Deep Quantum Graph Dreaming: Deciphering Neural Network Insights into Quantum Experiments

TL;DR

The paper tackles the interpretability gap in neural networks trained on quantum optical experiments by applying an inceptionism-based Deep Dreaming approach to quantum graphs. Quantum experiments are modeled as complete quadripartite graphs with edge weights $\omega_{a,b}$, and a feed-forward network learns to predict properties like state fidelities to $|\mathrm{GHZ}\rangle$ and $|\mathrm{W}\rangle$ and entanglement measures $\mathrm{Tr}(\rho^2_{\mathcal{M}})$. Inverse training (dreaming) with frozen weights reveals that the network shifts initial property distributions and uncovers a progression from simple feature detection in early layers to complex, entanglement-related structures in deeper layers, while also producing novel graphs beyond the training set. The study demonstrates that this approach can yield interpretable insights into AI-driven quantum experiment design and suggests paths to scaling to larger graphs and broader quantum tasks, with potential impact on reliable, graph-based discovery in quantum optics.

Abstract

Despite their promise to facilitate new scientific discoveries, the opaqueness of neural networks presents a challenge in interpreting the logic behind their findings. Here, we use a eXplainable-AI (XAI) technique called $inception$ or $deep$ $dreaming$, which has been invented in machine learning for computer vision. We use this technique to explore what neural networks learn about quantum optics experiments. Our story begins by training deep neural networks on the properties of quantum systems. Once trained, we "invert" the neural network -- effectively asking how it imagines a quantum system with a specific property, and how it would continuously modify the quantum system to change a property. We find that the network can shift the initial distribution of properties of the quantum system, and we can conceptualize the learned strategies of the neural network. Interestingly, we find that, in the first layers, the neural network identifies simple properties, while in the deeper ones, it can identify complex quantum structures and even quantum entanglement. This is in reminiscence of long-understood properties known in computer vision, which we now identify in a complex natural science task. Our approach could be useful in a more interpretable way to develop new advanced AI-based scientific discovery techniques in quantum physics.

Figures (6)

• Figure 1: Brief overview of quantum graphs. In this work, we consider complete graph representations of two-dimensional, quadripartite quantum graphs. We let $\omega_{a,b}$ denote the weight of the edge connecting vertex $a$ to vertex $b$. The weight's magnitude is indicated by the transparency of the edge and the presence of a diamond signifies a negative edge weight. The creation of every possible state is conditioned on three possible types of perfect matchings, which are distinguished in terms of their direction.
• Figure 2: Quantum Graph Deep Dreaming. (a) The weights and biases of a feed-forward neural network are continually updated during training to predict a property such as fidelity of a given input random quantum experiment represented by a graph. (b) In the deep dreaming process, the weights and biases of the network are frozen. The weights of an initial input graph are updated iteratively to maximize the output of the feed-forward network, which gives the network's prediction on the aforementioned property.
• Figure 3: Dreaming results for on the output layer of different neural network architectures. (a) Evolution of an input graph's fidelity with respect to (i) the GHZ state and (ii) the W state when dreaming on the $[400^{3}, 10]$ neural network; we also observe (iii) the evolution of an input graph's concurrence when dreaming on the $[800^{7}]$ network. For each case, we show the intermediate steps of the input graphs' evolution to its dreamed counterpart and only show edges whose weights are above a threshold of 0.4. These intermediate steps reveal that, in inverse-training, edges of perfect matchings which do not positively contribute to the target property are mitigated. (b) Distribution of initial vs. dreamed fidelities with respect to (i) the GHZ state and (ii) the W-state, as well as (iii) the mean value of tr$(\rho_{\mathcal{M}}^{2})$. We observe that most dreamed examples exceed the upper bound of the original dataset, attesting to our tool's ability to find quantum graphs that are novel to the original dataset.
• Figure 4: Extracted strategies from the evolution of certain states when dreaming on the output layer of the neural network. We discern the strategies employed by the inverse training routine when applied to a network tasked to optimize (a) the GHZ-State Fidelity, (b) the W-State Fidelity, and (c) the mean value of tr$(\rho_{M}^{2})$ by considering several initialisations for each case. For each graph, we only show edges with weights greater than 0.3. We find that the network attempts to construct perfect matchings (PMs) of terms which positively contribute to the property value and whose weights add up to 1. Conversely, we find that the network eliminates unwanted terms by either directly reducing the edge weights of the PM corresponding to that term, or by introducing negative, disjoint perfect matchings of that term. For (c), we observe that the network 'selects' a term in the initial state to be minimized, then creates terms that are separable across two or more bipartitions with respect to the remaining states.
• Figure 5: Information entropy throughout each different neural network architecture. (a) Workflow behind computing the mean information entropy for each layer of the trained neural network. We dream with multiple input graphs on each neuron in the neural network. To account for the diversity of structures that a neuron is interested in seeing, We compute the mean probability amplitudes for every possible perfect matching corresponding to each ket. We thereby observe the overall graph, which the neuron sees best. We may then compute the information entropy of each neuron, $H_{i,j}(p)$, and the mean information entropy of the layer, $\overline{H_{i,j}(p)}$. This gives us a measure of the complexity of structures seen by the neural network. As conveyed in the different $p_{i,j}$ for each dreamed graph, we note the variety of structures which the network over-interprets; this illustrates the multifactedness of the neurons. (b) Mean information entropy plots for the (i) $[400^{4}]$ (ii) $[49^{10}]$ and (iii) $[36^{26}]$ neural network architectures. A general trend that we may discern in all three cases is that the mean information entropy converges to a minimum in the lower layers and then gradually increases as we go deeper. We may attribute this to the intuition that the network initially learns to recognize simpler structures, then learns increasingly complicated ones as we go deeper within the network.