Solving constrained counting problems with tensor networks

Lei Zhang, Justin A. Reyes, Stefanos Kourtis +3 more

Yijia Wang, Xuanzhao Gao, pan zhang +2 more

Characterizing the ground-state properties of disordered systems, such as spin glasses and combinatorial optimization problems, is fundamental to science and engineering. However, computing exact ground states and counting their degeneracies are generally NP-hard and #P-hard problems, respectively, posing a formidable challenge for exact algorithms. Recently, Tensor Networks methods, which utilize high-dimensional linear algebra and achieve massive hardware parallelization, have emerged as a rapidly developing paradigm for efficiently solving these tasks. Despite their success, these methods are fundamentally constrained by the exponential growth of space complexity, which severely limits their scalability. To address this bottleneck, we introduce the Branch-and-Bound Tensor Network (BBTN) method, which seamlessly integrates the adaptive search framework of branch-and-bound with the efficient contraction of tropical tensor networks, significantly extending the reach of exact algorithms. We show that BBTN significantly surpasses existing state-of-the-art solvers, setting new benchmarks for exact computation. It pushes the boundaries of tractability to previously unreachable scales, enabling exact ground-state counting for $\pm J$ spin glasses up to $64 \times 64$ and solving Maximum Independent Set problems on King's subgraphs up to $100 \times 100$. For hard instances, BBTN dramatically reduces the computational cost of standard Tropical Tensor Networks, compressing years of runtime into minutes. Furthermore, it outperforms leading integer-programming solvers by over 30$\times$, establishing a versatile and scalable framework for solving hard problems in statistical physics and combinatorial optimization.

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker

Edwin Miles Stoudenmire

Ritvik Sharma, Cheng Peng, Siddharth Dangwal +1 more

Simulation of physics problems is one of the most important use cases of quantum computing. For this class of problems, the goal is typically to find the minimum energy state, or the ground state, of a physical system’s Hamiltonian. These problems frequently have constraints, such as symmetry conditions, which must also be satisfied. To solve such problems, researchers in computational physics use quantum-inspired algorithms that execute on classical computers. In particular, tensor network-based eigensolvers such as DMRG have become popular. However, to use these eigensolvers, the constrained optimization problem must first be encoded as a tensor network that implements a low-rank decomposition of the system’s Hamiltonian and state vector. These tensor network encodings are highly flexible, allowing for variables with ≥2 quantum states and supporting efficient constraint encodings that directly constrain the state vector. A critical challenge to developing tensor networkbased encodings is that, currently, the encoding process is manual; significant effort is required to identify an efficient encoding for a new physics problem. In this work, we introduce a quantum constrained optimization problem (QCOP), a general abstraction for describing minimization problems over quantum variables that are subject to hard constraints. We present Masq , the first constraint programming language for QCOPs implementable with tensor networks, and CoTenN , a compiler that automatically maps QCOPs specified with Masq programs to tensor networks. To demonstrate the utility of Masq and CoTenN , we formulate two physics problems in Masq and then use CoTenN to find their ground states. We find the CoTenN -generated tensor networks generally outperform SOTA problem formulations, providing between 2.05 × − 53.32 × total speedups across runs for QCOPs and yielding up to 2.49 ⋅ 10 7 × lower truncation errors for otherwise unconstrained problems.

Tianyi Hao, Xu-Xin Huang, Chunjing Jia +1 more

Combinatorial optimization is of general interest for both theoretical study and real-world applications. Fast-developing quantum algorithms provide a different perspective on solving combinatorial optimization problems. In this paper, we propose a quantum-inspired tensor-network-based algorithm for general locally constrained combinatorial optimization problems. Our algorithm constructs a Hamiltonian for the problem of interest, effectively mapping it to a quantum problem, then encodes the constraints directly into a tensor network state and solves the optimal solution by evolving the system to the ground state of the Hamiltonian. We demonstrate our algorithm with the open-pit mining problem, which results in a quadratic asymptotic time complexity. Our numerical results show the effectiveness of this construction and potential applications in further studies for general combinatorial optimization problems.