Dimensional Expressivity in Parametrized Models

• Dimensional expressivity is a measure of the real dimension of a parametrized model’s reachable set, capturing accessible states or predictive features.
• It uses differential geometric tools such as Jacobian and Gram matrices to identify and prune redundant parameter directions.
• Applications include quantum circuit controllability, neural ODE trajectory capacity, and behavioral predictive modeling.

Searching arXiv for papers on "dimensional expressivity" to ground the article and confirm relevant sources. Dimensional expressivity is a quantitative notion for how many independent directions of variation a parametrized model can realize. In its most explicit formulation, especially for parameterized quantum circuits, it is the real dimension of the manifold of reachable states or unitaries, computed from the maximal rank of a Jacobian or Gram matrix. Closely related constructions appear in controllability analysis for quantum devices, in trajectory-capacity studies for gated neural ODEs, and in predictive formulations of behavioral dimensionality; across these settings, the common theme is that expressivity is not identified with raw parameter count, but with the dimension of the actually accessible set of states, trajectories, or predictive features (Funcke et al., 2020, Gago-Encinas et al., 2023, Kim et al., 2023, Bialek, 2020).

1. Core concept and domain-specific meanings

In the PQC literature, a parametrized circuit $C(\theta)=U(\theta)\lvert \psi_0\rangle$ induces a differentiable manifold of reachable states, and dimensional expressivity is the local real dimension of that manifold, or equivalently the maximum number of independent directions in parameter space along which the circuit can move. The same literature also defines a unitary version in which $U(\theta)\in SU(d)$ is treated as the primary object, and expressivity is the maximal rank of a right-invariant Jacobian in $\mathfrak{su}(d)$ (Gago-Encinas et al., 2023).

In controllability theory, this dimension becomes a diagnostic criterion. Full operator controllability is equivalent to reaching the maximal unitary dimension $d^2-1$, while pure-state controllability is equivalent to reaching the maximal state-manifold dimension $2d-1$. Dimensional expressivity therefore functions as an operational test for universality rather than merely as a descriptive statistic (Gago-Encinas et al., 2023).

Outside quantum computing, analogous constructions appear with modified formal objects. For gated neural ODEs, expressivity is defined through the capacity to reproduce random trajectories, with the critical sequence length $T_c$ playing the role of a dimensional-capacity threshold. In the behavioral literature, dimensionality is defined either by the number of exponentials needed to represent an autocorrelation function or by the minimal number of real-valued features of the past needed to saturate predictive information about the future (Kim et al., 2023, Bialek, 2020).

A useful synthesis is that dimensional expressivity measures the dimension of a reachable or predictive object under a chosen representation. What varies across fields is the object itself: a circuit manifold, a reachable Lie-group orbit, a trajectory family, or a predictive feature space.

2. Manifold-rank formulation in parameterized quantum circuits

For a PQC with parameter manifold $\Theta\subset \mathbb{R}^m$, system Hilbert space $H\cong \mathbb{C}^n$, and initial state $\lvert \psi_0\rangle$, the ansatz defines a smooth map

$$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$

with $U(\theta)\in SU(d)$0. The image $U(\theta)\in SU(d)$1 is the circuit manifold, and the dimensional expressivity is

$U(\theta)\in SU(d)$2

After fixing a gauge and embedding the projective state into a real sphere, one forms the Jacobian $U(\theta)\in SU(d)$3; the key identity is

$U(\theta)\in SU(d)$4

Equivalent state-space formulations flatten real and imaginary parts of $U(\theta)\in SU(d)$5 into a real matrix and use the Gram matrix

$U(\theta)\in SU(d)$6

with $U(\theta)\in SU(d)$7 because $U(\theta)\in SU(d)$8. For unitary families $U(\theta)\in SU(d)$9, the right-invariant Jacobian is

$\mathfrak{su}(d)$0

and one defines $\mathfrak{su}(d)$1. The maximal possible expressivity is $\mathfrak{su}(d)$2 for state-manifolds and $\mathfrak{su}(d)$3 for $\mathfrak{su}(d)$4 (Funcke et al., 2020, Gago-Encinas et al., 2023).

Several standard consequences follow immediately. First, dimensional expressivity is generally smaller than the number of trainable parameters, because local parameter directions can be linearly dependent. Second, the relevant dimension is real rather than complex, reflecting the quotient by global phase. Third, the rank is a local property, but the global expressivity is the maximum local rank. Differential-topological results sharpen this picture: if the Jacobian has constant rank near a point, the image is locally an immersed submanifold, and Sard’s theorem implies that outside a measure-zero set of parameters one attains the maximal rank $\mathfrak{su}(d)$5, so that generically $\mathfrak{su}(d)$6 (Barzen et al., 22 Jul 2025).

Elementary one-qubit examples illustrate the distinction between parameters and physical degrees of freedom. The ansatz $\mathfrak{su}(d)$7 is physically constant and has $\mathfrak{su}(d)$8. The ansatz $\mathfrak{su}(d)$9 has rank $d^2-1$0. The two-parameter ansatz $d^2-1$1 reaches the full Bloch sphere and has generic rank $d^2-1$2 (Barzen et al., 22 Jul 2025).

3. Computation, redundancy detection, and ansatz design

Dimensional expressivity is not only definitional; it yields a concrete design workflow. In the state-based formulation, one computes or estimates the Gram matrix $d^2-1$3 and uses its rank to classify parameter directions as independent or redundant. An inductive pruning procedure initializes an empty set of kept parameters and then tests each new parameter by augmenting the current sub-Gram matrix. If the augmented matrix has full rank, the parameter is retained; otherwise it is redundant and can be fixed to a constant. Equivalent implementations monitor a determinant, a Schur complement, or the smallest eigenvalue against a threshold (Funcke et al., 2020, Funcke et al., 2021).

For gate families of the form

$d^2-1$4

the derivative satisfies

$d^2-1$5

so that

$d^2-1$6

These overlaps can be estimated on hardware by an ancilla-assisted circuit. If one prepares

$d^2-1$7

applies a Hadamard to the ancilla, and measures it, the probability of outcome $d^2-1$8 is

$d^2-1$9

Repeating this for all parameter pairs builds the Gram matrix. The reported resource scaling is $2d-1$0 for the quantum part and $2d-1$1 for the classical eigenvalue checks; in the controllability-oriented variant, each $2d-1$2 evaluation requires $2d-1$3 ancilla-flagged circuit runs (Funcke et al., 2020, Gago-Encinas et al., 2023).

This analysis supports both pruning and construction. In a proof-of-principle implementation on IBM hardware, the single-qubit ansatz

$2d-1$4

was found to have a redundant fourth parameter, while an eight-parameter two-qubit EfficientSU2 instance had only its last parameter redundant (Funcke et al., 2020). The same methodology also quantifies what is lost when the reachable manifold is too small: if $2d-1$5 has dimension $2d-1$6, then the best-approximation error for arbitrary target states cannot vanish, and lower bounds can be expressed in terms of the volume element $2d-1$7 over the circuit manifold (Funcke et al., 2021).

Symmetry sectors supply a further refinement. If the target problem lives in a physical subspace, maximal expressivity should be measured relative to that subspace rather than to the full Hilbert space. For translationally invariant $2d-1$8-qubit sectors, the proceedings article gives

$2d-1$9

with the corresponding complex-sector dimension derived from the eigenspace of the translation operator. This allows automated custom circuit construction: one adds generators until the rank saturates the physical dimension, then stops (Funcke et al., 2020, Funcke et al., 2021).

4. Controllability and universality in quantum devices

A central application of dimensional expressivity is the diagnosis of controllability. For a closed quantum system, operator controllability means the ability to implement every $T_c$0; equivalently, the dynamical Lie algebra generated by the drift and control Hamiltonians must be $T_c$1, of dimension $T_c$2. The dimensional-expressivity test reformulates this geometrically. By the Choi–Jamiołkowski isomorphism, one prepares the maximally entangled state

$T_c$3

and considers

$T_c$4

Then

$T_c$5

Similarly, pure-state controllability is tested by checking whether a parametrized state-preparation circuit reaches $T_c$6 independent directions (Gago-Encinas et al., 2023).

The method was illustrated on nearest-neighbour qubit arrays with local controls:

System	Expressivity result	Diagnosis
Four-qubit pure-state example with two local controls	$T_c$7 reached at layer $T_c$8	Pure-state controllable
Four-qubit non-controllable example	Plateau at $T_c$9	Not pure-state controllable
Three-qubit operator example	$\Theta\subset \mathbb{R}^m$0 reached at layer $\Theta\subset \mathbb{R}^m$1	Operator controllable
Three-qubit non-controllable example	Plateau at $\Theta\subset \mathbb{R}^m$2	Not operator controllable

The numerical details are specific. In the controllable four-qubit pure-state example, each layer had three parameters, $\Theta\subset \mathbb{R}^m$3, and the expressivity increased monotonically by $\Theta\subset \mathbb{R}^m$4 per layer until reaching $\Theta\subset \mathbb{R}^m$5 at layer $\Theta\subset \mathbb{R}^m$6. In the corresponding non-controllable four-qubit example, the Lie-rank was $\Theta\subset \mathbb{R}^m$7, the pure-state-Lie-rank was $\Theta\subset \mathbb{R}^m$8, the layered ansatz reached expressivity $\Theta\subset \mathbb{R}^m$9 already at layer $H\cong \mathbb{C}^n$0, and layer $H\cong \mathbb{C}^n$1 added no new independent directions. In the three-qubit operator-controllable case, $H\cong \mathbb{C}^n$2 so $H\cong \mathbb{C}^n$3, and the extended-system ansatz attained $H\cong \mathbb{C}^n$4 at the last parameter of layer $H\cong \mathbb{C}^n$5. Replacing the last control by $H\cong \mathbb{C}^n$6 caused the expressivity to stall at $H\cong \mathbb{C}^n$7, with no new independent directions beyond layer $H\cong \mathbb{C}^n$8 (Gago-Encinas et al., 2023).

This converts dimensional expressivity into a hardware-design tool. By omitting candidate controls and re-running the test, one can identify redundant drives, simplify calibration, and reduce footprint. The same paper notes a trade-off: fewer controls may preserve universality yet lengthen minimal-time implementations, so expressivity and control time need not be optimized by the same design.

5. Trajectory capacity and predictive dimensionality beyond quantum circuits

In gated neural ODEs, the same phrase is adapted to a dynamical setting in which the object of interest is not a state manifold but the capacity to generate complex trajectories. The model is written in discrete time as

$H\cong \mathbb{C}^n$9

and the authors define a Gardner-type solution volume

$\lvert \psi_0\rangle$0

for fitting an i.i.d. random trajectory $\lvert \psi_0\rangle$1. The critical sequence length $\lvert \psi_0\rangle$2 at which $\lvert \psi_0\rangle$3 vanishes is identified with expressivity, and in experiments this is probed by the training mean-squared error

$\lvert \psi_0\rangle$4

The governing variables are the phase-space dimension $\lvert \psi_0\rangle$5, the complexity of the flow-field parametrization, and the presence or absence of gating. Classical Gardner arguments suggest $\lvert \psi_0\rangle$6 for simple threshold architectures, whereas for multi-layer feedforward parametrizations of $\lvert \psi_0\rangle$7 the capacity can scale with the number of trainable parameters $\lvert \psi_0\rangle$8 rather than only with $\lvert \psi_0\rangle$9. Empirically, on an Ornstein–Uhlenbeck fitting task with $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$0 and $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$1 channels, gnODE achieved the lowest MSE when $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$2 was small; at $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$3 and $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$4, the reported best MSEs were $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$5 for RNN, $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$6 for GRU, $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$7 for nODE, and $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$8 for gnODE. Increasing hidden-layer width lowered MSE almost uniformly, while doubling depth yielded only marginal gains (Kim et al., 2023).

A different but structurally related framework appears in the analysis of behavioral dimensionality. For continuous trajectories, if an observed scalar $$C:\Theta \to \mathbb{P}(H), \qquad C(\theta)=[\,\lvert \psi(\theta)\rangle\,],$$9 is one component of a $U(\theta)\in SU(d)$00-dimensional linear Langevin system, then its autocorrelation takes the form

$U(\theta)\in SU(d)$01

and the minimal dynamical model consistent with that autocorrelation has rank $U(\theta)\in SU(d)$02. More generally, one defines predictive information

$U(\theta)\in SU(d)$03

and then

$U(\theta)\in SU(d)$04

where $U(\theta)\in SU(d)$05 maps the past into $U(\theta)\in SU(d)$06 real features. The behavioral dimensionality is the minimal $U(\theta)\in SU(d)$07 for which the predictive information saturates. If correlations decay as a power law, $U(\theta)\in SU(d)$08, then $U(\theta)\in SU(d)$09, and no finite $U(\theta)\in SU(d)$10 suffices; the effective dimensionality is then infinite (Bialek, 2020).

These examples broaden the concept without making the definitions identical. A plausible interpretation is that dimensional expressivity can be generalized from manifold dimension to the minimal number of independent degrees of freedom needed either to generate a trajectory family or to retain all predictive structure of a process.

6. Distinction from adjacent expressivity formalisms

The recent expressivity literature contains several mathematically precise notions that are adjacent to, but distinct from, dimensional expressivity in the strict manifold-rank sense. In transformer theory, self-attention has been modeled as a vector-valued tropical rational map, with zero-temperature self-attention exactly equivalent to a Power Voronoi Diagram. On that basis, multi-head aggregation increases polyhedral complexity from the $U(\theta)\in SU(d)$11 bottleneck of a single head to $U(\theta)\in SU(d)$12 in the regime $U(\theta)\in SU(d)$13, and deep transformers were shown to have $U(\theta)\in SU(d)$14 linear regions in the tropical limit (Su et al., 16 Apr 2026). This is a combinatorial and polyhedral notion of expressivity, not a manifold-dimension criterion.

A related tropical-geometric analysis of Top-$U(\theta)\in SU(d)$15 Mixture-of-Experts identifies the router with the $U(\theta)\in SU(d)$16-th elementary symmetric tropical polynomial, so that the normal fan of the $U(\theta)\in SU(d)$17-hypersimplex partitions input space into exactly $U(\theta)\in SU(d)$18 convex cones. Under the manifold hypothesis, the effective capacity of dense networks collapses to $U(\theta)\in SU(d)$19, whereas MoE retains a $U(\theta)\in SU(d)$20 combinatorial multiplier, a property described as “combinatorial resilience” (Su et al., 3 Feb 2026). Here again the central quantity is routing multiplicity and region count rather than the dimension of a reachable manifold.

In high-dimensional stochastic control, “expressivity” is used in an approximation-theoretic sense. For optimal switching, deep ReLU networks were shown to approximate the continuation-value recursion with size bounded by $U(\theta)\in SU(d)$21, where $U(\theta)\in SU(d)$22, $U(\theta)\in SU(d)$23, and $U(\theta)\in SU(d)$24 are independent of the state dimension $U(\theta)\in SU(d)$25 and the tolerance $U(\theta)\in SU(d)$26; the result is formulated as a curse-of-dimensionality-free expressivity theorem (Ye et al., 9 Apr 2026). DeepMartingale proves an analogous bound $U(\theta)\in SU(d)$27 for the dual optimal-stopping upper bound (Ye et al., 13 Oct 2025). These results quantify approximation complexity in high dimension, not dimensional expressivity as rank.

A third neighboring usage appears in latent-variable modeling. In hyperspherical VAEs, the von Mises–Fisher latent family was argued to suffer a “hyperspherical bottleneck” because a single scalar concentration parameter $U(\theta)\in SU(d)$28 must control concentration in all directions as ambient dimension grows. Replacing one high-dimensional sphere by a product of lower-dimensional spheres introduces multiple $U(\theta)\in SU(d)$29 and restores per-dimension expressivity (Davidson et al., 2019). This is dimension-sensitive, but it concerns flexibility of a latent distribution family rather than the manifold dimension of reachable states.

The broader implication is terminological as much as technical. “Expressivity” may refer to manifold dimension, Lie-orbit dimension, trajectory capacity, predictive-feature dimension, polyhedral region count, routing multiplicity, or approximation rate. Dimensional expressivity, in the narrowest and most stable usage, is the dimension of the actually accessible set under a parametrized representation; the surrounding literature shows how this core idea is repeatedly adapted to different mathematical objects and different operational questions.