Towards Symmetry-Aware Efficient Simulation of Quantum Systems and Beyond

TL;DR

The paper addresses the exponential growth of Hilbert space in quantum simulations and promotes symmetry-informed tensor networks as a scalable solution. It details a $U(1)$-symmetric MPS with right-charge indices and charge-conservation constraints that reduce memory and enable block-sparse updates, achievable with large speedups on modern hardware. It extends the framework to general Abelian/non-Abelian symmetries and connects to symmetry-preserving variational quantum circuits and equivariant neural networks, illustrating cross-domain benefits. It also discusses non-symmetry-guided approaches like hybrid tensor networks and parallel-sequential circuits, arguing for a unified, symmetry-aware strategy for scalable quantum simulation, computation, and machine learning.

Abstract

The efficient simulation of complex quantum systems remains a central challenge due to the exponential growth of Hilbert space with system size. Tensor network methods have long been established as powerful approximation schemes, and their efficiency can be further enhanced by incorporating physics-informed priors. A prominent example is symmetry: recent progress on $U(1)$-symmetric tensor networks, accelerated on GPUs and scaled to supercomputers, shows how conserved charges induce block-sparse structures that reduce computational cost and enable larger simulations. The same principle extends to general symmetries, inspiring equivariant neural networks in machine learning and guiding symmetry-preserving ansatze in variational quantum algorithms. Beyond symmetry, physics-informed design also includes strategies such as hybrid tensor networks and parallel sequential circuits, which pursue efficiency from complementary principles. This Perspective argues that physics-informed tensor networks, grounded in both symmetry and beyond-symmetry insights, provide unifying strategies for scalable approaches in quantum simulation, computation, and machine learning.