Quantum Reservoir Computing and Risk Bounds (2501.08640v1)

Abstract: We propose a way to bound the generalisation errors of several classes of quantum reservoirs using the Rademacher complexity. We give specific, parameter-dependent bounds for two particular quantum reservoir classes. We analyse how the generalisation bounds scale with growing numbers of qubits. Applying our results to classes with polynomial readout functions, we find that the risk bounds converge in the number of training samples. The explicit dependence on the quantum reservoir and readout parameters in our bounds can be used to control the generalisation error to a certain extent. It should be noted that the bounds scale exponentially with the number of qubits $n$. The upper bounds on the Rademacher complexity can be applied to other reservoir classes that fulfill a few hypotheses on the quantum dynamics and the readout function.

• The paper derives parameter-dependent generalization error bounds via Rademacher complexity for quantum reservoir classes.
• It employs polynomial readout functions to establish explicit bounds for both PTR and RRR reservoir models while analyzing qubit count scaling.
• The findings demonstrate convergence of risk bounds with increased training samples and highlight limitations due to exponential qubit scaling.

The paper addresses the problem of bounding the generalization errors in quantum reservoir computing using Rademacher complexity. The authors derive parameter-dependent bounds for two specific quantum reservoir classes and analyze how these bounds scale with an increasing number of qubits. The paper focuses on quantum reservoirs with polynomial readout functions, demonstrating that the risk bounds converge as the number of training samples grows.

The paper is structured as follows:

1. Introduction: Introduces the topic of quantum reservoir computing and the importance of bounding generalization errors.
2. Framework: Sets up the theoretical framework for analyzing quantum reservoirs.
3. General Assumptions: Establishes general assumptions on the data distribution and quantum reservoir classes.
4. Rademacher Complexity Bounds: Derives bounds on the Rademacher complexity for specific readout functions.
5. Specialized Subclasses: Introduces two specialized subclasses of quantum reservoirs.
6. Generalization Bounds: Presents generalization bounds for the quantum reservoir classes under consideration.
7. Discussion: Discusses the implications and limitations of the results.

Key aspects of the work include:

• The authors use Rademacher complexity to quantify the generalization ability of quantum reservoirs. Rademacher complexity is a measure of the richness of a function class, which in this case, is the class of functions implemented by the quantum reservoir.
• The generalization bounds are parameter-dependent, explicitly showing how the error scales with the quantum reservoir and readout parameters. This allows for a degree of control over the generalization error.
• The derived bounds scale exponentially with the number of qubits $n$, which is a limitation. This exponential scaling is a common challenge in quantum machine learning due to the exponentially growing Hilbert space.
• The upper bounds on the Rademacher complexity are applicable to other reservoir classes that satisfy certain hypotheses regarding quantum dynamics and readout functions, suggesting the potential for broader applications.

The authors derive explicit risk bounds for two quantum reservoir classes, parameterized by $R_{\text{max}}$ and $C_{\text{max}}$, denoted as $$\mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{PTR}}$$ and $$\mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{RRR}}$$. For the Polynomial Time Reservoir (PTR) class $$\mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{PTR}}$$, the risk bound is given by:

$$\sup_{H \in \mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{PTR}}} \left| R(H) - \hat{R}\_m(H) \right| \leq \left( C_0^P \left( n, R_{\text{max}}, \epsilon_{\text{PTR}} \right) + C_1 \left( n, R_{\text{max}}, \zeta_{\text{max}} \right) \right) \frac{1}{m} + C_2^P \left( n, R_{\text{max}}, \zeta_{\text{max}}, C_{\text{max}} \right) \frac{ \log{m} }{m} + C_3^P \left( n, R_{\text{max}}, \zeta_{\text{max}}, \left| \Theta_{\text{PTR}} \right|, C_{\text{max}} \right) \sqrt{\frac{ \log{m} }{m}} + C_4^P \left( n, R_{\text{max}}, \zeta_{\text{max}}, \epsilon_{\text{PTR}} \right) \sqrt{ \frac{\log{4/\delta}}{2 m} }$$

where:

• $R(H)$ is the true risk of the hypothesis $H$.
• $\hat{R}\_m(H)$ is the empirical risk on $m$ samples.
• $n$ is the number of qubits.
• $R_{\text{max}}$ and $C_{\text{max}}$ are parameters of the reservoir class.
• $\epsilon_{\text{PTR}}$ is a parameter related to the PTR reservoir.
• $\zeta_{\text{max}} = \max (r, D_{w^y}, D_{w^v})$ with $r$ being the contractivity, $D_{w^y}$ the dissipation of the output, and $D_{w^v}$ the dissipation of the input.
• $\Theta_{\text{PTR}}$ is the parameter set for the PTR reservoir.
• $C_0^P, C_1, C_2^P, C_3^P$, and $C_4^P$ are constants that depend on the parameters of the reservoir class, the data distribution, and the loss function.

For the Randomly Reed-Muller Reservoir (RRR) class $$\mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{RRR}}$$, two scenarios are considered based on the relationship between $r_0$ and $r_1$. If $r_0 = r_1 = r_{0,1}$, the risk bound is:

$$\sup_{H \in \mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{RRR}}} \left| R(H) - \hat{R}\_m(H) \right| \leq \left( C_0^{a,R} \left( n, R_{\text{max}}, \alpha_{\text{min}}, r_{0,1} \right) + C_1 \left( n, R_{\text{max}}, \zeta_{\text{max}} \right) \right) \frac{1}{m} + C_2^R \left( n, R_{\text{max}}, \zeta_{\text{max}}, C_{\text{max}}, \alpha_{\text{min}} \right) \frac{ \log{m} }{m} + C_3^R \left( n, R_{\text{max}}, \zeta_{\text{max}}, \left| \Theta_{\text{RRR}} \right|, C_{\text{max}} \right) \sqrt{\frac{ \log{m} }{m}} + C_4^{a, R} \left( n, R_{\text{max}}, \zeta_{\text{max}}, \alpha_{\text{min}}, r_{0,1} \right) \sqrt{ \frac{\log{4/\delta}}{2 m} }$$

If $r_0 > r_1$, the risk bound is:

$$\sup_{H \in \mathcal{H}_{n, R_{\text{max}}, C_{\text{max}}}^{\text{RRR}}} \left| R(H) - \hat{R}\_m(H) \right| \leq \left( C_0^{b, R} \left( n, R_{\text{max}}, \epsilon_{\text{RRR}} \right) + C_1 \left( n, R_{\text{max}}, \zeta_{\text{max}} \right) \right) \frac{1}{m} + C_2^R \left( n, R_{\text{max}}, \zeta_{\text{max}}, C_{\text{max}}, \alpha_{\text{min}} \right) \frac{ \log{m} }{m} + C_3^R \left( n, R_{\text{max}}, \zeta_{\text{max}}, \left| \Theta_{\text{RRR}} \right|, C_{\text{max}} \right) \sqrt{\frac{ \log{m} }{m}} + C_4^{b, R} \left( n, R_{\text{max}}, \zeta_{\text{max}}, \alpha_{\text{min}}, \epsilon_{\text{RRR}} \right) \sqrt{ \frac{\log{4/\delta}}{2 m} }$$

where:

• $\alpha_{\text{min}}$ is a parameter related to the RRR reservoir.
• $r_{0,1}$ is a contractivity parameter when $r_0 = r_1$.
• $\epsilon_{\text{RRR}}$ is a parameter related to the RRR reservoir when $r_0 > r_1$.
• $\Theta_{\text{RRR}}$ is the parameter set for the RRR reservoir.
• $C_0^{a, R}, C_0^{b, R}, C_2^R, C_3^R, C_4^{a, R}$, and $C_4^{b, R}$ are constants that depend on the parameters of the reservoir class, the data distribution, and the loss function.

The authors acknowledge support from European Union’s Horizon 2020, ANR projects Q-COAST and IGNITION, and the Institute for Mathematical and Statistical Innovation (IMSI).