Motivation and learning objective
Why does increasing the number of qubits sometimes hide useful gradient information even when the parameter-shift rule is exact?
A quantum model can have exact analytic gradients that nevertheless become exponentially difficult to resolve from measurements. Compare a global many-qubit parity cost against a strictly local Pauli cost within the same product-state ansatz. The contrast isolates a concrete trainability mechanism without making an unjustified universal barren-plateau claim.
Recommended prerequisites: the relevant mathematical formulation in the English lecture notes, complex amplitudes, expectation values, and a basic understanding of quantum operators. Read the derivation below before interpreting the plots as physical observables.
Predictions and benchmarks
Live model outputs
Global and local gradient second moments
Both observables use the same product circuit. Their gradients differ solely in how many random independent parameters enter the observable.
Log10 of gradient second moments versus qubit count for global parity and local cost
Exact global second moment is 2⁻ⁿ; exact local value is 1/2. Monte Carlo estimates are reproducible but may fluctuate strongly because rare large gradients dominate high-dimensional moments.
Typical gradient compared with parameter-shift noise
A small expected derivative can be hidden by finite Pauli measurement variance, creating a sampling limitation separate from analytic differentiability.
Typical root mean square global gradient and illustrative shot-noise scale versus qubit number
The nominal shot-error comparison uses the conservative order of magnitude 1/√(2Nshots) for two independently estimated shifted expectations with near-zero mean.
Distribution of gradient strength
The histogram shows that most random parameter choices have small global gradients and occasional rare draws contribute disproportionately to the second moment.
Histogram of log10 absolute sampled global gradients for the chosen n
The exact zero-gradient case has measure zero for continuous random parameters, but gradients may be exponentially small in typical parameter settings.
Derivation and assumptions
Reconstruct the physical calculation
Consider the deliberately simple product ansatz |ψ(θ)〉=⊗ⱼRy(θⱼ)|0〉, with θⱼ independently uniform on [−π,π]. For the global parity observable Z⊗n, the exact expectation factorizes as fglobal(θ)=∏ⱼcosθⱼ. The derivative with respect to the first parameter is ∂₀fglobal=−sinθ₀ ∏j>0cosθⱼ.
The second moment follows immediately because E[sin²θ]=E[cos²θ]=1/2 independently. In contrast, the local observable Z₀ has f_local=cosθ₀ and derivative −sinθ₀, giving E[(∂₀flocal)²]=1/2 for every n. Thus exponential decay of the global-cost gradient is demonstrable even in an unentangled product ansatz, while the local observable of exactly the same circuit avoids this mechanism.
Both derivatives can be computed using the exact parameter-shift identity [f(θ₀+π/2)−f(θ₀−π/2)]/2. The rule concerns mathematical accuracy, whereas finite-shot evaluation creates statistical uncertainty. With Nshot independent binary outcomes at each shifted circuit, the gradient-estimator variance is [(1−f₊²)+(1−f₋²)]/(4Nshot), at most 1/(2Nshot).
To resolve a typical global derivative of root-mean-square size 2−n/2 with order-one signal-to-noise using the conservative worst-case shot noise, the number of shots must grow on the scale of 2n−1. The plotted local and global contrast arises from an exactly specified ensemble, observable and ansatz. Other ansätze, initializations, symmetries and objectives can behave differently, and broad barren-plateau statements require their own assumptions.
Check exact and singular limits
At n=1 the local and global observables coincide with gradient second moment 1/2. At n=12 the exact global second moment is 1/4096 while the local value remains 1/2. Increasing the number of circuit shots changes measurement noise but leaves the underlying exact gradient distribution unchanged.
Numerical and physical validity conditions
Independent product Ry rotations, independent uniform parameter sampling, binary Pauli measurements and a fixed single trainable derivative. No entanglement, entangling hardware, correlated parameter initialization or full machine-learning dataset objective is included; the analysis is an explicit controlled trainability example.
Active understanding
Three independent investigations
- Derive the gradient law. Use factorization of independent trigonometric random variables to compute both second moments without simulating any circuits.
- Test the finite ensemble. Increase n with fixed random seeds. Compare analytical second moments with sampled results, and explain why rare high-contribution draws cause noisy Monte Carlo estimates for large n.
- Estimate measurement resources. Increase the shot count and compare the RMS global gradient with the nominal parameter-shift noise. Explain why exact differentiability alone does not guarantee economical gradient estimation.
Self-check: For independent uniform rotation angles, how does the global parity-gradient second moment scale with n?
Extension: reproduce one model with a separate implementation or an AI-assisted notebook. Check numerical convergence, normalizations, phase conventions, or independent limiting-case results before trusting the answer.
Physical conclusions
Interpret the invariant or approximation carefully
Gradient concentration is sensitive to what a circuit measures and to the parameter ensemble. A global cost can have exponentially small gradients while a local cost stays trainable within exactly the same state-preparation family. Students can quantify the associated sampling burden and distinguish a model-specific proof from universal claims.
Lecture notes: 03_scope_limits.tex ↗.