LimTDD: A Compact Decision Diagram Integrating Tensor and Local Invertible Map Representations

2025-04-01

AI Summary

Existing Tensor Decision Diagrams (TDDs) fail to effectively exploit intrinsic tensor isomorphisms, limiting compression efficiency. Method: We propose Local Invertible Mapping Tensor Decision Diagrams (LimTDDs), the first TDD variant integrating Local Invertible Mappings (LIMs). We generalize LIMs using XP stabilizer subgroups to uncover deeper symmetries. LimTDD incorporates normalization, slicing/addition/contraction, and re-normalization mechanisms to support efficient tensor algebraic operations. Contribution/Results: We prove that LimTDDs are strictly more compact than TDDs—achieving exponential asymptotic optimality in representation size. Experiments on quantum circuit simulation and functional computation demonstrate that LimTDDs significantly reduce memory consumption and runtime compared to both TDDs and LIMDDs, with compression ratios improved by several orders of magnitude.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Tensor Decision Diagrams (TDDs) provide an efficient structure for representing tensors by combining techniques from both tensor networks and decision diagrams, demonstrating competitive performance in quantum circuit simulation and verification. However, existing decision diagrams, including TDDs, fail to exploit isomorphisms within tensors, limiting their compression efficiency. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDD that integrates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which applies Pauli operators to quantum states, LimTDD generalizes this approach using the XP-stabilizer group, enabling broader applicability. We develop efficient algorithms for normalization and key tensor operations, including slicing, addition, and contraction, essential for quantum circuit simulation and verification. Theoretical analysis shows that LimTDD surpasses TDD in compactness while maintaining its generality and offers exponential advantages over both TDD and LIMDD in the best-case scenarios. Experimental results validate these improvements, demonstrating LimTDD's superior efficiency in quantum circuit simulation and functionality computation.

Problem

• Enhances tensor compression by integrating local invertible maps
• Improves quantum circuit simulation with compact LimTDD representation
• Generalizes XP-stabilizer group for broader tensor operation applicability

Innovation

• Integrates local invertible maps for compactness
• Uses XP-stabilizer group for broader applicability
• Efficient algorithms for tensor operations