Experimental memory control in continuous-variable optical quantum reservoir computing

Abstract

Forecasting complex processes requires efficient learning from temporal data. Reservoir computing platforms enable such learning with minimal training cost. Quantum reservoir computing (QRC) extends this framework into the quantum domain, offering promising capabilities for online, quantum-enhanced machine learning tailored to temporal tasks. As in the classical case, photonics provides a natural platform for QRC. However, implementing native memory capabilities in practical photonic quantum systems remains a major challenge. Here we demonstrate a photonic QRC platform based on deterministically generated multimode squeezed states, exploiting spectral and temporal multiplexing in a continuous-variable setting with controllable fading memory. Data is encoded via programmable pump phase shaping in an optical parametric process and retrieved through mode-selective homodyne detection. Real-time memory is implemented through feedback via electro-optic modulation, and expressivity is boosted via spatial multiplexing. This architecture enables nonlinear temporal tasks, including parity check at different delays and chaotic signal forecasting. All results are supported by a high-fidelity Digital Twin. Leveraging the entangled multimode structure enhances expressivity and memory capacity, establishing a scalable continuous-variable photonic platform for quantum-enhanced information processing.

Researchers demonstrate a photonic quantum reservoir computing platform that uses spectral and temporal multiplexing in a continuous-variable setting. Real-time memory is implemented, and nonlinear temporal tasks are enabled.

Main

Reservoir computing is a machine learning neuro-inspired paradigm that exploits the dynamics of an artificial network with minimal learning requirements. Unlike conventional neural networks, it avoids the computational cost of tuning internal weights during training; instead, only the output layer is trained, typically via linear regression^1,2. The untrained network, known as the reservoir, can be implemented as a physical process.

Although the term reservoir computing is now broadly used for tasks involving an untrained network (that is, a reservoir), its most distinctive feature is the ability to learn from temporal series (for example, predicting chaotic time series) by exploiting the memory properties of the system used as a reservoir. This temporal processing capability enables online learning and extends beyond static tasks such as classification traditionally addressed by architectures such as extreme learning machines^3,4.

Quantum reservoir computing (QRC) extends the concept of physical reservoirs into the quantum regime and has recently been extensively explored across several platforms^5–8 revealing both novel capabilities^9–15 and new challenges^16,17. Photonic platforms are particularly appealing as they inherit the capabilities of their classical counterparts^18,19, offer many exploitable degrees of freedom, possess high integration potential and, when information is encoded in quadrature continuous variables, can operate at room temperature using coherent detection for online monitoring²⁰.

Photonic systems have recently been tested in quantum machine learning tasks^21,22. In the context of QRC, following different proposals in either static^23,24 and temporal tasks^10,20,25, some pioneering experimental results have been demonstrated on quantum optical platforms; these have primarily focused on classification tasks^15,26,27 or classically assisted strategies²⁸. Only recently have efforts been directed at incorporating memory capabilities into general quantum platforms such as superconducting circuits^29–31, and into a memristor-based photonic platform³².

Here we present an experimental realization of memory-tailored QRC using a reconfigurable, multimode continuous-variable (CV) photonic platform. Memory is engineered through feedback conditioned on the history of the quantum system, enabling online temporal learning. Our approach exploits deterministically generated squeezed and reconfigurable entangled quantum states, multiplexed across both spectral and temporal modes^33–35 along with reconfigurable mode-selective homodyne detection. The experimental realization of temporal learning tasks via CV builds on theoretical results that demonstrate the potential of such systems in QRC^13,20,25,36. The experimental set-up supports controllable memory demonstrating a performance boost in online learning and does not require post-processing with a classical neural network platform.

Data is encoded either via modulation of the global phase or of the complex spectral shape of the pump pulses driving the nonlinear optical process. Global phase modulation imprints information into the angle of the squeezing ellipses, tuning quantum correlations in the measurement basis. Spectral shaping directly modifies quantum correlations among output modes by altering the nonlinear interaction itself.

Moreover, we construct a Digital Twin of the experiment by integrating accurate models of the quantum and nonlinear optical processes, along with the CV dynamics of the reservoir. Once calibrated with a noise model, this simulation framework accurately reproduces experimental results and enables predictive exploration of the platform’s capabilities.

We experimentally perform temporal tasks enabled by memory, using global phase encoding. Memory is achieved via a feedback signal applied to the electro-optic phase modulator (EOM)³⁷ that controls the pump phase; the feedback is derived from observables measured via mode-selective homodyne detection of the quantum reservoir’s multispectral components. For short-memory tasks, we implement real-time feedback control, while longer-delay target functions are experimentally realized using a spatial multiplexing strategy, that is, by running sequential experiments that act as spatially distributed copies of the quantum reservoir platform.

Finally, we demonstrate that fully exploiting the entangled multimode nature of the system yields better scaling of reservoir expressivity when compared with the use of multiple independent copies of the experiment (spatial multiplexing) and enhance the memory in temporal tasks.

This work demonstrates online temporal processing enhanced by controllable memory using a CV entangled reservoir. Gaussian states and homodyne detection have been theoretically shown to increase information processing capacity compared with classical encodings²⁵, allowing saturation of a polynomial advantage as the reservoir size grows²⁰. This regime is experimentally explored for the first time in this article, with potential implications for practical noisy intermediate-scale quantum implementations. Furthermore, our architecture is compatible with measurement-based approaches using cluster states¹³ and with the integration of non-Gaussian resources^15,38.

Defining quantum advantage in QRC or quantum machine learning remains an open challenge³⁹. Nevertheless, CV platforms enhanced with non-Gaussian resources are central to fault-tolerant photonic quantum computing⁴⁰. Our demonstration of memory-enhanced online temporal tasks enables scalable QRC using only deterministic Gaussian-state generation and room-temperature homodyne detection, providing a basis for exploring quantum advantage with non-Gaussian resources^41–44.

Results

The quantum resource

The core quantum resource is a multimode optical network generated via parametric down-conversion (PDC) in a nonlinear waveguide pumped by a pulsed laser, as shown as in Fig. 1. The interaction Hamiltonian describing the generation of signal and idler fields of wavelengths λ_S, λ_I reads

$$\widehat{H}\propto \int d{\lambda }_{\mathrm{S}}d{\lambda }_{\mathrm{I}}J({\lambda }_{\mathrm{S}},{\lambda }_{\mathrm{I}}){\widehat{a}}^{†}\left({\lambda }_{\mathrm{S}}\right){\widehat{b}}^{†}\left({\lambda }_{\mathrm{I}}\right)+\mathrm{h}.\mathrm{c}.,$$ 1

where J(λ_S, λ_I), the so-called joint spectral amplitude (JSA), can be written as^45–47

$$J({\lambda }_{\mathrm{S}},{\lambda }_{\mathrm{I}})=p({\lambda }_{\mathrm{S}},{\lambda }_{\mathrm{I}})\cdot \varphi ({\lambda }_{\mathrm{S}},{\lambda }_{\mathrm{I}}).$$ 2

Fig. 1. QRC: protocol, set-up and observable dynamics.

a, Quantum reservoir protocol. At each timestep 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 ${\widehat{x}}_i^{\prime }$ and their quantum correlations. The task output y_k is obtained from the observables. b, Experimental set-up. Information is encoded in the global phase of the pump, generated via second harmonic generation (SHG) and modulated by an EOM. The more general encoding involves phase shaping of the pump spectrum. The multimode entangled state from 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 δ, which is a linear combination of s_k and O^(k−1)). Left: experimental data (dots) in low- and average-noise regimes. The low-noise regime (first panel) is obtained by repeating the measurement ten times; the data are shown as mean values ± s.d. All the other panels present one-sample independent measurements and simulations to highlight the impact of the noise and therefore present no error bars. Right: simulated data (squares) using the Digital Twin model according to the experimental noise evaluation. The white panel spans a voltage range sufficient for observables to vary from 0 to 1 (phase difference slightly exceeding π/2). Yellow panels show observables recorded during training on a memory task.

The term p(λ_S, λ_I) is the complex pump spectral shape, while ϕ(λ_S, λ_I) is the phase-matching function, determined by the characteristics of the nonlinear medium. The multimode nature of the generated quantum system can be retrieved by performing a Schmidt decomposition of the JSA J(λ_S, λ_I) = ∑_kr_kh_k(λ_S)g_k(λ_I), where h_k(λ_S) and g_k(λ_I) are two sets of orthonormal modes, spectral profiles of signal and idler channels, and r_k are called the Schmidt coefficients. In case of degenerate signal and idler fields, as in our experimental set-up, h_k = g_k; and the Hamiltonian in equation (1) can be written as $\widehat{H}\propto {\mathrm{\sum }}_k{r}_k{\widehat{A}}_k^{†2}+\mathrm{h}.\mathrm{c}.$, with ${\widehat{A}}_k^{†}$ broadband creation operators associated to the modes h_k. Acting on the vacuum, this Hamiltonian generates a tensor product of single-mode squeezed vacuum states, one in each mode h_k(λ_S), also called supermodes. The used set-up can generate up to 40 Schmidt modes approximated by Hermite–Gauss polynomials as shown in Fig. 1.

The generated quantum state is Gaussian and fully characterized by the quadrature covariance matrix σ, measured via homodyne detection. Mode-selective homodyne detection enables probing in arbitrary mode bases. In the supermode basis, σ is diagonal; in any other basis ξ_j(λ_S) related by a unitary U, it transforms as ${\mathit{\sigma }}^{\prime }={\mathit{S}}_{\mathit{U}}^T\mathit{\sigma }{\mathit{S}}_{\mathit{U}}$, where S_U is the symplectic transformation induced by U (Supplementary Section B). The matrix ${\mathit{\sigma }}^{\prime }$ is non-diagonal and the off-diagonal terms are signatures of entanglement correlations between the measured modes. Measuring on such basis allows to experimentally access entanglement correlations for quantum protocols^35,48. The resource can then be pictured via an entangled network as shown in Fig. 2.

Fig. 2. Controllability of the resource by tailoring the local oscillator and the pump spectral shapes.

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 {ξ_i}, the modes show entanglement correlations represented as links in a network. The general mode quadrature is ${\widehat{x}}_i^{\prime }\left({\theta }_{\mathrm{LO}}\right)=\cos \left({\theta }_{\mathrm{LO}}\right){\widehat{q}}_i^{\prime }+\sin \left({\theta }_{\mathrm{LO}}\right){\widehat{p}}_i^{\prime }$ that corresponds to ${\widehat{q}}_i^{\prime }$ or ${\widehat{p}}_i^{\prime }$ if the local oscillator phase is set to be θ_LO = 0 or θ_LO = π/2. The correlations can be controlled by tailoring the pump of the process that modifies the Hamiltonian.

The phase-matching function term of the JSA in equation (2) is fixed and determined by the material and waveguide properties, and only the pump-envelope function can be modified in a reconfigurable way via mode-shaping techniques. This modifies the entanglement correlation between nodes as shown in Fig. 2. Therefore, we can explicitly write the dependence of the covariance matrix ${\mathit{\sigma }}^{\prime }$ on the pump p and the chosen basis U as ${\mathit{\sigma }}^{\prime }={\mathbf{S}}_{\mathbf{U}}^T\mathit{\sigma }\left(\mathbf{H}\left(\mathbf{p}\right)\right){\mathbf{S}}_{\mathbf{U}}$.

In the following, we exploit the control on the pump-envelope function to encode information in the quantum system in the configuration of a fixed measurement basis U. The expectation value of general observables $O_m=\langle \mathrm{\Delta }{\widehat{x}}_i\mathrm{\Delta }{\widehat{x}}_j\rangle$ that will be used in the reservoir task can be retrieved from the covariance matrix ${\mathit{\sigma }}^{\prime }(U,p)$. Given the pump shape, we have accurate numerical models of the process from the nonlinear interaction to the observables^45–47, setting a Digital Twin of the experimental set-up as fully explained in Supplementary Section A.

Information encoding and memory control of the quantum reservoir

Tuning the pump phase profile, the pump-envelope function can be written as

$$p\left({\lambda }_{\mathrm{P}},\mathit{\delta }\right)=\sum _{i=1}^N{f}_i\left({\lambda }_{\mathrm{P}}\right)e^{i{\delta }_i},$$ 3

where the f_i(λ_P) are a set of N pump modes and the $\mathit{\delta }={\left({\delta }_i\right)}_{i=1}^N\in {\mathbb{R}}^N$ are tunable phase parameters that allow precise shaping of the JSA to easily encode information. The pump input modes f_i that we consider are N frequency windows, delimiting a portion of our Gaussian pump profile f(λ_P), that we can precisely control with a spatial light modulator, for example. In the simple case of N = 1, when a unique global phase parameter is considered, an EOM is used as shown in Fig. 1. To perform the measurement, a set of n modes ξ_j(λ_S) is selected by the basis choice U via the local oscillator shaping in homodyne. The tailoring of N pump spectral components and the choice of the measurement basis composed by n elements are shown in the upper part of Fig. 2 (more details in Supplementary Section B).

Our system is endowed with memory through a real-time feedback mechanism: the results of previous measurements are fed back into the system to influence future states. Specifically, if we consider a discrete-time sequence of inputs {s_k}, where k denotes the timestep, the pump parameters are updated according to

$${\delta }_i^{\left(k\right)}={\alpha }_i{s}_k+{\beta }_i+\sum _m{M}_{im}{O}_m^{\left(k-1\right)}.$$ 4

Here i ∈ 1, …, N, α_i and β_i are real-valued coefficients, and M_im is a feedback mask (a real matrix) that determines how previous measurement outcomes influence the current pump parameters. This determines the fading memory of the QRC. To summarize, N is the number of different input components that we encode in a single instance by manipulating several components of the pump spectrum: N > 1 can encode multidimensional inputs as well as enrich the encoding of a scalar input, repeated across the pump spectrum with different parameters (α_i, β_i, M_im). The quantities $O_m^{(k-1)}={\langle \mathrm{\Delta }{\widehat{x}}_i\mathrm{\Delta }{\widehat{x}}_j\rangle }^{(k-1)}$ are the values of the mth observable measured at timestep k − 1, and are derived from a subset of elements of the covariance matrix (i, j ∈ 1, …, 2n). These are measured by accumulating data on the homodyne signal. The number n of measured modes determines then the size of the output layer of our reservoir. The reservoir computing is described by

$$\left\{\begin{array}{c}{O}_m^{\left(k\right)}\propto {\sigma }_{ij}^{\prime \left(k\right)}\mid {\mathbf{\sigma }}^{\prime \left(k\right)}={\mathbf{S}}_{\mathbf{U}}^T\mathbf{\sigma }\left(\mathbf{H}\left(\mathbf{p}\left({\mathit{s}}_{\mathbf{k}},{\mathbf{O}}^{\left(\mathbf{k}-\mathbf{1}\right)}\right)\right)\right){\mathbf{S}}_{\mathbf{U}}\hfill \\ y_k={\mathbf{w}}^T{\mathbf{O}}^{\left(k\right)}+b\hfill \end{array}\right).$$ 5

The reservoir is trained by adjusting the matrix w and parameter b to minimize the error in the chosen learning task. The scheme of the quantum reservoir with feedback is shown in the top part of Fig. 1. While we here focus on the case of feedback restricted to the last output at each cycle, the fading memory can be extended re-injecting a longer combination of past signals.

QRC tasks

We evaluate the performance of our system across a range of benchmark tasks considering two input encoding: the simple, experimentally implemented scheme with N = 1, and the more general encoding with N > 1. We begin with the XOR task, which assesses the system’s nonlinearity and short-term memory while requiring minimal resources. We then consider the memory task to characterize the system’s fading memory property. Finally, we turn to more complex benchmarks, including the double-scroll and parity check tasks, which challenge the system’s scalability and capacity to handle richer, higher-order nonlinear dynamics.

Experimental control of memory via global phase

The phase encoding via the EOM, as shown in Fig. 1, corresponds to N = 1, and the voltage applied changes linearly the pump phase δ:

$$p({\lambda }_{\mathrm{P}},\delta )=f\left({\lambda }_{\mathrm{P}}\right)e^{i\delta },$$ 6

where f(λ_P) is the natural Gaussian profile of the pump. As explained in Methods, with this type of encoding, the expressivity of the observables does not increase with the number of measured modes n; therefore, we use n = 1 as well. This means that we measure only the first supermode h₁ and we can use as observables {O₁, O₂, O₃} the set $\{\langle \mathrm{\Delta }{\widehat{q}}^2\rangle ,\langle \mathrm{\Delta }{\widehat{p}}^2\rangle ,\langle \mathrm{\Delta }\widehat{q}\mathrm{\Delta }\widehat{p}\rangle ~\langle \mathrm{\Delta }{\left(\frac{\widehat{q}+\widehat{p}}{\sqrt{2}}\right)}^2\rangle \}$. These three observables are sufficient for implementing several tasks, as they express three different nonlinear dependencies with respect to δ as shown in the lower part of Fig. 1. The first two panels show the three behaviours experimentally, in the case of a low-noise realization and a more noisy one, which is representative of the average noise in measurements. Assuming a Gaussian noise model, the typical variance of observables is recovered via a least square method. The result is used in the Digital Twin to simulate the expected behaviour as shown in the last two panels.

The unique non-zero term M that we use for the feedback mask is the one corresponding to ${\langle \mathrm{\Delta }{\widehat{q}}^2\rangle }^{(k-1)}$. At each timestep k, we set the global phase depending on the input s_k and the feedback as

$${\delta }^{\left(k\right)}=\alpha s_k+\beta +M{\langle \mathrm{\Delta }{\widehat{q}}^2\rangle }^{(k-1)}.$$ 7

M, α and β are randomly chosen within ranges that ensure both the input and the feedback contribute comparably to the phase and such to explore a nontrivial region of the observable space.

We begin with a binary task, the temporal XOR, where the target output is ${\widehat{y}}_k=s_k⨁s_{k-1}$ and the input is a random sequence of values s_k ∈ {0, 1}. This task requires both nonlinearity and memory of the previous encoded input. The reservoir is trained on a set of inputs by applying linear regression to the measured observables to maximize accuracy (Supplementary Section C).

The performance of the reservoir is then evaluated using the accuracy again, but on a test sequence of new data. In Fig. 3a, we present an example of the test target values and the corresponding experimental prediction. In Fig. 3b,c, we show experimental and simulated accuracies as a function of training set size, for different levels of Gaussian noise. Experimental instabilities can introduce Gaussian noise in the measurement of the observables (Fig. 1), which can be mitigated by averaging over repeated measurements. These instabilities arise from mechanical and thermal fluctuations and are not intrinsic limitations of the protocol, but rather reflect the current performance of the experimental set-up. We see that the training saturates for set sizes around 100 and that noise affects the performance. Nevertheless, the reservoir could achieve experimentally test accuracies of 98 ± 1% for realistic levels of noise, with only 70 training steps. The Digital Twin behaves similarly to the experiment, as shown in Fig. 3c.

Fig. 3. Experimental and simulated performance via global phase encoding.

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 set sizes are 99 and 49, respectively. b,c, Test accuracy versus training size: experimental data (dots) (b) and simulated data (squares) (c) under varying noise levels (light brown, brown, blue and light blue); these noise levels are determined by calculating the standard deviation around the expected behaviour of the observables (see the bottom of Fig. 1). Experimental error bars (shaded) represent mean ± s.d. over n = 50 independent reshufflings of the dataset (technical replicates). Simulated data points represent mean ± s.d. over n = 100 independent simulation runs (different random seeds; technical replicates). Error bars are shown only for the first and last noise levels for clarity. Larger training sets lead to smaller test sets, increasing variability owing to single misclassifications. Memory task (experimental and simulated). d, Continuous target (blue) and predicted (orange) output for a test set using 5 reservoirs with delay τ = 1 (capacity = 0.81). e, Capacities versus delay τ: experimental (dots) and simulated (squares) results for different reservoir numbers (light brown, dark purple and light blue). Experimental error bars represent mean ± s.d. over n = 100 reshufflings (technical replicates), and simulated error bars represent mean ± s.d. over n = 30 simulation runs (technical replicates). In both cases, simulations use a realistic experimental noise (for the memory task, we had a Gaussian noise of 0.057) model via the Digital Twin.

A way to further test the memory of the system is through the linear memory task, in which the reservoir is trained to reproduce past entries of the input series. The target function at timestep k is the input encoded τ steps in the past, ${\widehat{y}}_k=s_{k-\tau }$, the input s_k ∈ [−1, 1] being a random uniformly distributed sequence. Here the three previous observables are not sufficient. Therefore, we resort to spatial multiplexing, which is equivalent to use R different reservoirs at the same time. This is a classical reservoir computing technique used to enhance the expressivity of the system (see Methods for details), whereas the general encoding introduced in the next section leverages spectral multiplexing, an intrinsic resource of the quantum system. In Fig. 3d, we present an example of target and experimentally predicted values. In Fig. 3e, instead, we show the agreement between the experimental and simulated capacities (see Supplementary Section C for the definition) on test sets when varying τ and the number R of reservoirs. The simulations agree with the experimental results, confirming the advantage of multiplexing (better capacities for higher R) and the fading memory of the system (capacity gradually decreasing for higher delays).

Given the optimal agreement with the experiment, we use the Digital Twin to test the protocol on a more complex task. Concretely, we perform forecasting of chaotic time series such as the double-scroll electronic circuit⁴⁹ and also (Supplementary Section H) the Lorenz system. Since the input vectors are three-dimensional, for the global phase encoding, we require at least three spatially multiplexed reservoirs (one per input signal) with additional reservoirs possible to enhance expressivity. The target function is the input one timestep in the future: ${\widehat{\mathbf{y}}}_k={\mathbf{s}}_{k+1}$. Concerning the double-scroll, in Fig. 4a, we present an example of target and predicted values, obtained with simulations with realistic noise values. In Supplementary Fig. 2, we present three plots (corresponding to V₁, V₂ and I) where the capacity is calculated while varying the training size and the number of reservoirs R. We prove that even with the experimental noise, we saturate the prediction capacity for relatively small values of R (below 15) and of training size (below 400).

Fig. 4. Simulated reservoir computing performance: enhanced by general encoding.

Double-scroll task (simulated). a, Target (blue) and predicted (orange) trajectories using a realistic experimental noise model (Gaussian noise = 0.057), with 15 reservoirs and a training size of 350. Prediction capacities for V₁, V₂ and I are 93.1%, 85.3% and 90.7%, respectively. General encoding (simulated). b, Parity check accuracy versus delay τ for different combinations of N (the number of phase segments that we imprint on the pump) and n (the number of measured output modes). Data points represent mean accuracy over n = 10 independent trials with different random seeds. Error bars denote ±s.d. across these trials. c, Kernel quality (rank of the observables matrix) versus number of measured modes n for different N values, measuring reservoir expressivity. Data points represent mean rank over n = 25 independent trials; error bars denote ±s.d. For N = 1 (light brown), increasing n does not improve expressivity, while for higher N values, the rank approaches the saturation predicted by a polynomial scaling (n(n + 1)/2). d, Correlation matrices between observables for N = 1 and N = 5 at n = 6 ($n\widehat{q}$-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 behaviours are richer and more diverse. When the rank of the observables matrix is n(n + 1)/2, which is its dimension, it means that every observable is meaningful and not redundant.

Scaling the expressivity and boosting the memory with general encoding

In the global phase encoding, the expressivity is limited for any number of measured modes n > 1 (Methods). Therefore, to take advantage of the multimode nature of the system, we diversify the phase profile of the pump (N > 1). This means exploiting fully the phase encoding from equation (4), as shown in Fig. 4. In Fig. 4b, with this general encoding (N = 5), we achieve perfect capacities (100%) for a parity check task for τ ∈ {1, 2, 3} and prove the boost in memory when n, the number of measurement modes, is increased. Furthermore, as shown in Fig. 4c, the expressivity (see Supplementary Section C for definition) increases with n for N > 1. Specifically, for N > 5, we saturate a polynomial scaling with n as reported in ref. ²⁵: the expressivity scales with n(n + 1)/2, which is also the dimension of the observables matrix (considering only $\widehat{q}$ observables). This means that with a sufficiently large N, all the terms in our covariance matrix are independent. We highlight the increased diversity in behaviours of the observables in Fig. 4d. We also exploit the multidimensionality of the input (it can be a vector of size up to N), to implement again the double-scroll task, without multiplexing. The performances of the global phase encoding with multiplexing (R = 15 reservoirs) and the general encoding with N, n = 9, 9 are similar, but the gain in scalability is clear as with the general encoding use only one system, instead of multiplexing in 15 reservoirs.

Discussion

Starting from a multimode squeezed resource, we develop an optical QRC platform that leverages spectral shaping capabilities in femtosecond optics for both encoding and decoding information. Fading memory is enabled by linear feedback, even when limited to observables measured at the previous timestep. The CV setting is implemented via coherent homodyne detection, exploiting low-order quadrature moments as observables, an approach identified as offering polynomial advantage in QRC.

By tailoring the pump phase in the parametric process using an EOM, we experimentally show online temporal processing with memory, performing both binary tasks in real time and continuous tasks via successful spatial multiplexing. We demonstrate the nonlinear processing capabilities of the quantum reservoir with a temporal XOR task, assess its memory capacity using continuous input signals and evaluate its forecasting performance with the double-scroll benchmark. Results are reproduced by a Digital Twin, a detailed numerical model of the protocol that includes the properties of the nonlinear optical material, the optical processes, multimode and mode-selective detection schemes, and experimentally derived noise contributions. The Digital Twin enables exploration of the system’s capabilities with high fidelity.

The potential of the system’s multimode nature is further revealed by encoding information onto a richer phase profile of the pump. This allows consistent nonlinear processing, improved memory (as evidenced by a parity check task) and accurate forecasting of chaotic time series, confirming the advantages of utilizing the full multimode structure. Compared with a purely spatial multiplexing strategy, this approach shows superior scaling, revealing the potential of the system’s entangled multimode structure for enhanced performance.

We also provide, in Supplementary Section J, a detailed comparison with classical machine learning approaches (echo state networks and long short-term memory networks) and with another QRC model. Our approach compares favourably with state-of-the-art time-series processing techniques, while relying on inexpensive training (linear regression), requiring only short training sequences and exploiting polynomial scaling between the number of measured modes and the available observables.

Methods

The covariance matrix

The elements of the covariance matrix σ are ${\sigma }_{ij}=1/2\langle \mathrm{\Delta }{\widehat{x}}_i\mathrm{\Delta }{\widehat{x}}_j+\mathrm{\Delta }{\widehat{x}}_j\mathrm{\Delta }{\widehat{x}}_i\rangle$, being $\mathrm{\Delta }{\widehat{x}}_i={\widehat{x}}_i-\langle {\widehat{x}}_i\rangle$ and ${\widehat{x}}_i$ the quadrature ${\widehat{q}}_i$ or ${\widehat{p}}_i$ of the mode i with commutation relation $[{\hat{q}}_i,{\hat{p}}_i]=2\iota$. The covariance matrix is obtained by knowing the interaction Hamiltonian and, in the supermodes basis, takes the form of a diagonal matrix $\sigma \left(H\right)=\text{diag}\left\{e^{2r_1},e^{2r_2},...,e^{-2r_1},e^{-2r_2},...\right\}$.

Experiment

The experimental set-up is presented in Fig. 1. We use a pulsed laser with a repetition rate of 100 MHz centred at 1,560 nm, and a pulse duration of about 57 fs, which undergoes second harmonic generation in a periodically poled lithium niobate crystal to pump PDC in periodically poled potassium titanyl phosphate waveguides of type 0. More details about the source can be found in ref. ³⁴. In the set-up of this work, an EOM (iXblue NIR-800-LN-0.1) is used to set the phase of the pump beam for the global phase encoding. Phase lock is provided to encode information and measure the desired quadratures at the corresponding phases. The general multispectral phase encoding can be set via a pulse-shaping technique based on a spatial light modulator, as it is implemented in the homodyne set-up to select the spectral basis of the measurement. The average value of squeezing measured in this set-up is −0.45 dB, a reduced value compared with the one in ref. ³⁴ owing to the dispersion introduced in the pump pulse by the fibre-based phase modulator. Such dispersion can be notably reduced in future experiments by shortening the input and output fibre lengths. The anti-squeezing is around 0.7 dB because of losses.

Features and limits of global phase encoding

In experimental control of memory via global phase δ, only three observables that express different dependencies are used. For linear regressions indeed, observables that are correlated (linearly dependent) are redundant^50,51. For example, measuring $\langle \mathrm{\Delta }{\left(\frac{\widehat{q}+\widehat{p}}{\sqrt{2}}\right)}^2\rangle$ is equivalent to measuring $\langle \mathrm{\Delta }\widehat{q}\mathrm{\Delta }\widehat{p}\rangle$ since analytically, for a global phase encoding, $\langle \mathrm{\Delta }\widehat{q}\mathrm{\Delta }\widehat{p}\rangle \left(\delta \right)=\langle \mathrm{\Delta }{\left(\frac{\widehat{q}+\widehat{p}}{\sqrt{2}}\right)}^2\rangle \left(\delta \right)+cst$ (as shown in Supplementary Section D).

Moreover, when encoding on a global phase δ, even if we increase the number of measured modes n > 1, all the terms of the covariance matrix behave like either ${\sin }^2\delta$, ${\cos }^2\delta$ or $\cos \delta \sin \delta$. The expressivity of the reservoir is therefore limited for all n. This also happens when simulating the system with a detailed physical model that takes into account the characteristics of the PDC material as shown in the bottom part of Fig. 4, where we encode on a single pump band (N = 1) and measure on n output frequency bands (frexels). Indeed, to evaluate more quantitatively the expressivity of the system, we compute the rank of a matrix whose rows correspond to observables evaluated at different input states. This rank, called kernel quality in the literature^50–52, would indicate the number of linearly independent observables; however, since we scaled them beforehand, it also takes into account affine dependence. For linear regressions, an observable affine to another should not be useful. Since for a global phase, the observables behave like ${\sin }^2\delta =1-{\cos }^2\delta$, ${\cos }^2\delta$ or $\cos \delta \sin \delta$, analytically the rank should be equal to 2 for all n; however, numerical instabilities of the Schmidt decomposition yield small fluctuations. For example, we can see that there are two behaviours only ($O_A~{\cos }^2\delta$ and $O_B~-{\cos }^2\delta$) from the first subplot in Fig. 4d, where we consider correlations in $\widehat{q}$ quadratures alone.

Simulations with the Digital Twin

Our experiment is fully characterized and accurately reproducible via simulation. This allows for faithful reproduction of experimental results and reliable predictions for new configurations, including the impact of additions such as a spatial light modulator on the pump. Although simulations are much slower than the femtosecond-scale nonlinear processes in the waveguides, they constitute a useful Digital Twin of the experiment, enabling interpretation and hypothesis testing.

We begin by modelling the nonlinear PDC process in the waveguides. This involves computing the JSA, determined by the phase-matching function, derived from the waveguide geometry and material (via the Sellmeier equations), and by the complex pump beam profile, which in our protocols encodes the input information. From the JSA, we extract the Schmidt modes and squeezing coefficients via singular value decomposition, the most computationally intensive step. We model the measurement stage by computing overlaps between Schmidt and measurement modes, which allows us to finally reconstruct expected observables for arbitrary pump phase profiles. The core of the simulations stems from a detailed physical model shown in ref. ⁴⁶ and implemented in a Python code (all the results can be retrieved in ref. ⁵³).

Spatial multiplexing

At each timestep, the same input s_k is fed to all R reservoirs in parallel, each with their different parameters α_r and β_r (r ∈ 1, …, R); this ensures that the observables of the R reservoirs are not correlated. Concerning the feedback term, we collect the R observables ${\langle \mathrm{\Delta }{\widehat{q}}_{\left(r\right)}^2\rangle }^{(k-1)}$ and we distribute them in the R inputs with a mask M of size R × R. Each reservoir has then the following input at timestep k:

$${\delta }_r^{\left(k\right)}={\alpha }_r{s}_k+{\beta }_r+\sum _{r^{\prime }=1}^R{M}_{r,r^{\prime }}{⟨\mathrm{\Delta }{\widehat{q}}_{\left(r^{\prime }\right)}^2⟩}^{\left(k-1\right)}r\in 1,\dots ,R.$$ 8

The mask is a full matrix, meaning that the feedback term of each reservoir r carries the observable values from all R reservoirs. Experimentally, the parallelization of the process was implemented sequentially, sending the same input R times to the physical set-up but with different parameters for every repetition and collecting the observables to constitute the feedback term at the end of the cycle.

Pathway towards non-Gaussian features

While polynomial scaling is achievable with Gaussian states, surpassing this performance necessitates the inclusion of non-Gaussian elements. Implementable candidates include spectral mode-selective photon subtraction, already demonstrated for spectrally multimode light fields³⁸. Resources with stronger non-Gaussianity can be introduced via photon-number-resolving detection in similar spectral mode-selective processes. In fact, photon-number-resolving detectors have been used in integrated experimental set-ups to generate strongly non-Gaussian states such as Gottesman–Kitaev–Preskill states⁴⁴. They have also been shown to enable nearly deterministic schemes for the generation of Gottesman–Kitaev–Preskill⁴³ and Schrödinger cat states⁴². The experimental effort should go hand in hand with theoretical investigations of the advantages provided by non-Gaussianity in the CV setting, that is, based on homodyne data readout. We can reasonably expect that even weak non-Gaussianity, introduced by heralding on a low number of detected photons, would enhance the effective nonlinearity of the platform.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41566-026-01880-9.

Author contributions

I.P., J.H., G.L.G., M.C.S., R.Z. and V.P. conceived and designed the experiments. I.P. and J.H. performed the experiments. I.P., E.G. and D.M. analysed the data. I.P., J.G.-B., E.G., D.M., G.L.G., M.C.S., R.Z. and V.P. contributed materials and analysis tools. I.P., D.M., R.Z. and V.P. wrote the paper.

Peer review

Peer review information

Nature Photonics thanks Xiangdong Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Data availability

The experimental datasets generated and analysed during the current study are stored in a laboratory repository and are available from the corresponding authors on reasonable request owing to the large sizes. All of the simulated data that support the findings of this study are publicly available via GitHub at https://github.com/EQ15T/optical_qrc (ref. ⁵³).

Code availability

The code that supports the findings of this study is publicly available via GitHub at https://github.com/EQ15T/optical_qrc (ref. ⁵³).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41566-026-01880-9.