The minimal canonical form of a tensor network

TL;DR

A new canonical form is introduced, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and a corresponding fundamental theorem is proved to prove that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids.

Abstract

Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool, both for theoretical and numerical purposes. On the other hand, for tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. Here we introduce a new canonical form, the minimal canonical form, which applies to projected entangled pair states (PEPS) in any dimension, and prove a corresponding fundamental theorem. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. We show that two tensors have the same minimal canonical forms if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. In particular, this implies that the latter problem is decidable - in contrast to the well-known undecidability for PEPS on grids. We also provide rigorous algorithms for computing minimal canonical forms. To achieve this we draw on geometric invariant theory and recent progress in theoretical computer science in non-commutative group optimization.

Figures (8)

• Figure 1:
• Figure 2:
• Figure 3: Fundamental theorem: Two tensors $S$ and $T$ are gauge equivalent, meaning $\overline{G \cdot T}\cap \overline{G \cdot S}\neq\emptyset$ or that $\lim_{n\to\infty} \boldsymbol{g}^{(n)} \cdot T = \lim_{n\to\infty} \boldsymbol{h}^{(n)} \cdot S$ for certain $\boldsymbol{g}^{(n)}, \boldsymbol{h}^{(n)} \in G$ (equivalently, the two tensors have a common minimal canonical form), if and only if they contract to the same state on all contraction graphs.
• Figure 4: Matrix product state:$M = (M^{(i)})_{i=1}^d \in \mathop{\mathrm{Mat}}\nolimits_{D\times D}^d$ gives rise to a state $\ket{M_n}$ for any system size $n$.
• Figure 5: MPS gauge invariance: Tensors related by a gauge transformation give rise to the same MPS.

Theorems & Definitions (90)

• Example 2.1
• Definition 2.2: Minimum norm vectors
• Theorem 2.3
• Definition 2.4
• Theorem 2.5: Kempf--Ness
• Lemma 2.6
• Definition 2.7
• Lemma 2.8
• Theorem 2.9: Mumford
• Theorem 2.10: Hilbert finiteness