Efficient Learning of Long-Range and Equivariant Quantum Systems

TL;DR

The paper tackles learning ground-state properties of quantum Hamiltonians with long-range interactions, extending prior logarithmic data-efficiency results from short-range systems. It develops an equivarient, neighborhood-truncated ML framework that leverages Lieb-Robinson-type bounds to show that observables depend only on nearby Hamiltonian parameters, enabling efficient generalisation. Theoretical guarantees establish logarithmic sample complexity in system size for exponential and certain power-law decays (alpha>2D), with equivariant ML further reducing samples to a constant for sums of local observables in periodic systems; the approach tolerates p-local observables and uses LASSO with a structured feature map, optionally augmented by classical shadows. Experiments on 1D Heisenberg, Ising, and Rydberg chains up to 128 qubits corroborate the scaling predictions and demonstrate concentration phenomena via the CLT for global observables. The results offer a principled route to data-efficient prediction of ground-state properties in realistic, long-range quantum systems and highlight open questions for Coulomb-like interactions (alpha not exceeding 2D) and extensions to thermal or gapless phases.

Abstract

In this work, we consider a fundamental task in quantum many-body physics - finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. For short-range gapped Hamiltonians, a sample complexity that is logarithmic in the number of qubits and quasipolynomial in the error was obtained. Here we extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems. For interactions decaying as a power law with exponent greater than twice the dimension of the system, we recover the same efficient logarithmic scaling with respect to the number of qubits, but the dependence on the error worsens to exponential. Further, we show that learning algorithms equivariant under the automorphism group of the interaction hypergraph achieve a sample complexity reduction, leading in particular to a constant number of samples for learning sums of local observables in systems with periodic boundary conditions. We demonstrate the efficient scaling in practice by learning from DMRG simulations of $1$D long-range and disordered systems with up to $128$ qubits. Finally, we provide an analysis of the concentration of expectation values of global observables stemming from the central limit theorem, resulting in increased prediction accuracy.

Figures (8)

• Figure 1: Overview of the efficient machine learning algorithm. Given a vector $x \in [-1,1]^m$ parameterising a quantum Hamiltonian with long-range interactions, it gets separated into neighbourhoods of the summands in the observable $O$, which are then mapped to features $\vec{\phi}_i$, and concatenated into one full feature vector. This vector is then used as an input of the LASSO model, which is trained to predict $O$ for a new value of $x$.
• Figure 2: $S_{I,\delta}$ for two choices of $J$, once with cardinality $1$ (purple) and once with $2$ (red). Here $D=|I|=2$.
• Figure 3: (a)$1/(\nu-D)$ as a function of $x=\alpha-2D$ for $D=1,\dots,5$. (b)$1/\mu$ as a function of $x=\alpha-2D$ for $D=1,\dots,5$.
• Figure 4: (a)$1/(\nu-D)$ as a function of $D$ at $\alpha=2D+1-\zeta$ with $\zeta=10^{-12}$. (b)$1/\mu$ as a function of $D$ at $\alpha=2D+1-\zeta$ with $\zeta=10^{-12}$.
• Figure 5: Results for Heisenberg model. Each shown data point corresponds to a single trial with $N_\text{test} = 40$ randomly generated test samples. (a) Plot of the number of training samples needed to obtain a fixed additive RMS error of $\epsilon = 0.55$ when measuring $\frac{H}{\sqrt{n}}$ in the Heisenberg chain with open boundary conditions, using $\delta = 4$. The logarithmic fit is obtained from all of the data points starting from 8 qubits. (b) Plot of the average RMS error for a fixed number of training samples $N = 40$ when measuring all $C_{i,i+1} = \frac{1}{3} (X_i X_{i+1} + Y_{i} Y_{i+1} + Z_i Z_{i+1})$ individually, but training only on $C_{01}$, in the Heisenberg chain with periodic boundary conditions, using $\delta = 4$.

Theorems & Definitions (67)

• Theorem 1.1: Theorem \ref{['thm:sample_complexity_single_obs']} informal
• proof : Proof strategy
• Corollary 1.1.1: Corollary \ref{['cor:sample_complexity_equiv']}
• Lemma 2.1: Corollary 2.8 of Bachmann_2011
• Lemma 2.2: Lemma 2.6 (iv) of Bachmann_2011
• Proposition 2.3: Corollary \ref{['cor:delta_eps_exp']}
• Proposition 2.4: Corollary \ref{['cor:delta_eps_alpha_gt_2D']}
• Theorem 3.1
• proof
• Lemma 3.2: mohri2018foundations