Data-dependent generalization bounds for parameterized quantum models under noise

Abstract

Quantum machine learning offers a transformative approach to solving complex problems, but the inherent noise hinders its practical implementation in near-term quantum devices. This obstacle makes it challenging to understand the generalization capabilities of quantum circuit models. Designing robust quantum machine learning models under noise requires a principled understanding of complexity and generalization, extending beyond classical capacity measures. This study investigates the generalization properties of parameterized quantum machine learning models under the influence of noise. We present a data-dependent generalization bound grounded in the quantum Fisher information matrix. We leverage statistical learning theory to relate the parameter space volumes and training sizes to estimate the generalization capability of the trained model. By integrating local parameter neighborhoods and effective dimensions defined through quantum Fisher information matrix eigenvalues, we provide a structured characterization of complexity in quantum models. We analyze the tightness of the bound and discuss the trade-off between model expressiveness and generalization performance.

Appendices

Appendix A

1.1 A.1: Parameter space geometry

In the main text, we introduced the Fisher information matrix (FIM) ${\mathcal {F}}(\theta )$ as a Riemannian metric on the parameter space $\Theta \subset {\mathbb {R}}^d$. This induces a natural geometric structure on $\Theta$: the geodesic distance between $\theta$ and $\theta ^{\prime }$ measures how different these parameter points are in terms of their influence on the model’s predictions, rather than just their Euclidean distance.

For a parameterized model and its associated FIM, we consider the volume element induced by ${\mathcal {F}}(\theta )$:

$$\begin{aligned} dV(\theta ) = \sqrt{\det ({\mathcal {F}}(\theta ))}\, d\theta ^1 \cdots d\theta ^d. \end{aligned}$$

(A1)

A geodesic ball $B(\theta ,\epsilon )$ of radius $\epsilon$ centered at $\theta$ has volume

$$\begin{aligned} V(\theta ,\epsilon ) = \int _{B(\theta ,\epsilon )} dV(\theta ^{\prime }) = \int _{B(\theta ,\epsilon )} \sqrt{\det ({\mathcal {F}}(\theta ^{\prime }))}\, d\theta ^{\prime }. \end{aligned}$$

(A2)

If $\sqrt{\det ({\mathcal {F}}(\theta ))}$ is bounded by a positive constant $m>0$, then for small $\epsilon$, we can approximate:

$$\begin{aligned} V(\theta ,\epsilon ) \approx V_d \epsilon ^d \sqrt{\det ({\mathcal {F}}(\theta ))} \ge V_d \epsilon ^d m, \end{aligned}$$

(A3)

where

$$\begin{aligned} V_d = \frac{\pi ^{d/2}}{\Gamma \left( \tfrac{d}{2} + 1\right) } \end{aligned}$$

is the volume of the unit ball in ${\mathbb {R}}^d$. Intuitively, a lower bound on $\sqrt{\det ({\mathcal {F}}(\theta ))}$ ensures that geodesic balls are not “too small,” implying fewer truly distinct parameter configurations at $\epsilon$. This geometric insight underpins the covering number bound we discuss next.

1.2 A.2: Covering numbers of a parameter space

The covering number of a parameter space $\Theta$ measures how many balls of radius $\epsilon$ are required to cover the entire space. Since the FIM provides a natural measure of distinguishability, a lower bound on $\sqrt{\det ({\mathcal {F}}(\theta ))}$ relates volumes of small balls to parameter distinguishability.

Lemma A.1

(Covering Number and Volume) Let $\Theta \subset {\mathbb {R}}^d$ be a compact parameter space with volume $V_\Theta$. Assume that the determinant of the Fisher Information Matrix ${\mathcal {F}}(\theta )$ satisfies:

$$\begin{aligned} \sqrt{\det ({\mathcal {F}}(\theta ))} \ge m> 0, \quad \forall \theta \in \Theta . \end{aligned}$$

Then, for any $\epsilon> 0$, the covering number $N(\epsilon ,\Theta ,||\cdot ||)$ of $\Theta$ with respect to the Euclidean norm $||\cdot ||$ satisfies:

$$\begin{aligned} \log N(\epsilon ,\Theta ,||\cdot ||) \le C - d \log \epsilon , \end{aligned}$$

(A4)

where $C = \log V_\Theta - \log V_d - \log m$ and $V_d = \frac{\pi ^{\frac{d}{2}}}{\Gamma \left( \frac{d}{2} + 1\right) }$ is the volume of a unit ball in ${\mathbb {R}}^d$.

Proof

Consider an $\epsilon$-ball around $\theta$:

$$\begin{aligned} V(\theta ,\epsilon ) \ge V_d \epsilon ^d m. \end{aligned}$$

To cover $\Theta$, the total volume of N such balls must be at least $V_\Theta$. So we have:

$$\begin{aligned} N (\epsilon , \Theta , ||\cdot ||) \le \frac{V_\Theta }{V_d \epsilon ^d m}. \end{aligned}$$

(A5)

Taking the logarithm:

$$\begin{aligned} \log N(\epsilon , \Theta , | \cdot | )&\le \log V_\Theta - \log \left( V_d \epsilon ^d m \right) \\&= \log V_\Theta - \log V_d - \log m - d \log \epsilon \\&= C - d \log \epsilon . \end{aligned}$$

This completes the proof of Lemma (A.1). $\square$

The inequality (A4) shows how covering numbers scale with dimension d and resolution $\epsilon$. Smaller $\epsilon$ or larger d requires more balls, thus more complex model classes. Conversely, a larger m reduces complexity by ensuring a certain “geometric rigidity” in the parameter space.

Appendix B

1.1 B.1: Bounding empirical Rademacher complexity

We now relate covering numbers to the empirical Rademacher complexity $\hat{{\mathcal {R}}}_N({\mathcal {F}}_\Theta )$ of the model class ${\mathcal {F}}_\Theta = \{f_{\theta ,p}:\theta \in \Theta \}$. The Rademacher complexity quantifies the model’s capacity to fit random noise and thus provides an upper bound on generalization error.

Lemma B.1

(Empirical Rademacher Complexity Bound) Let ${\mathcal {F}}_\Theta = \{f_{\theta ,p}: \theta \in \Theta \}$ be a model function class. Let ${\mathcal {D}} = \{x_i,y_i\}_{i=1}^N$ be a dataset of N samples drawn from the distribution P. Assume that $f_{\theta ,p}$ is bounded by the Lipschitz constant $L_f^p$ and the determinant of the quantum Fisher information matrix is at least m. Then the empirical Rademacher complexity of the model class ${\mathcal {F}}_\Theta$ is bounded by:

$$\begin{aligned} {\hat{ \mathcal R}}_N({\mathcal {F}}_\Theta ) \le \frac{6 \sqrt{\pi d} \exp \left( \frac{C^\prime }{d} \right) }{\sqrt{N}} \end{aligned}$$

(B6)

where $C^\prime = \log V_\Theta - \log V_d - \log m + d \log L_f^p$.

Proof

Given a dataset ${\mathcal {D}} = \{x_i,y_i\}_{i=1}^N$ the empirical Rademacher complexity of a model class ${\mathcal {F}}_\Theta$ is defined as:

$$\begin{aligned} {\hat{\mathcal R}}_N({\mathcal {F}}_\Theta ) = {\mathbb {E}}_{\sigma } \left[ \sup \limits _{f \in F_\Theta } \sum \limits _{i=1}^N \sigma _i f(x_i) \right] , \end{aligned}$$

(B7)

where $\sigma = \{\sigma _i\}_{i=1}^N$ are i.i.d Rademacher random variables taking values in $\{-1,+1\}$ with equal probabilities. Dudley’s entropy integral provides an upper bound on the empirical Rademacher complexity in terms of the covering numbers. If we let $\Vert \cdot \Vert _{2,{\mathcal {D}}}$ be the empirical $L_2$-norm over the sample ${\mathcal {D}}$, we have:

$$\begin{aligned} {\hat{R}}_N({\mathcal {F}}_\Theta ) \le \frac{12}{\sqrt{N}} \int _0^{\epsilon _{max}} \sqrt{\log N(\epsilon , {\mathcal {F}}_\Theta , ||\cdot ||_{2,{\mathcal {D}}})} \, d\epsilon . \end{aligned}$$

(B8)

where $N(\epsilon , {\mathcal {F}}_\Theta , ||\cdot ||_{2,{\mathcal {D}}})$ is the covering number of ${\mathcal {F}}_\Theta$ with respect empirical $L_2$ metric: $||f - g ||_{2,{\mathcal {D}}} = \left( \frac{1}{N} \sum \limits _{i=1}^N (f(x_i) - g(x_i)) \right) ^{\frac{1}{2}}$ and $\epsilon _{max} = \sup \limits _{f \in {\mathcal {F}}_\Theta } ||f||_{2,{\mathcal {D}}}$. Since $f_{\theta ,p}$ is bounded by the Lipschitz constant $L_f^p$, we can write: $||f_{\theta ,p} - f_{\theta ^{\prime },p}||_{2,{\mathcal {D}}} \le L_f^p ||\theta - \theta ^{\prime }||, \quad \forall \theta , \theta ^{\prime } \in \Theta$. This implies that an $\epsilon$-cover of $\Theta$ with respect to the Euclidean norm $||\cdot ||$ is also an $\epsilon L_f^p$-cover of ${\mathcal {F}}_\Theta$ with respect to the empirical $L_2$ metric. Using Lemma A.1, we can write:

$$\begin{aligned}&\log N(\epsilon , {\mathcal {F}}_\Theta , ||\cdot ||_{2,{\mathcal {D}}}) \le \log N(\frac{\epsilon }{L_f^p}, \Theta , ||\cdot ||)\nonumber \\&\quad \le C - d \log \left( \frac{\epsilon }{L_f^p}\right) = C - d \left( \log \epsilon - \log L_f^p \right) \nonumber \\&\quad \implies \log N(\epsilon , {\mathcal {F}}_\Theta , ||\cdot ||_{2,{\mathcal {D}}}) \le C^\prime - d \log \epsilon , \end{aligned}$$

(B9)

Substituting this result into Eq. (B8) gives:

$$\begin{aligned} {\hat{R}}_N({\mathcal {F}}_\Theta ) \le \frac{12}{\sqrt{N}} \int _0^{\epsilon _{max}} \sqrt{C^\prime - d \log \epsilon } \, d\epsilon . \end{aligned}$$

(B10)

We perform the elementary calculus in the following derivations and if the reader trusts our calculations, they may skip the following steps.

Let $t = - \log \epsilon$, so $\epsilon = e^{-t}$ and $d\epsilon = -e^{-t} dt$. The limit changes as: when $\epsilon = \epsilon _{max}, t = \log \epsilon _{max}$ and as $\epsilon$ decreases from $\epsilon _{max}$ to 0, t increases from $-\log \epsilon _{max}$ to 0. Thus:

$$\begin{aligned} {\hat{R}}_N({\mathcal {F}}_\Theta )&\le \frac{12}{\sqrt{N}} \int _{-\log \epsilon _{max}}^0 \sqrt{C^\prime - d \log e^{-t}} \, -e^{-t} dt \end{aligned}$$

Assuming $\epsilon _{max} = 1$ (since $f_{\theta ,p}(x)$ is bounded in [0,1]), we have $t=0$ at $\epsilon = 1$, so:

$$\begin{aligned} {\hat{R}}_N({\mathcal {F}}_\Theta )&\le \frac{12}{\sqrt{N}} \int _{0}^{\infty } \sqrt{C^\prime + d t} \, e^{-t} dt \end{aligned}$$

To make the derivation easier let us define:

$$\begin{aligned} I = \int _{0}^{\infty } \sqrt{C'+dt} e^{-t} dt \end{aligned}$$

(B11)

Let $S = C' +dt$, then $t = \frac{S-C'}{d}$ and $\frac{dt}{dS} = \frac{d}{dS} \left( \frac{S - C'}{d} \right) \implies dt = \frac{dS}{d}$ and $e^{-t} = e^{-\left( \frac{S-C'}{d}\right) } = e^{\frac{C'}{d}}e^{\frac{-S}{d}}$. Since, $C' \text { and } d$ are constants, we can write:

$$\begin{aligned} I = \int _{C'}^{\infty } \sqrt{S} e^{\frac{C'}{d}} e^{\frac{-S}{d}} \frac{dS}{d} = \frac{e^{\frac{C'}{d}}}{d} \int _{C'}^{\infty } \sqrt{S} e^{\frac{-S}{d}} dS \end{aligned}$$

Let $u = \frac{s}{d} \implies s = du \text { and } ds = d du$. Thus,

$$\begin{aligned} I = \frac{e^{\frac{C'}{d}}}{d} \int _{\frac{C'}{d}}^{\infty } \sqrt{du} e^{-u} d(du) = e^{\frac{C'}{d}} \sqrt{d} \int _{\frac{C'}{d}}^{\infty } u^{\frac{1}{2}} e^{-u} du \end{aligned}$$

This integral is the gamma function and can be written as:

$I = {e^{\frac{C'}{d}}}\sqrt{d} \Gamma \left( \frac{3}{2}, \frac{C'}{d}\right)$.

Since $\Gamma (s,a) \le \Gamma (s)$ for $a> 0$, and $\Gamma \left( \frac{3}{2}\right) \ frac{\sqrt{\pi }}{2}$ we can write:

$$\begin{aligned} I \le {e^{\frac{C'}{d}}}\sqrt{d} \frac{\sqrt{\pi }}{2} \end{aligned}$$

(B12)

Substituting this back into Eq. (B8), we get:

$$\begin{aligned} \hat{{\mathcal {R}}}_N({\mathcal {F}}_\Theta ) \le \frac{6\sqrt{\pi d} \quad e^{\left( \frac{C'}{d}\right) }}{\sqrt{n}} \end{aligned}$$

(B13)

This completes the proof of the Lemma (B.1). $\square$

This lemma encapsulates how geometry (via quantum FIM) and parameter space volume control Rademacher complexity. As N grows, the complexity term diminishes, indicating improved generalization potential.

Next, we discuss how the Rademacher complexity bound can be used to derive a generalization bound for quantum machine learning models.

1.2 B.2: Generalization bound

We now derive the generalization bound proposed in Theorem 5.1 in the main text. The bound follows from standard statistical learning theory and the Rademacher complexity result obtained above.

Theorem B.2

(Restatement of Theorem 5.1) Let $d,N \in {\mathbb {N}}, \delta \in (0,1)$, and consider a parameter space $\Theta \subset {\mathbb {R}}^d$. The quantum model class ${\mathcal {F}}_\Theta =\{f{\theta ,p}:\theta \in \Theta \}$ with noise parameter $p\in [0,1)$ satisfies $f_{\theta ,p}(x)=\eta (p)f_\theta (x)$ with $\eta (0)=1$. Assume:

• The loss $l:{\mathcal {Y}}\times {\mathbb {R}}\rightarrow [0,1]$ is Lipschitz continuous in its second argument with constant $L \le 1$.

• The model gradients are bounded as: $\Vert \nabla _\theta f_{\theta ,p}(x)\Vert \le L_f^p$.

• The quantum FIM satisfies $\sqrt{\det ({\mathcal {F}}(\theta ))}\ge m>0$.

Define $C^{\prime } = \log V_\Theta - \log V_d - \log m + d \log L_f^p$.

Then with probability at least 1-$\delta$ over an i.i.d. sample $D=\{x_i,y_i\}_{i=1}^N$ of size N,

$$\begin{aligned} R(\theta ) \le {\hat{R}}_N(\theta ) + \frac{12\sqrt{\pi d}\exp (C^\prime /d)}{\sqrt{N}} + 3\sqrt{\frac{\log (2/\delta )}{2N}}, \end{aligned}$$

(B14)

uniformly for all $\theta \in \Theta$.

Proof

A standard result in learning theory states that with probability at least $1-\delta$:

$$\begin{aligned} R(\theta ) \le {\hat{R}}_N(\theta ) + 2\hat{{\mathcal {R}}}N({\mathcal {L}}) + 3\sqrt{\frac{\log (2/\delta )}{2N}}, \end{aligned}$$

(B15)

where ${\mathcal {L}}=\{(x,y)\mapsto l(y,f_{\theta ,p}(x)):\theta \in \Theta \}$.

Since $l(y,{\hat{y}})$ is Lipschitz with constant L and $f_{\theta ,p}(x)$ is bounded by $L_f^p$, it follows that ${\mathcal {L}}$ has Rademacher complexity at most $L\cdot L_f^p$. Applying Lemma B.1:

$$\begin{aligned} \hat{{\mathcal {R}}}_N({\mathcal {L}}) \le L L_f^p \frac{6\sqrt{\pi d}\text { exp}{\left( C^{\prime }/d\right) }}{\sqrt{N}}. \end{aligned}$$

For $L\le 1$, this simplifies directly to:

$$\begin{aligned} \hat{{\mathcal {R}}}_N({\mathcal {L}}) \le \frac{6\sqrt{\pi d}\exp{\left( C^{\prime }/d\right) }}{\sqrt{N}}. \end{aligned}$$

Substitute back into the generalization inequality:

$$\begin{aligned} R(\theta ) \le {\hat{R}}_N(\theta ) + 2\cdot \frac{6\sqrt{\pi d}\text { exp}{\left( C^{\prime }/d\right) }}{\sqrt{N}} + 3\sqrt{\frac{\log (2/\delta )}{2N}}. \end{aligned}$$

This gives:

$$\begin{aligned} R(\theta ) \le {\hat{R}}_N(\theta ) + \frac{12\sqrt{\pi d}\text { exp}{\left( C^{\prime }/d\right) }}{\sqrt{N}} + 3\sqrt{\frac{\log (2/\delta )}{2N}}, \end{aligned}$$

matching the statement of Theorem 5.1. $\square$

This completes the proof of the generalization bound for Theorem 5.1. The interplay between Fisher information geometry, covering numbers, and Rademacher complexity provides a path from parameter space properties to explicit generalization guarantees. In practice, after training, restricting to local regions or reducing dimension to the effective dimension often yields sharper bounds.