Normalized tensor train decomposition

TL;DR

The paper addresses the challenge of representing high-dimensional tensors under a unit-Frobenius-norm constraint and introduces the normalized tensor train (NTT) format. It establishes that fixed-rank NTT tensors form a smooth manifold, derives the associated Riemannian geometry, and introduces an efficient NTT-SVD projection along with an NTT-RCG optimization method. The authors demonstrate the utility of NTT across low-rank tensor recovery, tensor-product eigenvalue problems, approximate stabilizer-rank computation, and minimum output Rényi entropy of quantum channels, with numerical results indicating superior scalability and efficiency. This work provides a practical, geometry-based framework for unit-norm tensor optimization with potential impact in scientific computing and quantum information processing.

Abstract

Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition provides a powerful low-rank format for tackling high-dimensional problems, it does not intrinsically enforce the unit-norm constraint. To address this, we introduce the normalized tensor train (NTT) decomposition, which aims to approximate a tensor by unit-norm tensors in tensor train format. The low-rank structure of NTT decomposition not only saves storage and computational cost but also preserves the underlying unit-norm structure. We prove that the set of fixed-rank NTT tensors forms a smooth manifold, and the corresponding Riemannian geometry is derived, paving the way for geometric methods. We propose NTT-based methods for low-rank tensor recovery, high-dimensional eigenvalue problem, estimation of stabilizer rank, and calculation of the minimum output Rényi 2-entropy of quantum channels. Numerical experiments demonstrate the superior efficiency and scalability of the proposed NTT-based methods.

Figures (9)

• Figure 1: Normalized tensor train decomposition of a tensor.
• Figure 1: Flowchart of the NTT-SVD algorithm. Mat: matricization; ten: tensorization
• Figure 1: Left: phase plot of recovery results for five runs. The white block indicates successful recovery in all five runs, while the black block indicates failure of recovery in all five runs. Right: test error under noise levels $\lambda=10^{-4},10^{-6},\dots,10^{-12},0$.
• Figure 1: Numerical results on estimating approximate stabilizer rank of $|H^{\otimes n}\rangle$ for $n=5,6$ qubits. Left: infidelity. Right: the maximum $2$-stabilizer Rényi entropy among each component.
• Figure 2: An illustration of the geometry of $\mathcal{N}_\mathbf{r}$. $\mathcal{O}\in{{\mathbb C}}^{n_1\times n_2\times \cdots\times n_d}$: zero tensor.

Theorems & Definitions (14)

• Definition 3.1: normalized tensor train decomposition
• Proposition 3.2: quasi-optimality
• Proof 1
• Proposition 3.3: tangent space
• Proof 2
• Proposition 3.4
• Proof 3
• Corollary 3.5
• Proposition 3.6
• Proof 4