Generalization error of Transformer neural quantum states is proven to decay inversely with in-context examples and depth, with required depth scaling only linearly in system size

A rigorous inference-time generalization error bound is established for Transformer neural quantum states under ICL, with pointwise prediction error decreasing inversely with both the number of in-context examples and Transformer depth. Prior work demonstrated the generalization performance of Transformer neural quantum states only empirically, without a quantitative theoretical characterization; this work provides explicit MSE scaling laws and reveals a trade-off between depth and the amount of contextual information. Theorem 1 gives a high-probability bound under i.i.d. samples and queries whose components belong to a function class with bounded first absolute moment, with the full derivation in Appendix A; the proof is constructive, reducing nonlinear prediction to finite-dimensional linear estimation via feature representation, Lasso estimation, and the inexact proximal gradient method.

The Transformer depth required to achieve this generalization guarantee scales only linearly with system size, namely the number of particles in continuous systems or the number of qudits in discrete systems. In contrast to a possible exponential dependence for general quantum states, this result shows that architectural capacity requirements are polynomial (linear) in the stated setting, providing a theoretical basis for the scalability of Transformers in high-dimensional quantum systems. Theorem 1 provides a sufficient depth condition scaling with the number of particles and qudits in the continuous and discrete cases, respectively; Appendix A.5 gives a more precise layer-wise attention-head configuration as a sufficient condition.

The analysis is extended from unconstrained prediction to physically admissible normalized wavefunctions/states and their induced density operators, yielding MSE-based generalization bounds over continuous and discrete domains. Compared with the unnormalized setting of Theorem 1, Theorem 2 introduces normalization and operator-level error analyses, converting pointwise error control into Frobenius-norm error over the full state space. Theorem 2 gives high-probability bounds under the same setting as Theorem 1, with the proof in Appendix D; the text explains that the volume factor in the continuous case and the cardinality factor in the discrete case arise from the size of the configuration space rather than additional statistical or optimization complexity.

The framework extends to a broader class of quantum-property prediction tasks and yields polynomial depth requirements for structured quantum states represented as MPOs. The analysis is not restricted to wavefunction/state prediction and can cover quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and energy variances as deterministic functions; for an MPO with bond dimension, the required depth scales polynomially in the degrees of freedom rather than exponentially. Equations (29) and (30) give the depth condition and generalization bound for the MPO case; the authors note that this polynomial scaling relies on the structured MPO representation and does not cover general quantum states with exponentially large degrees of freedom. Numerical experiments (Figure 4) show NMSE decreasing monotonically with the number of in-context samples and higher NMSE for larger systems at fixed capacity, consistent with the theory.

The results target Transformer neural quantum states that use fixed parameters at inference and perform ICL through in-context examples, applicable to continuous bounded-energy domains and discrete many-body quantum configuration spaces, with samples and queries drawn i.i.d. from a distribution sharing the structural properties of the training distribution. The theorems provide existence guarantees: there exists a Transformer architecture achieving the stated bound, rather than uniform optimality over all parameter choices. For structured quantum states represented as MPOs with bond dimension, the depth requirement scales polynomially in the degrees of freedom, covering a broad but not exhaustive class of low-entanglement quantum states. The framework extends to prediction of quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and energy variances as deterministic functions of the underlying state.

The theoretical analysis is built on an augmented embedding matrix introduced for the proof, whereas the numerical experiments use the practical embedding matrix; the authors note that rigorously extending the theory to the practically relevant setting remains future work. The current assumptions are proof-oriented, and identifying assumptions intrinsic to quantum learning problems is an open question. Moreover, the theorems establish only the existence of architectures achieving the guarantees; whether practical training algorithms provably converge to such solutions remains unresolved. The apparent difference between the continuous and discrete volume factors stems from different normalizations of the configuration space, and under a more physical per-particle volume scaling the two become consistent, an interpretation that depends on the adopted normalization convention.