Universal 2-Local Symmetry-Preserving Quantum Neural Networks for Fermionic Systems

Abstract

Simulating quantum many-body systems represents a fundamental challenge where classical machine learning methods are severely bottlenecked by the exponential curse of dimensionality. Variational Quantum Algorithms (VQAs) offer a native paradigm to tackle this by optimizing parameterized unitary evolutions to find the ground states of problem Hamiltonians. However, the efficacy of these VQA is deeply hindered by the challenge of balancing the preservation of critical physical symmetries with the strict constraints of hardware implementability. In this work, we address this dilemma by proposing a hardware-efficient, symmetry-preserving ansatz fortified with complete theoretical guarantees for fermionic systems, termed the Hamming Weight Preserving (HWP) ansatz. We establish the necessary and sufficient conditions for 2-local HWP operators to achieve subspace universality, formally debunking the prevailing assumption that truncation-free simulation requires complex high-order interactions. Empirical validations corroborate our theoretical guarantees, showcasing the exact approximation of arbitrary unitary matrices within the HWP subspace. Crucially, we demonstrate the exceptional versatility of the proposed approach by deploying the exact same ansatz across distinct fermionic models, including diverse molecular electronic structures and the Fermi-Hubbard model. Our proposed HWP ansatz consistently suppresses ground-state energy errors below $1 \times 10^{-10}$ Ha, achieving a level of precision that surpasses the stringent threshold of chemical accuracy by multiple orders of magnitude. This work establishes a complete, theoretically fortified 2-local framework for symmetry-preserving computation, offering a highly universal and hardware-efficient building block for advancing quantum machine learning and fermionic many-body simulations.

Figures (8)

• Figure 1: Two-qubit HWP gates in related works with their unitary matrices and possible decomposition. The two-qubit elementary gates utilized in the decomposition include CX, CZ, and $\sqrt{\text{iSWAP}}$ (unitary matrix in the bottom right). We have RZZ farhi2014quantum, Givens Rotations wecker2015solvingjiang2018quantum, XY-interaction bacon2001encodedterhal2002classical, onsite stanisic2022observing, $A$ gate gard2020efficient, and non-parameterized SWAP gate.
• Figure 2: Generative relationships of the basis matrices. Directed arrows indicate the generative pathways via commutators. Although successive commutations progressively tensor additional Pauli-Z ($\sigma_z$) operators, they strictly do not yield any new type of basis matrix, thereby explicitly demonstrating the closure of the generated algebra.
• Figure 3: Implementation details of the BS ansatz.a. A possible circuit implementation of the proposed BS gate. b. The implementation of the BS ansatz with both NN and FC connectivity. The physical qubit layout is depicted on the left. FC allows more generators per layer than NN connectivity. Additionally, certain HWP gates, e.g., the BS gate, are directional. To maximize expressivity, we alternate between layers containing BS gates and their reverse counterparts.
• Figure 4: Results for unitary approximation. We iterate through all the cases for $d_k=\{\tbinom{5}{1},\tbinom{5}{2},\tbinom{6}{2},\tbinom{6}{3}\}=\{5,10,15,20\}$, with both GR and BS gates for NN and FC connectivity. For each $d_k$, 100 unitary matrices are randomly sampled based on Haar measure zyczkowski1994randommezzadri2006generate. a. The training curves for BS with NN connectivity. b. The training curves for BS with FC connectivity. c. The loss function is plotted versus the number of parameters. Both BS-NN and BS-FC show a similar decreasing pattern with the number of parameters needed for exact approximation at around $d_k^2$. Inset shows the results for GR-NN and GR-FC, with neither method getting close to $\mathcal{L}(\theta)=0$.
• Figure 5: Results for simulating molecular electronic structures.a. Potential energy curves of four molecules w.r.t. the bond length, with the corresponding absolute errors compared to the exact energy shown in b. HF stands for the energy with Hartree-Fock state, and GS stands for the exact energy. The grey region shows results within chemical accuracy (less than 1.6 milli-hartrees). c. Potential energy surface of H$_2$O, and the energy error of $100$ sampled molecule structures for both UCCSD and BS. All data points are a minimum of 10 random seeds, with error bars indicating the range from minimum to maximum.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Lemma 2
• Theorem 3
• proof
• Lemma 4
• proof
• Corollary 5