Expressibility of neural quantum states: a Walsh-complexity perspective

Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in $\mathbb{C}P^{2^n-1}$ and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that aCLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.

Neural Quantum States (NQS) are a remarkably expressive class of variational ansätze for quantum many-body wavefunctions, yet little is understood about their internal mechanisms: trained on variational objectives alone, how do NQS accurately capture physical observables that they have never been explicitly optimized for? In this work, we present a systematic approach to analyze the internal activations of NQS using sparse autoencoders. We extract features from the residual stream and demonstrate that these features strongly correlate with physical observables such as order parameters, staggered magnetization, and half-chain correlators, across both ground state representation and real-time dynamics. Remarkably, the discovery of these features is entirely unsupervised, with no physical labels provided. We further establish that such features causally affect the corresponding observables predicted by NQS, by showing that targeted, post-training intervention on a single feature smoothly and monotonically steers the corresponding observable, while leaving the variational energy nearly unchanged. These results demonstrate that NQS are not merely functional approximators, but encode rich, interpretable internal representations of physical information. Our approach provides both a diagnostic and an intervention tool for NQS, and serves as a foundation for using mechanistic interpretability towards more reliable, transparent NQS.

Neural quantum states (NQS) provide a flexible and scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoiding the autocorrelation and mixing issues of Markov chain methods. Yet their optimization remains comparatively underexplored: Adam is a scalable method but ignores function space geometry, while stochastic reconfiguration is principled but costly and numerically fragile in large models. To address this gap, we show that variational energy minimization can be viewed as an advantage policy-gradient problem over the Born distribution, motivating trust-region optimization for NQS training. We introduce Proximal Wavefunction Optimization (PWO), a principled trust-region algorithm that clips probability-ratio changes in the amplitude channel and phase increments in the phase channel. PWO avoids explicit matrix inversion, reuses samples across multiple updates, and combines the scalability of first-order optimization with theoretical guarantees. Across Ising and frustrated $J_1$-$J_2$ one- and two-dimensional spin systems, PWO improves stability and wall-clock convergence over Adam, minSR, and SPRING. Finally, we fine-tune a $1.5$B-parameter RWKV-7 model, demonstrating NQS optimization at a scale over three orders of magnitude beyond prior work.