Computational complexity of isometric tensor network states

TL;DR

The paper establishes a precise complexity landscape for isometric tensor network states (isoTNS) by mapping 2D isoTNS contractions to 1+1D quantum-channel dynamics. It proves that computing local expectation values is BQP-complete in general, while injectivity introduces a depolarizing noise rate $η=δ^2D^2$ that can render strongly injective isoTNS classically contractible in poly$(1/ε)$ time; weakly injective isoTNS remain hard. Sampling from isoTNS is also hard in general, but a family exhibits a measurement-induced transition to an easy phase under monitoring, linking 2D tensor-network contraction to 1D monitored dynamics. These results yield provable contraction algorithms and deepen the connection between tensor-network representations of ground states and dynamical, noisy quantum circuits, with implications for both classical simulability and quantum algorithm design.

Abstract

We determine the computational power of isometric tensor network states (isoTNS), a variational ansatz originally developed to numerically find and compute properties of gapped ground states and topological states in two dimensions. By mapping 2D isoTNS to 1+1D unitary quantum circuits, we find that computing local expectation values in isoTNS is $\textsf{BQP}$-complete. We then introduce injective isoTNS, which are those isoTNS that are the unique ground states of frustration-free Hamiltonians, and which are characterized by an injectivity parameter $δ\in(0,1/D]$, where $D$ is the bond dimension of the isoTNS. We show that injectivity necessarily adds depolarizing noise to the circuit at a rate $η=δ^2D^2$. We show that weakly injective isoTNS (small $δ$) are still $\textsf{BQP}$-complete, but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNS ($η\geq0.41$). Sampling from isoTNS corresponds to monitored quantum dynamics and we exhibit a family of isoTNS that undergo a phase transition from a hard regime to an easy phase where the monitored circuit can be sampled efficiently. Our results can be used to design provable algorithms to contract isoTNS. Our mapping between ground states of certain frustration-free Hamiltonians to open circuit dynamics in one dimension fewer may be of independent interest.

Figures (3)

• Figure 1: Illustration of one sample in the algorithm to compute local expectation values. Each site is randomly assigned the depolarizing channel (empty crossing) or the channel $\mathcal{E}_1$ (blue tensor). In the easy phase, sites with tensors do not percolate and clusters are typically small. As a result, to compute an observable on the red site, we only need to contract the cluster connected to the sites (blue) and in fact only the part of the cluster in the past light cone (grey background).
• Figure 2: For concreteness, we analyse an isoTNS with rectangular geometry with subsystems $A$, $B$, $C$. The screening region $B$ has width $W$ and length $l$ and is divided in two equally sized parts, a left half $B_L$ and right half $B_R$.
• Figure 3: An oblique brickwork circuit generated by the same tensor $V$ and a translationally varying product state that effectively implements a quantum Turing machine. This circuit can equivalently be read as an isoTNS with bulk tensor with a translation-invariant bulk and a translationally varying boundary.

Theorems & Definitions (2)

• Definition 1: PEPS
• Definition 2: Uniform Markov property Brandao2019