Experimental quantum reservoir computing with a circuit quantum electrodynamics system

TL;DR

This work delivers an experimental quantum reservoir computing implementation on a circuit QED platform using a single bosonic mode. Inputs are encoded into the cavity displacement amplitude $\alpha_{\rm in}(t)$ and outputs are obtained from Fock-state occupations $P_n$, enabling a rich nonlinear feature set through measurement. The system achieves high-accuracy temporal tasks, including sine/square waveform classification with up to $99.8\%$ test accuracy using 20 features (and $99.5\%$ with eight features), and Mackey–Glass time-series prediction, with Kerr nonlinearity $K_{cc}$ shown to enhance performance in simulations. The results highlight a hardware-efficient pathway to scalable quantum neural networks, feasible to extend to multi-resonator architectures and quantum data processing by leveraging measurement-induced nonlinearity and Kerr effects.

Abstract

Quantum reservoir computing is a machine learning framework that offers ease of training compared to other quantum neural networks, as it does not rely on gradient-based optimization. Learning is performed in a single step on the output features measured from the quantum system. Various implementations of quantum reservoir computing have been explored in simulations, with different measured features. Although simulations have shown that quantum reservoirs present advantages in performance compared to classical reservoirs, experimental implementations have remained scarce. This is due to the challenge of obtaining a large number of output features that are nonlinear transformations of the input data. In this work, we show that even with a circuit quantum electrodynamics system as simple as a single transmon coupled to a readout resonator, we can implement a proof-of-concept realization of quantum reservoir computing. We obtain a large number of nonlinear features from a single physical system by encoding the input data in the amplitude of a coherent drive and measuring the cavity state in the Fock basis. We demonstrate classification of two classical tasks with significantly smaller hardware resources and fewer measured features compared to classical neural networks. Our experimental results are supported by numerical simulations that show additional Kerr nonlinearity is beneficial to reservoir performance. Our work demonstrates a hardware-efficient quantum neural network implementation that can be further scaled up and generalized to other quantum machine learning models.

Figures (9)

• Figure 1: Quantum reservoir computing with a superconducting circuit. (a) Input data are encoded in the amplitude of the resonant drive (green) of the quantum system acting as a reservoir (blue). Output features (purple circles) are obtained by measuring the occupation probabilities of Fock states $|0\rangle$ to $|n\rangle$ at different times $t_1$ and $t_2$, yielding a total of $2n$ output features. Features are classified by training a layer of linear weights (dashed lines). (b) Schematic and (c) design of the quantum circuit used to implement the quantum reservoir. Tantalum coplanar waveguide resonator (blue) is capacitively coupled to a transmon qubit (purple) and to a transmission line through a Purcell filter (green).
• Figure 2: Fock state probabilities as neurons. (a) Two tone spectroscopy of the cavity. The phase shift of the reflection coefficient at the cavity resonance frequency, as a function of the qubit drive frequency and the cavity drive amplitude $\alpha_{\rm{in}}(t)$. Different resonances from right to left correspond to the qubit frequency dressed by 0, 1, 2, 3 and 4 photons in the cavity. Yellow horizontal lines indicate the amplitude range used for data encoding. (b) Top: pulse sequence used to measure the photon number in the cavity. A 200 ns displacement pulse $D_{\alpha}$ at cavity frequency $\omega_c$ is followed by a 200 ns $\pi_n$ pulse conditioned on $n$ photons in the cavity, and a high power readout (HPR) after the 1 $\mu$s waiting time. Bottom: Measured occupation probabilities (dots) $P_0$ to $P_4$ for Fock states $|0\rangle$ to $|4\rangle$ as a function of the cavity displacement amplitude $\alpha_{\rm{in}}$. Each point is averaged 10 000 times. Lines correspond to the probabilities obtained from the simulations using the Lindbladt master equation, with dissipation rates and Kerr effect as free parameters.
• Figure 3: Sine and square waveform classification task. (a) Input data are obtained from a random series of 400 sine and square periods, discretized into 8 points per period. Only 4 periods are shown for clarity. (b) Top: Pulse sequence consists of two displacement pulses $D_{\alpha_{i-1}}$ and $D_{\alpha_{i}}$, whose amplitudes encode two consecutive data points, followed by a $\pi_{n}$ pulse and high-power readout. To increase the number of extracted features, the occupation probabilities are sampled at four times $t_1$ to $t_4$ during the second displacement pulse. Bottom: Cavity drive $\mathrm{Re}(\alpha_\mathrm{in})$ whose amplitude encodes the input data. (c) Prediction of the quantum reservoir with the 8 most informative features among probabilities of states $|0\rangle$ to $|4\rangle$ at times $t_i=i\times50$ ns, $i \in [ 1, 4 ]$. (d) Experimental accuracy (purple crosses) as a function of the number of measured states $N$, with 4 time samples per state, giving $4N$ features. For each $N$, the plot shows the best accuracy obtained with the $N$ most informative features. The solid lines connect simulated data points (dots) corresponding to different qubit dephasing rates $\kappa_{\phi}$. (e) Experimental accuracy with 5 measured features (states from $|0\rangle$ to $|4\rangle$ as a function of the number of time sampling steps, with sampling times indicated in brackets). (f) Accuracy as a function of the number of measurement shots and (g) as a function of the encoding range.
• Figure 4: Mackey-Glass chaotic time series prediction. (a) Pulse sequence used for the task. (b) Part of the input data (green) and corresponding target for a delay $d=25$ (gold). Quantum reservoir predictions for delays $d=1$ and $d=25$ are shown respectively in pink and purple. (c) Normalized root mean square error (NRMSE) as a function of delay $d$ for the experimental quantum reservoir for the lag parameter $\tau = 17$. (d) NRMSE as a function of the delay $d$ for a simulated quantum reservoir for different parameters $\tau$.
• Figure 5: Impact of the Kerr effect on performance. (a) Simulated mean photon number $\langle n \rangle$ in the cavity after a 400 ns drive as a function of detuning and input amplitude $\alpha_{\rm{in}}$, for different Kerr rates $K_{cc}$. Contour lines indicate regions of equal mean photon number. (b) NRMSE for sine-square classification using 20 neurons, shown as a function of Kerr coefficient and detuning, for various input encoding ranges [$\alpha_\mathrm{in}^\mathrm{min}$, $\alpha_\mathrm{in}^\mathrm{max}$]. (c) Simulated NRMSE as a function of the Kerr coefficient $K_{cc}$ for $\Delta = 0$ and the encoding range [0.7 $\alpha_0$, 1.3 $\alpha_0$], where $\alpha_0$ is the drive amplitude required to reach $\langle n \rangle \in$ {1, 2, 3} after a 400 ns drive (see panel (a)). Because multiple combinations of detuning and amplitude can yield the same mean photon number, the NRMSE values for different amplitudes are shown with distinct line styles.