Experimental memory control in continuous variable optical quantum reservoir computing

TL;DR

This work introduces a continuous-variable optical quantum reservoir computing platform that leverages multimode squeezed states and spectral–temporal multiplexing to achieve memory-enabled online processing. Memory is realized via real-time feedback on pump phase and extended through spatial multiplexing, while information encoding is performed through flexible pump shaping; a Digital Twin validates experimental results and guides exploration. The authors demonstrate nonlinearity and memory with tasks such as XOR, parity checks, and chaotic time-series forecasting (double-scroll, Lorenz), achieving high accuracies and capacities, and showing that entangled multimode structure enhances expressivity with polynomial scaling in the number of measured modes. The approach offers a scalable, room-temperature photonic route to quantum-enhanced temporal processing, with potential extensions to non-Gaussian resources for further advantages in CV quantum information processing.

Abstract

Quantum reservoir computing (QRC) offers a promising framework for online quantum-enhanced machine learning tailored to temporal tasks, yet practical implementations with native memory capabilities remain limited. Here, we demonstrate an optical QRC platform based on deterministically generated multimode squeezed states, exploiting both spectral and temporal multiplexing in a fully continuous-variable (CV) setting, and enabling controlled fading memory. Data is encoded via programmable phase shaping of the pump in an optical parametric process and retrieved through mode-selective homodyne detection. Real-time memory is achieved through feedback using electro-optic phase modulation, while long-term dependencies are achieved via spatial multiplexing. This architecture with minimal post-processing performs nonlinear temporal tasks, including parity checking and chaotic signal forecasting, with results corroborated by a high-fidelity Digital Twin. We show that leveraging the entangled multimode structure significantly enhances the expressivity and memory capacity of the quantum reservoir. This work establishes a scalable photonic platform for quantum machine learning, operating in CV encoding and supporting practical quantum-enhanced information processing.

Figures (10)

• Figure 1: (a) Quantum reservoir protocol. At each time step $k$, the input $s_k$ is added to a feedback signal derived from the previous observables $O^{(k-1)}$. This sets the pump of the parametric process, determining the operators $\hat{x}'_i$ and their quantum correlations. The task output $y_k$ is obtained from the observables. (b) Experimental setup. Information is encoded in the global phase of the pump, generated via second harmonic generation (SHG) and modulated by an electro-optic modulator (EOM). The more general encoding involve amplitude and phase shaping of the pump spectrum. The multimode entangled state from parametric down-conversion (PDC), and thus its observables, is tailored by this encoding. The control and data acquisition (C&DA) module includes electronics to set the measurement basis ($U$) via local oscillator shaping in homodyne detection (HD), measure the observables $O_m$, and generate the feedback. Phase locking is not shown. (c) Nonlinear dependence of observables with the EOM voltage (proportional to the encoding phase $\delta$, which is a linear combination of $s_k$ and $O^{(k-1)}$
• Figure 2: Controllability of the resource by tailoring the local oscillator spectral shape and the pump spectral. The nodes of the network are different spectro-temporal modes of the field. The set of modes are defined by the basis choice $U$. For convention when $U = I$ the supermode basis $\{h_i\}$ is chosen, and the modes are independently squeezed. In all the other mode bases $\{\xi_i\}$ the modes show entanglement correlations represented as links in a network. The general mode quadrature is $\hat{x}'_i(\theta_{LO})=\cos{(\theta_{LO})}\hat{q}_i'+\sin{(\theta_{LO})}\hat{p}_i'$ that corresponds to $\hat{q}_i'$ or $\hat{p}_i'$ if the local oscillator phase is set to be $\theta_{LO}=0$ or $\theta_{LO}=\pi/2$. The correlations can be controlled by tailoring the pump of the process that modifies the Hamiltonian.
• Figure 3: XOR task (experimental and simulated). (a) Example of binary target (blue) and predicted (orange) outputs for the test set, with a test accuracy of $95.9\%$ (correct predictions over total). Training and test sizes are 99 and 49, respectively. (b,c) Test accuracy versus training size: experimental (dots) and simulated (squares) results under varying noise levels (light brown, brown, blue, light blue), these noise levels are determined by calculating the standard deviation around the expected behavior of the observables (see the bottom of Fig.\ref{['fig:setup']}). Experimental error bars (shaded) are obtained by reshuffling data before the train/test split and calculating a standard deviation of the accuracy on these different realizations; simulated ones from repeated sampling. Larger training sets lead to smaller test sets, increasing variability due to single misclassifications. For clarity, error bars are shown only for the first and last noise levels. Memory task (experimental and simulated). (d) Continuous target (blue) and predicted (orange) output for a test set using $5$ reservoirs with delay $\tau=1$ (capacity $= 0.81$). (e) Capacities vs. delay $\tau$: experimental (dots) and simulated (squares) results for different reservoir numbers (light brown, dark purple, light blue). Error bars are computed as for the XOR task. In both cases, simulations use a realistic experimental noise (for the memory task we had a Gaussian noise of 0.057)
• Figure 4: Double-scroll task (simulated). (a) Target (blue) and predicted (orange) trajectories using a realistic experimental noise model (gaussian noise equal to $0.057$), with $15$ reservoirs and a training size of $350$. Prediction accuracies for $V_1$, $V_2$, and $I$ are $93.1\%$, $85.3\%$, and $90.7\%$, respectively. General encoding (simulated).(b) Parity check accuracy for different delays $\tau$ and for different combinations of $N$ (the number of phase segments we imprint on the pump) and $n$ (the number of measured output modes). The error bars are obtained by repeating the simulation with random parameters $10$ times. (c) Kernel quality (rank of the observables matrix) vs. number of measured modes $n$ (with $N$ input pump frexels), measuring reservoir expressivity. For $N=1$ (light brown), increasing $n$ does not improve expressivity, while for higher values of $N$ we get closer to the saturation of a polynomial scaling with $n$ (quadratic in $n(n+1)/2$). Error bars reflect variations across random input sequences and parameters; fluctuations at $N=1$ mainly stem from numerical noise. (d) Correlation matrices between observables for $N=1$ and $N=5$ at $n=6$ ($n$$\hat{q}$-quadrature operators, yielding $n(n+1)/2=21$ observables). For $N=1$, we can see the high redundancy; for $N=5$, correlations and thus observables behaviors are richer and more diverse. When the rank of the observables matrix is $n(n+1)/2$, which is its dimension, it means every observable is meaningful and not redundant.
• Figure 5: Experimental acquisition process and phase dependence. On the left oscilloscope (top) and spectrum analyzer (bottom) normalized acquisitions for a single global pump phase. The visibility, acquired with the oscilloscope, is normalized between -1 and 1 as a cosine. The variance of the homodyne signal from the spectrum analyzer is normalized between 0 and 1 (to have already scaled observables). The values of the visibility cosine dictate what quadrature we are measuring ($\hat{x}_{\theta_{\text{LO}}} = \cos(\theta_{LO})\hat{q} + \sin(\theta_{\text{LO}})\hat{p}$), we take the corresponding times on the spectrum analyzer signal to reconstruct an average value of the covariance elements for the selected global phase (in dB). These averages become our observables. On the right the results of these acquisitions when scanning the global phase of the pump with the modulator and by repeating $10$ times the acquisition on the left.