The Universal Suppression Law identifies causal depth, acting through entanglement, as the mechanism governing observable suppression. This paper tests its circuit-level consequences and does not rederive it. Exact identity cancellation separates nominal depth from the effective initialisation query: increasing nominal depth fiftyfold to 300 leaves the initialisation gradient moments unchanged within sampling scatter and the output purity at one, whereas a random control loses purity and gradient together by nominal depth 48. That purity is a diagnostic of net uncomputation and is not identified with the purity entering the gradient bound. Raw gate count is therefore not a sufficient diagnostic of suppression. Before saturation, R´enyi-2 entropy grows approximately linearly with causal depth; fits over depths 6–14, at fixed register size, over predict held-out entropy at depth 16 by 5–6% for registers of 8, 10 and 12 qubits, with an effective growth rate that resolves with register size and does not resolve with gate set at the sample sizes used. This supports identifying the framework’s area with circuit entanglement entropy, but does not establish equality with the law’s structural constant. The law’s third link is a Hilbert–Schmidt inequality that is valid but weak: it is slack by two to three orders of magnitude, and the ratio of gradient variance to purity is not constant over the tested range. Bounding causal depth removes an upper-bound exponential factor; it does not guarantee a nonzero gradient, since state–observable symmetries can still force exact cancellation. In the tested families, exactly cancelling fragments compose at initialisation without accumulated suppression, and all site-resolved local-cost gradients of a fully site-dependent 20-site ring are reproduced from width-17 windows to machine precision, whereas width 15 fails
The Universal Suppression Law identifies causal depth, acting through entanglement, as the mechanism governing observable suppression. This paper tests its circuit-level consequences and does not rederive it. Exact identity cancellation separates nominal depth from the effective initialisation query: increasing nominal depth fiftyfold to 300 leaves the initialisation gradient moments unchanged within sampling scatter and the output purity at one, whereas a random control loses purity and gradient together by nominal depth 48. That purity is a diagnostic of net uncomputation and is not identified with the purity entering the gradient bound. Raw gate count is therefore not a sufficient diagnostic of suppression. Before saturation, R´enyi-2 entropy grows approximately linearly with causal depth; fits over depths 6–14, at fixed register size, over predict held-out entropy at depth 16 by 5–6% for registers of 8, 10 and 12 qubits, with an effective growth rate that resolves with register size and does not resolve with gate set at the sample sizes used. This supports identifying the framework’s area with circuit entanglement entropy, but does not establish equality with the law’s structural constant. The law’s third link is a Hilbert–Schmidt inequality that is valid but weak: it is slack by two to three orders of magnitude, and the ratio of gradient variance to purity is not constant over the tested range. Bounding causal depth removes an upper-bound exponential factor; it does not guarantee a nonzero gradient, since state–observable symmetries can still force exact cancellation. In the tested families, exactly cancelling fragments compose at initialisation without accumulated suppression, and all site-resolved local-cost gradients of a fully site-dependent 20-site ring are reproduced from width-17 windows to machine precision, whereas width 15 fails
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
The Universal Suppression Law, derived within the Quantum Entanglement Spacetime Theory (QuEST) framewor, is applied to the gradient observable of parameterized quantum circuits. Conditional on a QuEST-compliant embedding of the gradient observable into a history of finite causal depth and on an operator-norm reformulation of the suppression bound introduced as an application-level inequality, the variance of the cost-function gradient under symmetric parameter initialization is shown to be bounded by a product of an operator-norm prefactor and an exponential factor in the embedded causal depth. For ansatz families in which the causal depth of the gradient observable grows linearly with qubit count, this bound reproduces the barren plateau scaling of the Variational Quantum Eigensolver (VQE). Five previously distinct analyses of the barren plateau --- random-circuit concentration, cost-function locality, causal cone width, noise-induced collapse, and Lie-algebraic dimension --- are interpreted within the same suppression framework under explicit identification of their respective regimes. The interpretation extends to the Quantum Approximate Optimization Algorithm (QAOA), Adaptive Derivative-Assembled Pseudo-Trotter VQE (ADAPT-VQE), Variational Adiabatic Quantum Computing (VAQC), and quantum machine learning, whenever the effective causal region of the gradient query grows extensively with problem size. A structural criterion for evaluating proposed mitigations is obtained within this framework: a mitigation can alleviate trainability collapse only by reducing causal depth or the suppression constant.
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.
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.
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.
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.