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<p>The rapid advancement of <i>quantum-inspired</i> classical algorithms, particularly Quantics Tensor Trains (QTT), has demonstrated that exponentially large vectors can be manipulated efficiently via Matrix Product States (MPS). Conventionally, the compression of these tensor networks relies exclusively on Singular Value Decomposition (SVD)-based truncation. While SVD is optimal for minimizing the Frobenius norm error, it remains structurally blind to exact algebraic correlations, such as reversible logic gates or Clifford symmetries, that do not require numerical approximation to factor out. In this work, we propose a hybrid compression protocol that integrates the topological rigor of ZX-Calculus with the numerical power of SVD. We construct an explicit isomorphism between Rank-3 MPS tensors and ZX-diagrams, allowing us to subject QTT representations to formal diagrammatic rewriting rules prior to numerical truncation. We establish that this Topological Preconditioning can algebraically collapse the effective bond dimension (<InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\chi\)</EquationSource></InlineEquation>) and T-count of the underlying tensor network without information loss. By benchmarking the algorithmic implementation against hardware-accelerated SVD on discretized functions and stabilizer states, we demonstrate that ZX-driven algebraic erasure fundamentally bypasses standard <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(\mathcal {O}(\chi ^3)\)</EquationSource></InlineEquation> bottlenecks, yielding up to a computational speedup. Finally, we formalize this advantage by defining the structural complexity class <InlineEquation ID="IEq3"><EquationSource Format="TEX">\(\mathsf {ZX\text {-}R}\)</EquationSource></InlineEquation>, proving that for systems characterized by algebraic symmetry, this work reflects our ongoing effort to deepen understanding of Categorical Quantum Mechanics, aiming to contribute to the optimization of high-dimensional linear algebra.</p>

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Topological approaches to quantum tensor train compression via ZX-calculus and SVD

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月5日

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  • Luis Gerardo Ayala Bertel,
  • Srinjoy Ganguly

摘要

The rapid advancement of quantum-inspired classical algorithms, particularly Quantics Tensor Trains (QTT), has demonstrated that exponentially large vectors can be manipulated efficiently via Matrix Product States (MPS). Conventionally, the compression of these tensor networks relies exclusively on Singular Value Decomposition (SVD)-based truncation. While SVD is optimal for minimizing the Frobenius norm error, it remains structurally blind to exact algebraic correlations, such as reversible logic gates or Clifford symmetries, that do not require numerical approximation to factor out. In this work, we propose a hybrid compression protocol that integrates the topological rigor of ZX-Calculus with the numerical power of SVD. We construct an explicit isomorphism between Rank-3 MPS tensors and ZX-diagrams, allowing us to subject QTT representations to formal diagrammatic rewriting rules prior to numerical truncation. We establish that this Topological Preconditioning can algebraically collapse the effective bond dimension (\(\chi\)) and T-count of the underlying tensor network without information loss. By benchmarking the algorithmic implementation against hardware-accelerated SVD on discretized functions and stabilizer states, we demonstrate that ZX-driven algebraic erasure fundamentally bypasses standard \(\mathcal {O}(\chi ^3)\) bottlenecks, yielding up to a computational speedup. Finally, we formalize this advantage by defining the structural complexity class \(\mathsf {ZX\text {-}R}\), proving that for systems characterized by algebraic symmetry, this work reflects our ongoing effort to deepen understanding of Categorical Quantum Mechanics, aiming to contribute to the optimization of high-dimensional linear algebra.