Quantum inference on a classically trained quantum extreme learning machine

Abstract

Quantum extreme learning machines (QELMs) are unconventional computing architectures that bear remarkable promise in both classical and quantum machine-learning tasks, such as the estimate of quantum state properties. However, the probabilistic nature of quantum measurements demands extensive repetitions for training to precisely estimate expectation values, imposing stringent trade-offs among experimental resources, acquisition time, and signal-to-noise ratio, particularly for large datasets. Here we introduce a paradigm shift by harnessing the correspondence between stimulated and spontaneous emission. The QELM is trained exclusively with intense classical fields, yet it performs inference directly on previously unseen quantum input states to predict their quantum properties. This strategy dramatically reduces acquisition times while substantially enhancing the signal-to-noise ratio. Using frequency-bin-encoded biphoton states, implemented here for the first time in a quantum machine-learning architecture, we demonstrate entanglement witnessing of two-qubit states with (93 +- 4)% accuracy, multi-dimensional entanglement detection, and learning of the Hamiltonian governing photon-pair generation with a fidelity of (96 +- 4)%. By establishing classical training as a scalable route to quantum feature extraction, our results bridge macroscopic observables and nonclassical correlations, opening a new pathway toward faster and more robust quantum neural networks

Figures (9)

• Figure 1: Photonic quantum extreme learning machine and its classical training. (a) In the paradigm shift introduced here, the QELM is trained using classical light by exploiting the correspondence between stimulated and spontaneous emission. A properly amplitude-shaped coherent state (blue) seeds a third-order parametric nonlinear process, such as four-wave mixing. The resulting stimulated coherent beam (red) then propagates through the reservoir (here represented as a network of interconnected nodes without recurrent memory), and its output spectrum is recorded by a spectrum analyzer. If the seed is prepared in one of the asymptotic states of the full system, for example, in spectral mode $j$, the spectral intensity of the stimulated mode $k$ at the output becomes proportional to the photon-pair emission rate in the mode pairs $(k,j)$ in the corresponding spontaneous process. This duality is exploited to train the QELM, thereby determining the weights used in the linear regression. (b) Once the classical training phase is complete, inference is performed on quantum input states generated via the corresponding spontaneous process, such as spontaneous four-wave mixing. This process involves the inelastic scattering of two pump photons, leading to the generation of entangled biphoton states. After generation, the quantum state propagates through the same reservoir used during classical training. The photon-pair detection probabilities are then processed through the linear regression model previously trained with classical signals (transfer learning) to infer the expectation value of a property of the input quantum state, such as the entanglement witness $\langle \mathcal{W}\rangle$. Using classical beams as seeds significantly increases the output photon number compared to coincidence measurements, thereby reducing the time required to train the QELM by several orders of magnitude.
• Figure 2: Experimental implementation of a frequency-bin QELM. (a) Training of the QELM is performed by stimulated FWM. To obtain an output intensity of the stimulated beam which is proportional to the coincidence probability $C_{kj}$ of detecting a signal-lder photon-pair in frequency modes $k$ and $j$ in the corresponding spontaneous process, a laser seed (blue) is shaped into a superposition of frequency-bins that mimics the asymptotic output state of the $j^{\mathrm{th}}$ frequency mode of the system (see liscidini2012asymptotic and Supplementary Materials). The stimulated beam (red) is a coherent state that undergoes evolution through the reservoir, and its spectrum is acquired by an OSA. The output intensity $I_{kj}$ on the $k^{\mathrm{th}}$ bin is proportional to $C_{kj}$. The intensities $I_{kj}$ are used to train the QELM using linear regression, from which the weights are computed. (b) Inference is performed on quantum input states generated by SpFWM. Light from a pump laser (green) is sent through a sequence of an EOM and a WS, tailoring its spectrum. The pump is routed to an integrated silicon photonic waveguide, where broad-band entangled photon-pairs are generated by SpFWM. A second WS completes the state synthesis by defining the output frequency-bin encoded biphoton state by setting amplitude and phase to the frequency-bins. The state evolves through the reservoir, consisting of two EOMs acting on the demultiplexed signal and idler modes, realizing a tight-binding-type interaction, whose intensity and phase depend on the RF driving of the two EOMs. The reservoir mixes the frequency-bins, and expands the dimensionality of the Hilbert space. After narrowband filtering, which selects one of the possible signal-idler frequency-bins combinations $(k,j)$, coincidence detection is performed by a pair of SNSPDs, yielding the value $C_{kj}$. The multiple frequency-bin combinations represent the output layer of the QELM. These are used as inputs to the linear regression, which is applied using the weights assessed in the training phase. EOM: electro-optic phase modulator. WS: waveshaper. SpFWM: spontaneous four-wave mixing. StFWM: stimulated four-wave mixing. SNSPD: superconducting nanowire single-photon detector. OSA: optical spectrum analyzer.
• Figure 3: Synthesis and characterization of two-qubit states. a) Box-plots showing the fidelity of the reconstructed density matrices with respect to the expected targets. Random states are prepared in the SP (Eq.(\ref{['eq:SP_states']})) and DP Eq.(\ref{['eq:DP_states']}) configurations. Datasets are acquired either using StFWM or coincidence measurements (SpFWM). (b) Distribution of the fidelity between the expected $6\times6$ matrices describing the frequency correlations at the output of the QELM and the corresponding experimental patterns acquired via coincidence measurements (SpFWM) and stimulated emission (StFWM). (b) A representative example of frequency correlations at the output of the QELM for a state prepared in the SP configuration. From left to right: the expected pattern, the pattern obtained via coincidence measurements of spontaneously emitted photon-pairs (SpFWM), and the pattern obtained via StFWM by measuring light intensities with an optical spectrum analyzer. (c) Representative frequency correlation matrix for a state prepared in the DP configuration.
• Figure 4: Entanglement witness for two qubits. Inferred versus true values of the entanglement witness $\langle \mathcal{W} \rangle$ on a representative test set acquired through coincidence measurements of spontaneously emitted photon-pairs. Diamond markers indicate separable states. The confusion matrix in the inset illustrates the binary classification of states with $\langle \mathcal{W}\rangle \le 0$ and $\langle \mathcal{W}\rangle \ge 0$.
• Figure 5: Entanglement witness in high-dimensions. (a-c) Representative examples of frequency-bin correlation matrices at the output of the QELM. The panels show the expected patterns (Expected) for randomly chosen qubit (a), qutrit (b), and ququart (c) states. The rightmost column panels display the corresponding patterns measured via stimulated emission (StFWM).