Computational Complexity of Isometric Tensor-Network States

We determine the computational power of isometric tensor-network states (isoTNSs), a variational ansatz originally developed to numerically find and compute properties of gapped ground states and topological states in two dimensions. By mapping two-dimensional isoTNSs to (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>)D unitary quantum circuits, we find that the computation of local expectation values in isoTNSs (including those with translation-invariant bulk) is -complete. We then introduce isoTNSs, which are those isoTNSs that are the unique ground states of frustration-free Hamiltonians and which are characterized by an injectivity parameter <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>δ</d:mi><d:mo>∈</d:mo><d:mo stretchy="false">(</d:mo><d:mn>0</d:mn><d:mo>,</d:mo><d:mn>1</d:mn><d:mo>/</d:mo><d:mi>D</d:mi><d:mo stretchy="false">]</d:mo></d:math>, where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><i:mi>D</i:mi></i:math> is the bond dimension of the isoTNS. We show that injectivity necessarily adds depolarizing noise to the circuit at a rate <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><l:mi>η</l:mi><l:mo>=</l:mo><l:msup><l:mi>δ</l:mi><l:mn>2</l:mn></l:msup><l:msup><l:mi>D</l:mi><l:mn>2</l:mn></l:msup></l:math>. We show that weakly injective isoTNS (small <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><o:mi>δ</o:mi></o:math>) are still -complete but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNSs (<r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><r:mi>η</r:mi><r:mo>≥</r:mo><r:mn>0.41</r:mn></r:math>). We also show that while weakly injective isoTNSs can support long-range correlations, strongly injective isoTNSs are approximate Markov states with exponentially decaying correlations. Sampling from isoTNSs corresponds to monitored quantum dynamics and we exhibit a family of isoTNSs that undergo a phase transition from a hard regime to an easy phase in which the monitored circuit can be sampled efficiently. Our results can be used to design provable algorithms to contract isoTNSs. Our mapping between ground states of certain frustration-free Hamiltonians to open circuit dynamics in one dimension fewer may be of independent interest.