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

TL;DR

LimTDD delivers a compact tensor decision diagram by merging Tensor Decision Diagrams with Local Invertible Maps through the XP-stabilizer group, enabling compression that exploits tensor isomorphisms beyond Pauli-based LIMDDs. The framework provides rigorous normalization, slicing, addition, and contraction algorithms to operate on general tensor networks while preserving canonicity. Theoretical results show LimTDD is at least as compact as TDDs and LIMDDs and can achieve exponential gains in best-case scenarios such as QFT- and CP-phase–based circuits. Empirical evidence on random Clifford+T circuits and quantum circuit functionality confirms substantial reductions in node counts and runtimes, demonstrating LimTDD’s practical impact for quantum circuit simulation and tensor-network computations.

Abstract

Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient framework for tensor representation by leveraging decision diagram techniques. However, the current implementation of TDDs and other decision diagrams fail to exploit tensor isomorphisms, limiting their compression potential. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDDs that incorporates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which uses Pauli operators for quantum states, LimTDD employs the $XP$-stabilizer group, enabling broader applicability across tensor-based tasks. We present efficient algorithms for normalization, slicing, addition, and contraction, critical for tensor network applications. Theoretical analysis demonstrates that LimTDDs achieve greater compactness than TDDs and, in best-case scenarios and for quantum state representations, offer exponential compression advantages over both TDDs and LIMDDs. Experimental results in quantum circuit tensor computation and simulation confirm LimTDD's superior efficiency. Open-source code is available at https://github.com/Veriqc/LimTDD.

Figures (10)

• Figure 1: Tensor isomorphism, where $\lambda\neq 0$ is a complex number and each $O_i$ is an invertible operator. The same edge color corresponds to the same index. After applying a local operator, we rename the corresponding index to keep it consistent with the original one.
• Figure 2: A quantum circuit with 3 qubits and 6 gates.
• Figure 3: An intuitive comparison among (a) LIMDD, (b) TDD, and (c) LimTDD. All decision diagrams represent the output state resulted from simulating the circuit shown in Fig. \ref{['fig:qc']} with input state $\ket{000}$. The dotted red lines correspond to low edges and the solid blue lines correspond to high edges. The amplitude of the state can be obtained by multiplying the weights along a path. When $Z$ or $P$ is encountered, a phase $-1$ or $e^{2\pi i/8}$ should be added when going along the high edges. Here, $P = P_8$.
• Figure 4: The addition of two tensors using LimTDD.
• Figure 5: The contraction of two tensors using LimTDD, where $\times$ denotes tensor contraction.

Theorems & Definitions (32)

• Example 1
• Example 2
• Definition 1: Local Invertible Maps vinkhuijzen2023limdd
• Definition 2: Quantum State Isomorphism vinkhuijzen2023limdd
• Definition 3: LIMDD vinkhuijzen2023limdd
• Example 3
• Definition 4: TDD hong2020tensor
• Example 4
• Definition 5: Tensor Vectorization biamonte2019lectures
• Example 5