Time-series forecasting with multiphoton quantum states and integrated photonics

TL;DR

This work demonstrates photonic quantum reservoir computing on a reconfigurable integrated circuit using single- and multiphoton inputs to forecast time-series data. A four-mode PIC implements a quantum reservoir whose dynamics are steered by adaptive feedback and encoded inputs, with a classical linear readout trained by Ridge regression. Crucially, indistinguishable two-photon inputs significantly enhance expressivity and nonlinear function approximation, yielding better performance on nonlinear benchmarks like NARMA and Mackey-Glass while preserving memory through feedback. The results indicate quantum correlations and multiphoton states can enrich reservoir dynamics and improve temporal processing in compact photonic hardware, offering a path toward efficient quantum-enhanced neuromorphic computing. The work also shows that classical correlations alone offer limited gains, highlighting the value of quantum resources in this context.

Abstract

Quantum machine learning algorithms have very recently attracted significant attention in photonic platforms. In particular, reconfigurable integrated photonic circuits offer a promising route, thanks to the possibility of implementing adaptive feedback loops, which is an essential ingredient for achieving the necessary nonlinear behavior characteristic of neural networks. Here, we implement a quantum reservoir computing protocol in which information is processed through a reconfigurable linear optical integrated photonic circuit and measured using single-photon detectors. We exploit a multiphoton-based setup for time-series forecasting tasks in a variety of scenarios, where the input signal is encoded in one of the circuit's optical phases, thus modulating the quantum reservoir state. The resulting output probabilities are used to set the feedback phases and, at the end of the computation, are fed to a classical digital layer trained via linear regression to perform predictions. We then focus on the investigation of the role of input photon indistinguishability in the reservoir's capabilities of predicting time-series. We experimentally demonstrate that two-photon indistinguishable input states lead to significantly better performance compared to distinguishable ones. This enhancement arises from the quantum correlations present in indistinguishable states, which enable the system to approximate higher-order nonlinear functions when using comparable physical resources, highlighting the importance of quantum interference and indistinguishability as a resource in photonic quantum reservoir computing.

Figures (5)

• Figure 1: Schematic overview of the realized photonic quantum reservoir computing.(a) Scheme of a classical echo state network (ESN) for time-series forecasting jaeger2001echo. The input data is injected into a fixed, recurrent reservoir of nonlinear nodes. The reservoir output is linearly combined and trained via linear regression to predict future values. (b) Photonic implementation of a quantum reservoir computing scheme. Two-photon input states, either indistinguishable ($2_{ph,I}$) or distinguishable ($2_{ph,D}$) with a control on the delay $\tau$, are injected into a four-mode reconfigurable photonic integrated platform. This includes the tunable phases $\phi_i$ (with $i=1,2,A,B,C,D,3,4$). The input data is encoded into one phase of the central layer. The output statistics are reconstructed using single-photon detectors. A feedback-loop mechanism is based on the reinjection of the past measurement outcomes and is implemented via phase reconfiguration of $\phi_D$ and $\phi_4$. On the right, the plot of an example of a component of the output probability distribution as a function of the internal phases $\phi_A$, $\phi_D$ and $\phi_4$.
• Figure 2: Characterization of the quantum reservoir computing model: short-term memory and expressivity. We report the quantum reservoir performance achieved for different photonic input states: two indistinguishable photons ($n_{\text{ph}, I} = 2$, light blue dots), two distinguishable photons ($n_{\text{ph}, D} = 2$, pink dots), and a single photon ($n_{\text{ph}} = 1$, green dots). (a) The short-term memory capacity is measured via the coefficient of determination $R^2_d$ as a function of the delay $d \in [0, 6]$. We report the results obtained when injecting into the device two indistinguishable photons but without feedback dynamics (yellow dots). The dataset contains 497 points in total. (b) The expressivity of the QRC model is measured by looking at the reservoir performance, in terms of the mean squared error (MSE) on the test set, in reconstructing $f_n(x) = x^n$ for $n = 2, \ldots, 13$. Insets display the predicted outputs for $n = 5$ and $n = 7$ for different photon-input configurations compared to the target (black lines). It is also plotted the corresponding training output for $n_{\text{ph}, I} = 2$ (dark blue line). The datasets contain 150 points in total. (c) The expressivity of the model is further explored through the approximation of polynomial functions defined as $f_N(x)=\sum_{n=1}^N (-1)^n x^n$, with $N=1, \ldots, 7$. The MSE is plotted as a function of $N$. (d) Three examples of polynomial function approximations for selected values of $N=4, 5, \text{and } 6$. Error bars and shaded areas in the plot refer to the statistical fluctuations evaluated from 100 Monte Carlo extractions to account for the presence of Poissonian sampling noise.
• Figure 3: Quantum reservoir performance on machine learning benchmarks: temporal XOR and NARMA-$N$. The tasks are evaluated for different input configurations: two indistinguishable photons ($n_{\text{ph}, I} = 2$, light blue), two distinguishable photons ($n_{\text{ph}, D} = 2$, pink), and single photons ($n_\text{ph} = 1$, green). (a) The temporal XOR is defined as $s_k \bigoplus s_{k-d}$, where $s_k \in \{0, 1\}$ is the input sequence and $d$ is the temporal delay. The task is evaluated using classification accuracy for delays $d = 1,\ldots,4$. The datasets contain 300 points in total. (b) The output predictions are compared with the targets for both $n_{\text{ph}, I} = 2$ and $n_{\text{ph}, D} = 2$ configurations, and for $d = 1$ and $d = 3$. While both perform well at $d = 1$, only the $n_{\text{ph}, I} = 2$ configuration successfully addresses the increased memory and nonlinearity demands at $d = 3$. (c) Performance on the test set for the NARMA-$N$ task atiya2000new. Also this testbed shows the enhancement due to quantum correlations; indeed, the configuration with $n_{\text{ph}, I}=2$ always outperforms the one with $n_{\text{ph}, D}=2$ and $n_\text{ph}=1$. The datasets contain 500 points in total. (d) The test with respect to the target for the three configurations investigated is plotted for the function NARMA-$5$. Both the predicted and the target values $y_k$ are normalized to ensure a fair comparison. (e) The different normalized mean absolute errors between the predicted and the target values are compared in a histogram. Error bars and shaded areas in the plot refer to the statistical fluctuations evaluated from 100 Monte Carlo extractions to account for the presence of Poissonian sampling noise.
• Figure 4: Mackey-Glass time series forecasting. (a) Chaotic attractor of the Mackey–Glass (MG) time-delay differential equation, simulated with parameters $\alpha = 0.2$, $\beta = 10$, $\gamma = 0.1$, and delay $\tau = 17$, which induces chaotic dynamics. The differential equation is numerically integrated, then time-discretized, normalized, and directly encoded as input to the quantum reservoir system. (b) and (c) Forecasting of future time steps with prediction horizon $t_f = 3$, comparing the target signal (black) with the predicted outputs during both training and testing phases. Two photon-input configurations are considered: (b) injection of two distinguishable photons ($n_{\text{ph}, D} = 2$, shown in salmon for training and pink for test), and (c) injection of two indistinguishable photons ($n_{\text{ph}, I} = 2$, shown in dark blue for training and light blue for test). The achieved MSE is reported in the title of the corresponding plot. Each dataset contains 390 points. Shaded areas in the plot refer to the statistical fluctuations evaluated from 100 Monte Carlo extractions to account for the presence of Poissonian sampling noise.
• Figure 5: Performance of NARMA-5 task using reservoir without feedback-loop mechanism. (a) Output prediction (dark green) versus target (black) for the configuration with $n_{\text{ph}, D}=2$. (b) Output prediction (orange) versus target for the configuration with $n_{\text{ph}, I}=2$. In both cases, the datasets are reshuffled to eliminate any temporal correlations, and thus preventing the reservoir from retaining memory of past inputs. The resulting mismatch confirms that memory is essential for solving time-dependent tasks. Both the test and the target outputs $y_k$ are min–max normalized to the range $[0,1]$ to ensure a fair comparison between $n_{\text{ph}, D} = 2$ and $n_{\text{ph}, I} = 2$. Error bars and shaded areas in the plot refer to the statistical fluctuations evaluated from 100 Monte Carlo extractions to account for the presence of Poissonian sampling noise.