avoid the barren plateau
on a 6-qubit EfficientSU2 ansatz at L=10 and L=20 ansatz depth, as
predicted by Cerezo et al. 2021, when compared against the
global-cost baseline at the same depth and against the shallow local-cost
control?*

Hypotheses:
statement: At ansatz depth L=10, the empirical variance of the gradient under local cost is at least 20 times larger than under global cost (3-seed mean of variance per condition), confirming Cerezo et al. 2021's prediction that locality changes the barren-plateau scaling., statement: Under local cost, the gradient variance at L=20 is at least 1/100 of the gradient variance at L=2, i.e. the polynomial (sub-exponential) decay regime is maintained even at deep ansatz, with log10(var_L20) - log10(var_L2) > -2.0 (3-seed mean)., statement: When trained with Adam for 100 gradient steps starting from random initialization, the global-cost-L10 condition produces a loss change of at most 1 percent of the initial loss (3-seed mean), demonstrating that the predicted barren plateau actively prevents practical training, while the local-cost-L10 condition produces a loss change of at least 10 percent.

Experiment design:
research_question: Does the local-cost trick avoid the barren plateau on a 6-qubit EfficientSU2 ansatz at L=10 and L=20, as predicted by Cerezo et al. 2021, when compared against the global-cost baseline and a shallow local-cost control?
conditions: name: local_cost_L10
description: Proposed-method core. EfficientSU2(num_qubits=6, reps=10, entanglement='linear') ansatz. Cost function: <Z_0> (expectation of Pauli Z on qubit 0 only). 100 random parameter vectors theta drawn from Uniform[-pi, pi]^{n_params}. For each theta, compute parameter-shift gradient dL/dtheta_0 by evaluating L at theta + (pi/2) * e_0 and theta - (pi/2) * e_0. Report variance of these 100 gradient values., name: local_cost_L20
description: Proposed-method stress test. EfficientSU2 reps=20, same local cost <Z_0>. Tests whether the polynomial-decay prediction holds at deeper ansatz.
baselines: local_cost_L2: trainable-control baseline. EfficientSU2 reps=2, local cost <Z_0>. Known to have non-vanishing gradient because the ansatz is too shallow to enter the barren-plateau regime., global_cost_L10: barren-plateau negative-control baseline. EfficientSU2 reps=10, global cost <Z_0 Z_1 Z_2 Z_3 Z_4 Z_5> (product of Z on all 6 qubits). Known from Cerezo 2021 to exhibit exponentially small gradient variance even though the ansatz depth matches local_cost_L10.
metrics: name: log_gradient_variance
direction: maximize
description: log10 of the empirical variance of dL/dtheta_0 over 100 random parameter samples. Higher means more trainable gradient signal. Primary metric. Less negative numbers are better (closer to 0)., name: mean_absolute_gradient
direction: maximize
description: Mean of |dL/dtheta_0| over the 100 random samples., name: training_loss_change_fraction
direction: maximize
description: Relative loss change after 100 Adam steps starting from a random initialization: (L_initial - L_after_100_steps) / |L_initial|. Practical training-feasibility probe. Larger positive means training actually moved the loss.
datasets: name: random_parameter_samples
source: Synthetic, generated in-script via numpy.random.RandomState(ansatz_structure_seed).uniform(-pi, pi, size=(100, n_params))
description: Each cell samples 100 random parameter vectors in the parameter space of EfficientSU2 at the given (num_qubits, reps). These are the inputs over which gradient variance is estimated. No external data is involved; this is a measurement experiment on the ansatz family.
compute_requirements: gpu_required: no
estimated_wall_clock_sec: 600

Requirements:

Scoring rubric:
requirements: An empirical study of barren plateau onset on a 6-qubit EfficientSU2 ansatz under local vs global cost functions and shallow vs deep ansatz. The agent must (a) build the EfficientSU2 ansatz with reps in {2, 10, 20} via qiskit.circuit.library, (b) for each (cost, depth) condition sample 100 random parameter vectors and compute parameter-shift gradient dL/dtheta_0, (c) report the empirical variance of those 100 gradient values per cell, (d) run a 100-step Adam training as a practical sanity probe of trainability, and (e) score H1/H2/H3 with numerical evidence.
judging_note: Barren plateau studies are scored on (i) correctness of the EfficientSU2 / cost construction (using qiskit.circuit.library, not hand-rolled), (ii) whether the 100-sample gradient distribution actually captures the variance (cells reporting std=0 across 100 samples indicate a broken sampler), (iii) the local_cost_L10 vs global_cost_L10 contrast being clean (these have IDENTICAL ansatz, only cost differs, so any large variance gap directly tests Cerezo 2021), and (iv) the training-feasibility probe correlating with the gradient-variance prediction (BP cell should not train; trainable cell should train). Quantitative result leaves use a graded scale: 100% if cleanly met, 67% if in predicted direction without clearing threshold, 33% if ambiguous, 0% if contradicted.
sub_tasks: requirements: Code-development bucket: EfficientSU2 ansatz built correctly via qiskit.circuit.library, parameter-shift gradient implemented correctly, and the 100-sample gradient distribution actually populated per cell.
sub_tasks: requirements: EfficientSU2(num_qubits=6, reps=reps_value, entanglement='linear') from qiskit.circuit.library used directly (NOT hand-coded H/CNOT/RY stacks). Parameter-shift gradient of <O>(theta) with respect to theta_0 computed via the qiskit standard pattern: dL/dtheta_0 = (L(theta + (pi/2)*e_0) - L(theta - (pi/2)*e_0)) / 2. Cost observable for local_cost is Pauli Z on qubit 0; for global_cost is the product Z_0 Z_1 Z_2 Z_3 Z_4 Z_5 (a SparsePauliOp). A sanity check confirms that the gradient is nonzero for at least one random parameter sample at L=2 local cost.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: For each (condition, ansatz-structure seed) cell, 100 independent random parameter vectors are drawn from Uniform[-pi, pi]^{n_params} using a per-cell seeded RNG. For each parameter vector, the gradient dL/dtheta_0 is computed. The 100 gradient values are stored and the empirical variance is reported. Reusing the same parameter vector across conditions in the same cell is allowed (paired comparison) but NOT required.
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: After the gradient-variance measurement, each cell runs a 100-step Adam optimization (qiskit_algorithms.optimizers.ADAM or pytorch Adam wrapping a parameter-shift gradient closure) starting from a random initialization. The initial loss and the loss after step 100 are recorded; the training_loss_change_fraction = (L_initial - L_final) / |L_initial| is computed per cell.
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: Execution-validity bucket: all 12 cells ran and produced numerically valid outputs.
sub_tasks: requirements: At least 11 cells out of 12 expected (4 conditions x 3 ansatz-structure seeds) completed and produced a non-null log_gradient_variance value. Missing more than 1 cell without documented justification fails this leaf.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Numerical validity: no NaN or Inf in gradient values; empirical std of the 100 gradients is strictly > 0 in every cell (cells reporting std=0 indicate a broken sampler and fail this leaf); log10(variance) values are finite. The training_loss_change_fraction is in the range [-1.0, 1.0] (otherwise loss diverged or wasn't properly normalized).
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. Is var(local_cost_L10) / var(global_cost_L10) >= 20 (3-seed mean of variances)? 100% if ratio >= 20, 67% if 5 <= ratio < 20, 33% if 1 < ratio < 5, 0% if global cost has equal or larger variance.
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H2. Is log10(var_local_L20) - log10(var_local_L2) > -2.0 (3-seed mean)? 100% if difference > -2.0 (polynomial decay), 67% if difference > -3.0, 33% if difference > -4.0, 0% otherwise (exponential decay observed).
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Quantitative test of H3. Is global_cost_L10 training_loss_change_fraction <= 0.01 AND local_cost_L10 training_loss_change_fraction >= 0.10 (3-seed mean)? 100% if both conditions hold cleanly, 67% if local trains but global change is 1-5 percent, 33% if local trains but global has any change, 0% if global cost also trains (would contradict the BP prediction).
task_category: Result Analysis
finegrained_task_category: Evaluation, Metrics & Benchmarking, 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 variance values per (condition, seed) and training_loss_change_fraction values. Explicitly references Cerezo et al. 2021 'Cost function dependent barren plateaus' and discusses whether the observed variance ratio matches their theoretical prediction. Identifies dominant uncertainty (ansatz-structure seed variance, finite-sample variance estimate from 100 gradient samples).
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation