Quantum Reservoir Computing and Risk Bounds

TL;DR

This work develops a formal framework to bound the generalisation error of quantum reservoir computing (QRC) models using Rademacher complexity. It derives parameter-dependent risk bounds for a general class of QRCs and analyzes how the bounds scale with the number of qubits $n$, showing exponential dependence but convergence with training size $m$ for polynomial readouts; linear or fixed-degree readouts can mitigate the bound growth. The paper introduces two specialised quantum reservoir subclasses, Partial Trace Reservoir (PTR) and Random Reinitialisation Reservoir (RRR), and provides explicit bounds that incorporate reservoir contraction constants and readout parameters, guiding practical design choices for QRC hardware. It also discusses the implications for universality, the potential of spatial multiplexing, and directions toward device-specific, verifiable reservoir classes, aiming to bridge theory with near-term quantum implementations.

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.

Figures (3)

• Figure 1: Schematic of the temporal evolution of a quantum reservoir. The purple dots designate qubits, the outline determines the reservoir. At time $t$, the reservoir is in state $\rho_t$. A new input $v_{t+1}$ is injected and the full reservoir is made to evolve according to the dynamics induced by the CPTP map $T$. After all the data has been injected in this way, the final reservoir state $\rho_0$ is processed in a readout function $h$ which produces the prediction.
• Figure 2: Schematic of the CPTP map of a partial trace reservoir. The purple dots designate the reservoir qubits, the solid outline determines the reservoir. The single pink dot designates the ancillary "input qubit", and the dashed line designates the system on which we apply the unitary evolution determined by the XY-Hamiltonian. At time $t$, the reservoir is in state $\rho_t$. A new input $v_{t+1}$ is injected in the ancillary qubit by setting it to the mixed state $v_{t+1} \ket{0}\bra{0} + (1-v_{t+1}) \ket{1}\bra{1}$ and the reservoir qubits along with the ancillary qubit are made to evolve according to the unitary map induced by the Hamiltonian. After time $\tau$, the new reservoir state $\rho_{t+1}$ is obtained by tracing out the ancilla qubit. Finally, the next input is injected into the ancilla qubit.
• Figure 3: Schematic of the CPTP map of a Random Reintialisation Reservoir. The purple dots designate the reservoir qubits, the solid outline determines the reservoir. At time $t$, the reservoir is in state $\rho_t$. With probability $v_{t+1} (1 - \alpha)$ we apply the map $T_0$ to the reservoir to obtain the new reservoir state $\rho_{t+1} = T_0(\rho_t)$, with probability $(1 - v_{t+1}) (1 - \alpha)$ we apply the map $T_1$ to the reservoir to obtain the new reservoir state $\rho_{t+1} = T_1(\rho_t)$, and with probability $\alpha$ we reintialise the reservoir to some fixed quantum state $\sigma$.

Theorems & Definitions (14)

• Remark 1
• Lemma 2
• Proposition 3
• Theorem 4
• Corollary 5
• Proposition 6
• Corollary 7
• Remark 8
• Lemma 9
• Lemma 10