  Solving the ground state and the ground-state properties of quantum many-body
systems is generically a hard task for classical algorithms. For a family of
Hamiltonians defined on an $m$-dimensional space of physical parameters, the
ground state and its properties at an arbitrary parameter configuration can be
predicted via a machine learning protocol up to a prescribed prediction error
$\varepsilon$, provided that a sample set (of size $N$) of the states can be
efficiently prepared and measured. In a recent work [Huang et al., Science 377,
eabk3333 (2022)], a rigorous guarantee for such a generalization was proved.
Unfortunately, an exponential scaling for the provable sample complexity,
$N=m^{{\cal{O}}\left(\frac{1}{\varepsilon}\right)}$, was found to be universal
for generic gapped Hamiltonians. This result applies to the situation where the
dimension of the parameter space is large while the scaling with the accuracy
is not an urgent factor. In this work, we consider an alternative scenario
where $m$ is a finite, not necessarily large constant while the scaling with
the prediction error becomes the central concern. By jointly preserving the
fundamental properties of density matrices in the learning protocol and
utilizing the continuity of quantum states in the parameter range of interest,
we rigorously obtain a polynomial sample complexity for predicting quantum
many-body states and their properties, with respect to the uniform prediction
error $\varepsilon$ and the number of qubits $n$. Moreover, if restricted to
learning local quantum-state properties, the number of samples with respect to
$n$ can be further reduced exponentially. Our results provide theoretical
guarantees for efficient and accurate learning of quantum many-body states and
their properties, with model-independent applications not restricted to ground
states of gapped Hamiltonians.
