Replication and Scale Extension of Identity-Fragmented Gradient Evaluation: Exact Local Gradients and Restricted Training of a 10^6-Qubit Variational Circuit
The measurements reported here separate maximum contiguous depth from nominal depth as the variable governing initialization gradient visibility in identity-fragmented hardware-efficient variational circuits, and extend the fragmented branch to 106 qubits. Three separately written implementations of a reduced active-window evaluator agree pairwise, to displayed precision, on every per-sample gradient, and the window evaluation is verified against full-register simulation at small size, against closed forms for the two shallowest prefix depths, and by window-width convergence. At fixed parameter location, the quantity barren-plateau statements concern, the prefix-depth-one gradient variance is one eighth analytically and is measured consistent with that value at both 65,536 and 10^6 qubits. A matched-nominal-depth comparison over the tested sizes yields suppression exponents 0.688 unfragmented against 0.011 fragmented; these are finite-size values and are not asymptotic exponents. A dense-topology control, in which the backward cone fills the register for both branches, retains the separation, which rules out cone width as the controlling variable. At exactly simulable sizes the fragmented circuit trains to 10^−3 relative energy error while the matched monolithic baseline collapses; asymptotic trainability is not established. The single-block 10^6-qubit circuit then trains off the identity point under translation-invariant parameters, exactly on certified causal-cone windows, and multistart search over the six shared angles reaches an energy density within 1.8% of the infinite-chain value. That residual is not a boundary effect, and is not the expressivity limit of identity fragmentation: relaxing translation invariance to period-two angles - twelve parameters, the same window - raises the correlation energy recovered beyond the best product state from 35.1% to 87.8%. Optima are best-found under local search; global optimality is not certified. The regimes trained here are classically simulable by construction; no claim of quantum advantage is made.
The measurements reported here separate maximum contiguous depth from nominal depth as the variable governing initialization gradient visibility in identity-fragmented hardware-efficient variational circuits, and extend the fragmented branch to 106 qubits. Three separately written implementations of a reduced active-window evaluator agree pairwise, to displayed precision, on every per-sample gradient, and the window evaluation is verified against full-register simulation at small size, against closed forms for the two shallowest prefix depths, and by window-width convergence. At fixed parameter location, the quantity barren-plateau statements concern, the prefix-depth-one gradient variance is one eighth analytically and is measured consistent with that value at both 65,536 and 10^6 qubits. A matched-nominal-depth comparison over the tested sizes yields suppression exponents 0.688 unfragmented against 0.011 fragmented; these are finite-size values and are not asymptotic exponents. A dense-topology control, in which the backward cone fills the register for both branches, retains the separation, which rules out cone width as the controlling variable. At exactly simulable sizes the fragmented circuit trains to 10^−3 relative energy error while the matched monolithic baseline collapses; asymptotic trainability is not established. The single-block 10^6-qubit circuit then trains off the identity point under translation-invariant parameters, exactly on certified causal-cone windows, and multistart search over the six shared angles reaches an energy density within 1.8% of the infinite-chain value. That residual is not a boundary effect, and is not the expressivity limit of identity fragmentation: relaxing translation invariance to period-two angles - twelve parameters, the same window - raises the correlation energy recovered beyond the best product state from 35.1% to 87.8%. Optima are best-found under local search; global optimality is not certified. The regimes trained here are classically simulable by construction; no claim of quantum advantage is made.
The barren plateau (BP) phenomenon—exponential decay of gradient variance in parameterized quantum circuits (PQCs) with system size—is a primary obstacle to scalingvariational quantum algorithms (VQAs). This paper reports a set of empirical benchmarks on hardware-efficient ansatz (HEA) circuits up to n = 20 qubits, showing that identity fragmentation—partitioning a circuit of nominal depth D into pairs of forward and inverse blocks of size k—substantially increases the variance of cost-function gradients, relative to unfragmented HEA at the same nominal depth. The fragmented design exhibits a suppression exponent αfrag = 0.225 ± 0.008 in Var(∂θ C) ∼ e−αn, versus αunfrag = 0.711 ± 0.025 for the unfragmented baseline (R2 ≥ 0.996 in both cases), giving a measured ratio of approximately 4 × 104 at n = 20, D = 24. In the tested fragmented HEA settings, the advantage is monotone in the maximum contiguous depth Lcontig, persists under finite-shot estimation (8000 shots), and is partially robust to depolarizing noise at gate-error rates relevant to current superconducting hardware. A direct training-drift test over ten gradient-descent steps on an X-basis cost at n = 8 shows the fragmentation gradient-variance advantage persists (50× at t = 0, 40× at t = 10) while the fragmented circuit reduces the cost ∼ 50× more than the unfragmented baseline over the same window. The design rule is derived from the Universal Suppression Law in companion theoretical work; the present paper’s role is to establish benchmark-level empirical support across the circuit families and noise settings studied here.
Direct measurements are reported for the causal support of initialization gradients in identity-fragmented variational circuits. Evaluation restricted to the computed causal support reproduces full-register simulation to 1.4 × 10^−16, and gradients for parameters whose support excludes the observable vanish to 2.75×10^−17 rather than merely becoming small. Support cardinality is min(N, 4ℓ) on nearest-neighbour connectivity and min(N, 8ℓ) with an added antipodal coupling, where ℓ is prefix depth and N the register size, and is independent of N once the register exceeds the support. For a local observable at fixed fragment depth K, the number of parameters surviving the support criterion is 2K^2 −2K +1 on the nearest-neighbour topology, also independent of N . A stacked-block control separates nominal circuit depth from effective causal depth: a 256-amplitude prefix tile reproduces a circuit of 96 macro-layers to 2.1 × 10^−16, while the corresponding cone of a plain circuit grows linearly in depth. Controls establish the limits of the construction, including failure of tile reuse under general input states. Two corrections to the support bound and the tile-count figure of the replication manuscript are reported. A finer object is then measured: the set of gate transitions, rather than qubits, on which the answer depends. At N = 14 and prefix depth 2, eighteen of fifty-six gates suffice. All figures are reproducible from the accompanying scripts.
A RING-only finite-size and large-logical-size test is presented using the terminology of the Universal Suppression Law (USL) interpretation of variational trainability collapse. The differentiated quantity is treated as a gradient observable Ai, and the structural variable is the embedded causal depth L of the history of Ai. The baseline circuit is an unfragmented RING with depth D = N . The comparison circuit is an identity-block RING evaluated at the initialization point B∧−1B = I, where a parameter-shift query opens one parameter inside the forward block. The headline statistic is the variance of a local gradient observable for a local cost component Hj = Xj , so no 1/N-normalized global observable is used as central evidence. Small full-statevector simulations show rapid suppression in the unfragmented RING, while identity-block RING circuits with bounded embedded causal depth retain approximately constant local gradient-observable variance. A large logical calculation with N = 16384, identity-block depth k = 3, and measured support of 8 qubits, implemented with a 9-qubit evaluation register, gives local gradient variance 2.646675805284 × 10∧−1 using 512 amplitudes. A full-state effective-causal-region sweep confirms that restricted evaluations fail when the active region under-contains the history of Ai, and reach floating-point precision only when the measured history is contained. The result is an initialization-time trainability-structure test of the embedded history of Ai, not a quantum-advantage claim.
A companion application of the Universal Suppression Law (USL) to variational quantum algorithms (VQAs) is presented as a Category II structural consistency study rather than as a parameter-free validation. The imported theory input is the Quantum Entanglement Spacetime Theory (QuEST)-derived claim that operator visibility is exponentially suppressed with embedded causal depth. A literature synthesis across barren-plateau mechanisms shows that cost locality, ansatz expressibility, dynamical Lie algebra growth, pooling, adaptive construction, Hamiltonian structure, localization, and noise can be reinterpreted as controls on an effective observable-coupled depth. This motivates the design claim that gradient visibility is governed less by nominal gate count than by the largest active suppressive history coupled to the measured observable. Finite-size statevector benchmarks for circuits with Ry /Rz rotations and controlled-NOT entanglers at n ≤ 12 support this interpretation. Local observables suppress gradients more slowly than a global projector, circular connectivity suppresses more strongly than brick-wall connectivity, and identity fragmentation substantially weakens suppression at fixed nominal depth. At fixed total depth, increasing contiguous block size reduces gradient variance. These results support a compact design program: preserve trainability by bounding, fragmenting, or filtering irrelevant observable-coupled history
A naturally fragmented PAIRS circuit is used as an analytic validation case for gradient-observable evaluation in large logical quantum circuits. The circuit consists of independent two-qubit fragments (0, 1), (2, 3), (4, 5), . . ., so the logical unitary factorizes as a tensor product of two-qubit blocks. In the terminology of the Universal Suppression Law (USL) interpretation of variational trainability collapse, the differentiated quantity is the gradient observable Ai, and the relevant computationalobject is the measured embedded history of Ai. For the PAIRS circuit, this history is exactly contained in a two-qubit natural fragment. A closed-form expression is derived for the two-layer PAIRS observable ⟨Z0⟩ and its gradient. Small-system tests show agreement between ordinary full-register simulation, two-qubit iterative evaluation, and the analytic expression to floating-point precision. A 16384-logical-qubit PAIRS circuit is then evaluated by iterating a two-qubit register over 8192 natural fragments. The large logical calculation agrees tile-by-tile with the analytic formula, with maximum observable and gradient errors 6.661338×10∧−16. The result is not a barren-plateau mitigation experiment and not a quantum-advantage claim. It is a calibration example showing that, in a naturally fragmented circuit, the measured history of Ai rather than the nominal logical Hilbert space determines the exact evaluation.
Variational quantum circuits train poorly on chaotic forecasting, usually blamed on barren plateaus (exponentially vanishing gradients). Using an exactly simulable four-qubit variational quantum physics-informed circuit fit to Lorenz-63, we show the barren-plateau explanation fails: the failure is an architectural capacity ceiling fixed by the circuit time-encoding, not its trainable depth. Four measurements support this. (i) A McClean-comparable gradient-variance estimator sits at the local-cost Haar/2-design scale 2^(-2n)=3.9e-3 at n=4; on structurally live parameters it decays about ninefold with depth then saturates there, large enough to train, not an exponential collapse. (ii) At a common budget of 200 optimiser iterations (600, in three stages, for layer-wise), gradient descent, layer-wise, and SPSA reach the same order of magnitude of loss, so no optimiser unlocks a better basin. (iii) The output-Jacobian rank saturates at 33 from five layers on, so depth buys no new output directions. (iv) A Fourier analysis explains why: the qubit-1 phase encoding acts on the initial |0> and is inert, so the maximum accessible frequency is 2.5/t_max=0.83 Hz, identical at every depth and about 4.4x below the narrowest Lorenz component bandwidth. The corrected band has dimension 1+2x5=11 per observable, and 3x11=33 equals the measured rank ceiling exactly, unifying the two diagnostics. A trained depth sweep agrees: mean loss improves with depth then flattens once the rank saturates. We correct our earlier preprint diagnosis, which compared unnormalised gradient norms to the McClean threshold, and place the advantage of fixed reservoirs and classical echo-state networks in architecture, not quantum mechanics.