A scalable advantage in multi-photon quantum machine learning

TL;DR

The paper investigates how increasing photon number in linear optical quantum circuits affects learning in photonic quantum machine learning (QML). By framing learning capacity with the data quantum Fisher information matrix $D_L=\mathrm{rank}(\mathcal{Q}_{ij})$ and deriving the maximal capacity $R_L(n)$ along with the critical data size $L_c(n,m)=\lceil(m-1)/n\rceil+1$, it shows a polynomial growth in $n$, enabling reduced training data and improved generalization. The authors validate these results through unitary learning and quantum metric learning tasks on a programmable photonic chip, demonstrating that multi-photon states (e.g., $n=2$) generalize with far fewer training samples and achieve lower test losses than single-photon states. This work provides a rigorous theoretical and experimental demonstration of scalable quantum advantage in photonic QML and points to practical pathways for near-term quantum-enhanced learning on photonic hardware, including extensions to nonlinear optics and non-separable states.

Abstract

Photons are promising candidates for quantum information technology due to their high robustness and long coherence time at room temperature. Inspired by the prosperous development of photonic computing techniques, recent research has turned attention to performing quantum machine learning on photonic platforms. Although photons possess a high-dimensional quantum feature space suitable for computation, a general understanding of how to harness it for learning tasks remains blank. Here, we establish both theoretically and experimentally a scalable advantage in quantum machine learning with multi-photon states. Firstly, we prove that the learning capacity of linear optical circuits scales polynomially with the photon number, enabling generalization from smaller training datasets and yielding lower test loss values. Moreover, we experimentally corroborate these findings through unitary learning and metric learning tasks, by performing online training on a fully programmable photonic integrated platform. Our work highlights the potential of photonic quantum machine learning and paves the way for achieving quantum enhancement in practical machine learning applications.

Figures (4)

• Figure 1: Architecture of multi-photon quantum machine learning.a. Our model consists of multi-photon states with $n$ photons (shown as dots) propagating through a parameterized linear optical circuit $\mathcal{U}(\vb*{x, \theta})$ with $m$ modes, which encodes feature vectors $\vb*{x}$ and trainable parameters $\vb*{\theta}$. Photons are measured at the output modes, where the number of possible outcomes increases with $n$ and leads to different learning results. b. Illustration of the accessible quantum state space during training with respect to $\vb*{\theta}$. The size of the learnable state space increases with $n$, enhancing the expressivity and trainability of the QML model. c. The scaling trend of a photonic QML model describes the relationship between test error and the training dataset size. A higher photon number $n$ corresponds to a larger learning capacity of the model, leading to lower test error and less training data required for learning. d. Our multi-photon QML setup using a programmable integrated photonic circuit. Identical photons are generated by multi-photon sources and injected into a photonic chip. The chip consists of tunable optical interferometers which imprint either feature vector data $\vb*{x}$ or trainable parameters $\vb*{\theta}$ as phase shifts onto the photon state. Each block in the circuit represents a $2\times2$ Mach-Zehnder interferometer containing two tunable phase shifters. The final states are measured via coincidence counting of photons in each mode, and used to update the parameters during the learning process.
• Figure 2: Learning capacity of linear optical circuits.a. A photonic QML model is trained on $L$ training data, consisting of $n$-photon states $\{\ket{\phi_\ell}\equiv \ket{\phi(\vb*{x}_\ell)}\}_{\ell=1}^L$ generated by a data-encoding unitary $S(\vb*{x}_\ell)$ carrying feature vector $\vb*{x}_\ell$. A parameterized unitary $U(\vb*{\theta})$ with $K$ trainable circuit parameters $\vb*{\theta}$ is optimized with respect to the loss function $C(\vb*{\theta})$. b. Sketch of the state space that can be accessed by $U(\vb*{\theta})$ constrained to $L$ training data. The size of this space is characterized by the learning capacity $D_L(n)$ (the rank of DQFIM $\mathcal{Q}_{ij}$), which increases with $n$ and correlates with learning performance. c. Learning capacity $D_L$ as a function of the number of trainable parameters $K$ for different photon numbers $n$ with $m=6$ and $L=1$. $D_L$ increases with $K$, reaching the theoretical maximum value predicted by Eq. \ref{['eq:capacity']} as horizontal dashed lines. d. Maximal learning capacity $R_L(n)$ as a function of the training dataset size $L$ for various $n$ with $m=10$. $R_L(n)$ increases with $L$ and saturates at the critical dataset size $L_\text{c}$, indicated by vertical dashed lines.
• Figure 3: Experimental unitary learning with multi-photon states.a. We aim to learn an unknown unitary $V$ by training a parameterized unitary $U(\vb*{\theta})$ such that $U(\vb*{\theta})=V$. b. Experimental results of learning a $5\times5$ unitary $V$. We plot the matrix closeness $C_\text{M}(U(\vb*{\theta}),V)$ up to local phases and permutations. Perfect generalization corresponds to $C_\text{M}=0$. For $n=2$ photons, we achieve nearly optimal closeness $C_\text{M}\approx0$ using only $L=2$ training data (blue curve). In contrast, $n=1$ photon requires at least $L\geq4$ training data (red curve), and fails with $C_\text{M}\approx 0.3$ for $L=3$ (orange curve). With one additional training state, we can learn the full unitary $V$ including local phases and mode permutations, as experimentally verified in Supplementary Note 4.
• Figure 4: Experimental quantum metric learning with multi-photon states.a. Two input data points $\vb*{x}_1, \vb*{x}_2$ are encoded into the circuit and processed by a trainable parameterized unitary $U(\vb*{\theta})$. The output probability distributions of photon configurations are measured in the Fock space as $p$ and $q$, respectively. We use a cost function based on the cosine similarity $S_\text{C}(p,q)=\sum_i \sqrt{p_i q_i}$ to guide the learning process, as detailed in Supplementary Note 5. After training, data points from the same class are close to each other on the hypersphere, and points from different classes are well separated. b. Simulation of test loss for different photon numbers $n=1,2,3,4,5$. For each $n$, the model is trained until convergence with 100 random seeds, and the average loss on the test data is plotted with standard deviation (shaded region). c. Experimentally measured training and test loss for single-photon and two-photon cases during the training process. d. Measured Gram matrices on the test data at selected epochs (0, 90 and 180), where darker blue implies higher similarity $S_\text{C}$. The top and bottom rows show the results for single-photon and two-photon experiments, respectively. Groups of data from the same classes are highlighted in red boxes. Above each matrix we show the pairwise accuracy, which is defined as the probability of correctly identifying whether a pair of test data belongs to the same or different classes.