Quantum Reservoir Computing for Statistical Classification in a Superconducting Quantum Circuit

TL;DR

This paper investigates Quantum Reservoir Computing (QRC) implemented in a Bose-Hubbard-type superconducting circuit to perform statistical inference on distributions and time-series data. It uses a two-capacitively coupled superconducting island reservoir driven by gate voltage and a linear readout trained with the Moore–Penrose pseudoinverse, evaluating on distribution discrimination between $f_{\mathcal{N}}$ and $f_{\mathcal{L}}$, tail-parameter inference for a Student-$t$ distribution with parameters $(\mu=0,\sigma=1)$, and volatility-band classification for $\text{GARCH}(1,1)$ sequences. The results show QRC can match or exceed classical baselines in the limited-data regime, notably for heavy-tailed and persistent-volatility tasks, with performance improving when increasing Hilbert-space size and reservoir nonlinearity. The work demonstrates a concrete, analog quantum hardware platform for QRC on real-world problems and outlines concrete hardware routes to scale the reservoir for enhanced quantum learning.

Abstract

We analyze numerically the performance of Quantum Reservoir Computing (QRC) for statistical and financial problems. We use a reservoir composed of two superconducting islands coupled via their charge degrees of freedom. The key non-linear elements that provide the reservoir with rich and complex dynamics are the Josephson junctions that connect each island to the ground. We show that QRC implemented in this circuit can accurately classify complex probability distributions, including those with heavy tails, and identify regimes in correlated time series, such as periods of high volatility generated by standard econometric models. We find QRC to outperform some of the best classical methods when the amount of information is limited. This demonstrates its potential to be a noise-resilient quantum learning approach capable of tackling real-world problems within currently available superconducting platforms. We further discuss how to improve our QRC algorithm in real superconducting hardware to benefit from a much larger Hilbert space.

Figures (4)

• Figure 1: (a) The quantum reservoir is a superconducting circuit composed of two capacitively coupled superconducting islands, with phase and charge degrees of freedom $\theta_j,N_j$, which are connected to the ground via Josephson junctions. Both islands are connected to an external voltage $V_g.$ The coupling capacitance is $C_{12},$ while each gate capacitance is $C_{g_{i}}.$ (b) Representation of the QRC framework. The quantum reservoir (large blue circle) evolves in response to the input (green circle). At specific times, a set of observables $O_i$ is measured from the reservoir and sent to the readout layer. The reservoir outputs (orange square) are subsequently mapped to the reservoir prediction (yellow square) through a set of trained weights (represented by the weight matrix $W$).
• Figure 2: Normal vs. Laplace discrimination. (a) Prediction accuracy of QRC and a generalized likelihood ratio test versus the number of data points per sample $T$. (b) Accuracy scaling of QRC with input length $T$ fitted by stretched-exponential laws $A(T)=A_{\infty}-c\,e^{-kT^{p}}$, together with the corresponding linearized representation (inset: classical case). For this task, the driving amplitude is fixed to $\epsilon_0 = 3.8\,\mathrm{GHz}$.
• Figure 3: Student-t degrees-of-freedom prediction. (a) Test-set RMSE of the $1/\nu$ estimator versus sequence length $T$, comparing the QRC readout with a classical maximum-likelihood estimator. Both methods improve monotonically with $T$, with QRC achieving lower RMSE and the gap narrowing at large $T$. (b) Scaling analysis of data in panel (a), which are fitted to power-law decays $RMSE(T)=c\,T^{-p}$, together with the corresponding linearized representation (inset: classical case). For this task, the driving amplitude is fixed to $\epsilon_0 = 1\,\mathrm{GHz}$.
• Figure 4: GARCH(1,1) volatility-regime classification. (a) Test accuracy versus input length $T$ for QRC and a classical feature-based classifier. (b) Scaling analysis of the accuracy versus $T$ using the fitted scaling laws for $A(T).$ The classical method is best described by a stretched-exponential approach to a plateau, $\hat{A}_c(T)=A_{\infty,c}-c_c e^{-k_c T^{p_c}}$, whereas the QRC data favor a power-law approach, $\hat{A}_q(T)=A_{\infty,q}-c_q T^{-p_q}$. For this task, the driving amplitude is fixed to $\epsilon_0 = 1.9\,\mathrm{GHz}$.