Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems

TL;DR

The paper studies the superposition-of-product-states (SPS) variational ansatz as a geometry-agnostic, CP-decomposition-based tool for quantum many-body ground-state problems. It characterizes SPS statistics, entanglement capacity, and trainability, revealing restricted typicality and the absence of barren plateaus, with gradient variances scaling as $1/M^2$. Through ground-state searches in tilted Ising models across 1D and 3D geometries with nearest-neighbor, long-range, and random couplings, SPS achieves high accuracy with moderate $M$ and can outperform DMRG in higher dimensions and disordered settings. The work highlights SPS as a scalable, parallelizable approach with analytic gradient access, and outlines extensions to spatially modulated, complex-valued, and time-evolving states as promising directions.

Abstract

Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-$1/2$ systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models -- including one-dimensional and three-dimensional with short- and long-range interaction, and a random network -- demonstrating that SPS can attain high accuracy.

Figures (6)

• Figure 1: Distributions of 10,000 SPS norms $\mathcal{Z}$ over $\mathcal{D}$ for a system size $L=20$ at different values of $M$. The norms are centered around $M/3$, with the spread increasing as $M$ grows. The inset highlights the small-$M$ regime, where clear deviations from the theoretical Gaussian form can be observed (dashed curves).
• Figure 2: The variances of 10,000 values of $\expval{\sigma^x_{L/2}}$ over $\mathcal{D}$ as a function of $L$ for different $M$. The variances remain constant as $L$ increases. This is an emergence of restricted typicality for the SPS ansatz in the large-$M$ limit.
• Figure 3: Typical 2-Rényi entropies obtained from 10,000 random SPS samples. (a) The distribution of the typical 2-Rényi entropies for $L=20$. The entropies typically concentrate far from the theoretical maximum, even as $M$ increases. (b) The minimum relative errors from the theoretical maximum saturate for larger $L$.
• Figure 4: Trainability of the SPS ansatz is analyzed by evaluating the gradients of a local observable with respect to the variational parameters across 10,000 random samples. Panels (a) and (b) correspond to the case of $L=20$ and $M=32$. In panel (a), the gradients with respect to the coefficients $c_m$ are identical across all $m$. In panel (b), fixing $m=1$, the gradients with respect to the site index $l$ vanish everywhere except at the site where the local observable is applied. Panels (c) and (d) display the gradient variances as functions of system size $L$ and the number of product states $M$, focusing on derivatives with respect to $c_1$ and $\theta^1_{L/2}$, respectively. Although the case $M=2$ in panel (c) shows fluctuations due to finite-sample effects, overall the variances decrease and saturate for large $L$, while decaying polynomially with $M^2$. These results demonstrate that the SPS ansatz avoids barren plateaus, thereby ensuring efficient optimization even for large quantum systems.
• Figure 5: The performances in the ground state searches task with the SPS ansatz. The performances with different $M\in \{1,2,4,\ldots,512\}$ are quantified by the minimum relative error, from 20 different initialized SPS parameters, in the SPS ansatz’s approximation of the ground state energy compared to the target ground state energy $E_{\text{GS}}$ for a tilted Ising model with $J=-1$, $h_z = 0.25$ on (a) 40-qubit 1D nearest-neighbor connection for $h_x \in \{0.05,0.5,1.0,1.5,2.0,3.0,4.0,5.0\}$, (b) ($4\times4\times4$)-qubit 3D nearest-neighbor connection for $h_x = \{0.05,1.0,3.0,5.0,7.0\}$, (c) 40-qubit 1D all-to-all connection for $\alpha\in\{0.0,0.5,1.0,1.5,2.0\}$ with $(J,h_x,h_z) = (-1,3.0,0.25)$, (d) ($4\times4\times4$)-qubit 3D all-to-all connection for $\alpha\in\{0.0,0.5,1.0,1.5,2.0\}$ with $(J,h_x,h_z) = (-1,7.0,0.25)$, and (e) 40-qubit random connected graph for $p\in\{0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95\}$. The gradient background is shaded according to the ferromagnetic correlator $C_F$, with lighter shading corresponding to lower values.