Exponentially improved efficient machine learning for quantum many-body states with provable guarantees

TL;DR

This work shows that when the parameter-space dimension m is a fixed constant, one can achieve polynomial-sample learning of quantum many-body states and their properties by enforcing physical constraints through Positive Good Kernels (PGKs) in a reproducing-kernel framework. The key idea is to construct kernel-based estimators σ_N(x) that preserve density-matrix properties while exploiting the continuity of ρ(x) over the parameter space, enabling guarantees with sample complexity N = poly(ε^{−1}, n, log(1/δ)). Moreover, for local quantum-state properties, the dependence on the number of qubits n can be reduced to poly(log n). Theoretical results are complemented by numerical demonstrations on the quantum XY model, illustrating polynomial scaling in ε and successful learning of ground-state energies and long-range order, with extensions to RKHS-based generalization bounds and non-uniform input distributions. Overall, the work provides model-independent, provable guarantees for efficient quantum-state learning beyond the prior focus on gapped Hamiltonians, with practical implications for quantum simulation and metrology.

Abstract

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 learning of quantum many-body states and their properties, with model-independent applications not restricted to ground states of gapped Hamiltonians.

Figures (3)

• Figure 1: (a) Comparison between the exact (red dashed curve) ground-state energies per qubit and the positive good kernel (PGK) predicted (blue dots) ones, for a finite-size one-dimensional quantum $XY$ model with sample size $N = 10^6$. The black arrows indicate the locations where the energy curve is continuous but not smooth due to the vacua competition PasqualePRA2009OkuyamaPRE2015. Note that there are two additional but not obvious points in the vicinity of $h/J = \pm 1$ where the energy curve is also continuous but not smooth due to the vacua competition, which are not indicated in this figure. (b) Double-logarithmic plot of $\varepsilon^{-1}$ vs $N$ (blue dots), and a linear regression (red dashed line) with a slope $\approx 0.45$ and an R-squared score $\approx 0.99$, confirming a polynomial scaling of the sample complexity. We use the rectangular Fejér kernel with $\Lambda = 50$, $n=5$ qubits, $\gamma = 1/3$, and samples from a uniform distribution in both (a) and (b). The error $\varepsilon$ in (b) is obtained as the maximal error between the predicted and true energies. The mean values and error bars in both panels are calculated with $30$ independent runs.
• Figure 2: Predicting the ground-state long-range order (spin-spin correlation function) $\lim_{r \rightarrow \infty}(-1)^r \langle S_0^x S_r^x\rangle$ of the quantum $XY$ model in the thermodynamic limit. (a) Double-logarithmic plot of the positive good kernel (PGK) prediction accuracy $\varepsilon^{-1}$ vs $N$ (blue dots) within the two-dimensional parameter space $-3/2 \le h/J \le 3/2$ and $0 \le \gamma \le 1$. A linear regression (red dashed line) with a slope $\approx 0.45$ and an R-squared score $\approx 0.98$ confirms a polynomial scaling of the sample complexity. (b) Comparison between the exact (red dashed curve) long-range spin-spin correlation function and the PGK predicted (blue dots) one with $N = 10^5$, for a quantum $XY$ chain with one-dimensional parameter space $-3/2 \le h/J \le 3/2$ and a fixed value of $\gamma = 1/3$. We use uniform distributions, a Fejér kernel with $\Lambda = 50$, and $30$ independent runs as in Fig. \ref{['fig:XY_model']}.
• Figure A1: Schematics of bounding the prediction error of positive good kernel (PGK) predictors, with polynomial number of samples. The target function to be learned (e.g., entries of the density matrix, expectation values of operators with respect to a target quantum state, etc.), is a continuous function defined on the compact parameter space ${\cal{X}}$, the space of which is denoted by ${\cal{C}} ({\cal{X}})$. In the left bottom of the Approximation-Generalization-Prediction (A-G-P) triangle, the model space is obtained via the convolution (denoted by a $*$) between the function itself and a PGK $K_{\Lambda}$. In the right bottom, the kernel predictor $f_N$ is defined as a discrete convolution over the sample set ${\cal{S}}$ of size $N$. With the physical constraints and proper PGKs, if both the approximation error $d_{\Lambda}(f)$ and the generalization error $d_N(f)$ can be bounded from above up to a small error bound of ${\cal{O}}(\varepsilon)$, in a consistent manner and with polynomial number of samples $N$, then the prediction error can be bounded at ${\cal{O}}(\varepsilon)$, according to the triangle inequality for distances and norms. Note that $B$ is twice the upper bound of the target function $f(x)$ for $x \in {\cal{X}}$ (related to the number of qubits $n$), and $\delta$ is the probability of success with respect to ${\cal{S}}$.

Theorems & Definitions (7)

• Theorem 1: Efficient learning of quantum-state representations with positive good kernels (PGKs)
• Theorem 2: Efficient learning of quantum-state properties with positive good kernels (PGKs)
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1