Zimu Li

The generation of k -designs (pseudorandom distributions that emulate the Haar measure up to k moments) with local quantum circuit ensembles is a problem of fundamental importance in quantum information and physics. Despite the extensive understanding of this problem for ordinary random circuits, the crucial situations in which symmetries or conservation laws are in play are known to pose fundamental challenges and remain little understood. Here, we construct explicit local unitary ensembles that can achieve high-order unitary k -designs under transversal continuous symmetry, in the particularly important SU ( d ) case. Specifically, we define the convolutional quantum alternating (CQA) group generated by 4-local SU ( d ) -symmetric Hamiltonians as well as associated 4-local SU ( d ) -symmetric random unitary circuit ensembles and prove that they form and converge to SU ( d ) -symmetric k -designs, respectively, for all k < n ( n − 3 ) / 2 , with n being the number of qudits. A key technique that we employ to obtain the results is the Okounkov-Vershik approach to S n representation theory. To study the convergence time of the CQA ensemble, we develop a numerical method using the Young orthogonal form and the S n branching rule. We provide strong evidence for a subconstant spectral gap and certain convergence time scales of various important circuit architectures, which contrast with the symmetry-free case. We also provide comprehensive explanations of the difficulties and limitations in rigorously analyzing the convergence time using methods that have been effective for cases without symmetries, including Knabe’s local gap threshold and Nachtergaele’s martingale methods. This suggests that a novel approach is likely necessary for understanding the convergence time of SU ( d ) -symmetric local random circuits. Published by the American Physical Society 2024

Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU(2)-symmetric quantum many-body problems, to design new spatial equivariant components for neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term ‘fusion blocks,’ serve as universal approximators of any continuous equivariant function defined on the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.

Graph Transformers have emerged as a powerful alternative to Message-Passing Graph Neural Networks (MP-GNNs) to address limitations such as over-squashing of information exchange. However, incorporating graph inductive bias into transformer architectures remains a significant challenge. In this report, we propose the Graph Spectral Token, a novel approach to directly encode graph spectral information, which captures the global structure of the graph, into the transformer architecture. By parameterizing the auxiliary [CLS] token and leaving other tokens representing graph nodes, our method seamlessly integrates spectral information into the learning process. We benchmark the effectiveness of our approach by enhancing two existing graph transformers, GraphTrans and SubFormer. The improved GraphTrans, dubbed GraphTrans-Spec, achieves over 10% improvements on large graph benchmark datasets while maintaining efficiency comparable to MP-GNNs. SubFormer-Spec demonstrates strong performance across various datasets.

Biological networks are commonly used in biomedical and healthcare domains to effectively model the structure of complex biological systems with interactions linking biological entities. However, due to their characteristics of high dimensionality and low sample size, directly applying deep learning models on biological networks usually faces severe overfitting. In this work, we propose R-MIXUP, a Mixup-based data augmentation technique that suits the symmetric positive definite (SPD) property of adjacency matrices from biological networks with optimized training efficiency. The interpolation process in R-MIXUP leverages the log-Euclidean distance metrics from the Riemannian manifold, effectively addressing the swelling effect and arbitrarily incorrect label issues of vanilla Mixup. We demonstrate the effectiveness of R-MIXUP with five real-world biological network datasets on both regression and classification tasks. Besides, we derive a commonly ignored necessary condition for identifying the SPD matrices of biological networks and empirically study its influence on the model performance. The code implementation can be found in Appendix D.

Rapid developments of quantum information technology show promising opportunities for simulating quantum field theory in near-term quantum devices. In this work, we formulate the theory of (time-dependent) variational quantum simulation of the 1 + 1 dimensional λ ϕ 4 quantum field theory including encoding, state preparation, and time evolution, with several numerical simulation results. These algorithms could be understood as near-term variational quantum circuit (quantum neural network) analogs of the Jordan–Lee–Preskill algorithm, the basic algorithm for simulating quantum field theory using universal quantum devices. Besides, we highlight the advantages of encoding with harmonic oscillator basis based on the Lehmann—Symanzik—Zimmermann reduction formula and several computational efficiency such as when implementing a bosonic version of the unitary coupled cluster ansatz to prepare initial states. We also discuss how to circumvent the ‘spectral crowding’ problem in the quantum field theory simulation and appraise our algorithm by both state and subspace fidelities.