Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information $F_Q$, the Fisher information $F_{\rm full}$ in the complete bitstring distribution, and the largest variance-normalized response $\mathcal I_{\mathcal A}$ available to a diagonal readout space $\mathcal A$. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are $1/2$ and $r/(2^n-1)$, where $r$ is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(n^k2^{-n})$ of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.
Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information FQ, the Fisher information Ffull in the complete bitstring distribution, and the largest variance-normalized response IA available to a diagonal readout space A. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are 1/2 and r/(2n−1), where r is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight k retain only O(nk2−n) of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.
A fixed quantum measurement can expose substantially more tangent information than a restricted observable readout retains. We study this second restriction. For a normalized covariance C⪰0, TrC=1, of measurement-induced tangent scores in an N-dimensional centered score space, and a rank-r readout projector P, we quantify retained tangent mass by R=Tr(PC). The ratio ρ=R/(r/N) separates the actual retained mass from a rank-only random-orientation reference. Standard Grassmann averaging gives Eρ=1 and Var(ρ)≤2/(rdeff), where deff=1/Tr(C2). We use this identity as a null model rather than as a new random-projection theorem. Numerically, family-balanced one- and two-body readouts remain close to the rank reference through n=16 even as the tangent covariance becomes strongly anisotropic. The decisive equal-rank comparison holds the circuit, measurement record, readout rank, and evaluation shot budget fixed. For Haar-U(4) at n=12, cross-fitted alignment increases the mean directional gradient-energy proxy by a factor 9.584 and the finite-shot signal-to-noise ratio by a factor 3.111 relative to the physical one-body readout, while a random rank-matched subspace remains near the rank baseline. A half-filled U(1)-conserving family provides a structured counterexample to generic orientation: physical low-weight Z readouts are already strongly aligned with leading tangent directions over the tested finite-size range. We treat this symmetry result as a case study, not as a claim that U(1) symmetry generically prevents barren plateaus or that hydrodynamics is the established mechanism. The results isolate readout orientation as a degree of freedom invisible to rank alone that directly controls how much measured tangent information remains usable after readout restriction.
Barren plateaus hinder training of parameterized quantum circuits by making loss gradients exponentially small in the number of qubits. For angle-encoded product states, we show how the input distribution affects output variation through algebraic input purity, i.e. the input's overlap with the circuit's dynamical Lie algebra. With the specified readouts and Haar or exact group 2-design sampling, matchgate circuits on n qubits retain output variance of order 1/n for every pure product input. The off-diagonal family instead has zero output on computational-basis inputs at any depth and for every parameter choice. Independent uniform angles yield mean variance of the same order as matchgates. A count of diagonal Pauli strings identifies these zero-output endpoints. For any fixed dataset of nonzero inputs with at least 2n coordinates, we construct a classical preprocessing certificate. A shared random rotation is accepted only when the dataset's mean purity passes a computable threshold. This certifies mean output variance of order 1/n for the off-diagonal family before circuit execution, with a constant expected number of trials. Binary and ternary weighted encodings also recover the independent-angle mean purity from one uniform scalar input. Numerical experiments examine training behavior and extensions to larger algebras. The guarantee concerns output variance averaged over inputs, while successful learning also depends on the task and information preserved by the encoding.
Quantum Fisher information (QFI) determines the ultimate precision of a quantum sensor after optimization over all allowed measurements, but it does not quantify how difficult the optimal readout is to implement. We introduce a moment Fisher hierarchy FM(k), which measures the fraction of QFI accessible to collective spin moments up to degree k, together with a readout-complexity scale k90 defined by the minimum degree required to recover 90% of the QFI. Applying this framework to a nonlinear collective-spin sensor, we show that many-body dynamics can preserve large QFI while transferring the optimal readout into high-rank SU(2) tensor sectors inaccessible to low-order collective observables. For system sizes up to N=100, first moments access only ∼10−2 of the available QFI, while the degree required to recover 90% of it grows to k90≃28. A rank-resolved analysis of the symmetric logarithmic derivative (SLD) under weak damping shows that the nonlinear protocol pushes the optimal readout toward the highest available tensor ranks while strongly depleting its low-rank weight. These results identify nonlinear tangent-space shear as the mechanism behind a finite-size readout-complexity bottleneck in many-body quantum metrology.
Understanding how the performance of parameterized quantum circuits scales with available resources is important for characterizing their trainability and effective model capacity. In this study, we numerically investigate data scaling, model scaling, and compute scaling in parameterized quantum circuits and examine Lie-algebraic quantities as alternative measures of model size. In addition to the number of circuit parameters, we consider the dimension of the dynamical Lie algebra, the observable-orbit dimension, and a Jacobian effective dimension defined as the rank of the Jacobian of the parameterized observable orbit. Using a regression task with randomly generated Pauli-string generators, we observe decreasing loss with increasing training dataset size, parameter size, and number of optimization iterations over the ranges investigated. For model scaling, the dynamical Lie algebra and observable orbit dimensions rapidly saturate as the parameter size increases, whereas the Jacobian effective dimension remains strongly correlated with the parameter size and exhibits comparable scaling behavior. These results suggest that the Jacobian effective dimension provides a geometry-aware measure of the locally accessible observable degrees of freedom of finite-depth parameterized quantum circuits and may serve as a useful scaling parameter beyond the nominal parameter size.
How a quantum state responds to small parameter changes underlies state distinguishability, many-body response, and metrological sensitivity. The Pauli probability spectrum provides a natural description of many-body quantum states, but it is not evident how much of this response survives in the corresponding classical distribution. Here we show that, for pure states, the complete labeled Pauli distribution retains the full local metric of the pure-state manifold, reproducing the quantum Fisher metric up to a fixed normalization. This correspondence has a direct measurement realization: preparing a state together with its complex conjugate and performing pairwise Bell measurements between corresponding degrees of freedom converts the quantum response into an ordinary classical distribution without losing any of the sensitivity available in the two inputs. The readout is fixed, factorizes into local pairwise measurements even for strongly entangled many-body states, and preserves the response along all parameter directions simultaneously. We trace this correspondence to the reality of the underlying operator amplitudes and show that the same mechanism extends beyond qubits. When the correspondence fails, we identify explicitly where the missing information resides: part of the response can remain hidden in phases for identical-copy and complex-coordinate readouts, while mixed states develop transverse operator-space motion that is invisible to the Bell probabilities. These results provide a direct link between Pauli/Bell statistics, many-body response, nonstabilizerness, and quantum geometry.
Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every n, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared n-qubit fiducial register A with parallel Bell readout of A and the unknown system S, producing one 2n-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the 3n local-Pauli settings conventionally used for complete reconstruction while retaining the minimum d2 outcomes with d=2n. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For n≥3, worst-state Pauli variances remain bounded in the axial sector and scale as Θ(d) in the mixed sector. Fiducial preparation before system contact uses n−1 arbitrary two-qubit gates, no work qubits beyond A, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves O(1) adaptive quantum depth using n+1 extra qubits in the heralded route, or O(nlogn) auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.