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 $\delta\in(0,1/D]$, where $D$ is the bond dimension of the isoTNS. We show that injectivity necessarily adds depolarizing noise to the circuit at a rate $\eta=\delta^2D^2$. We show that weakly injective isoTNS (small $\delta$) are still $\textsf{BQP}$-complete, but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNS ($\eta\geq0.41$). 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.

翻译：我们确定了等距张量网络态（isoTNS）的计算能力，这是一种最初为数值求解二维有能隙基态和拓扑态及其性质而发展的变分拟设。通过将二维isoTNS映射到1+1维幺正量子电路，我们发现计算isoTNS中的局域期望值是完全BQP问题。我们随后引入了单射isoTNS，即那些作为无阻挫哈密顿量的唯一基态的isoTNS，其特征由单射参数$\delta\in(0,1/D]$刻画，其中$D$是isoTNS的键维数。我们证明单射性必然以速率$\eta=\delta^2D^2$向电路引入去极化噪声。我们展示了弱单射isoTNS（小$\delta$）仍然完全属于BQP，但对于强单射isoTNS（$\eta\geq0.41$），存在一种有效的经典算法来计算局域期望值。从isoTNS采样对应于受监测的量子动力学，我们展示了一族isoTNS经历从困难相到易相（其中受监测电路可被高效采样）的相变。我们的结果可用于设计可验证的isoTNS收缩算法。将某些无阻挫哈密顿量的基态映射到低一维度的开放电路动力学这一结论可能具有独立的研究意义。