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arXiv:2402.07975 · PRX Quantum

Computational Complexity of Isometric Tensor-Network States

Daniel Malz, Rahul Trivedi

Published Apr 14, 2025• 6 citations

Abstract

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 ( 1 + 1 )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 δ ∈ ( 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 2 D 2 . We show that weakly injective isoTNS (small δ ) are still -complete but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNSs ( η ≥ 0.41 ). 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.

DOI: 10.1103/prxquantum.6.020310

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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 ( 1 + 1 )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 δ ∈ ( 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 2 D 2 . We show that weakly injective isoTNS (small δ ) are still -complete but that there exists an efficient classical algorithm to compute local expectation values in strongly injective isoTNSs ( η ≥ 0.41 ). 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.

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We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits. Inspired by the density matrix renormalization group, we optimize tensors sequentially by diagonalizing a series of effective Hamiltonians. These are constructed using a tomography-inspired method on a qubit subset whose size depends only on the bond dimension. Our approach leverages quantum computers to enable accurate solutions without relying on approximate contractions, circumventing the exponential complexity faced by classical techniques. We demonstrate our method on the two-dimensional (2D) transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead -- significantly less than standard solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both noisy and fault-tolerant regimes.

The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an ``orthogonality column'' of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (

{\mathrm{iTEBD}}^{2}

) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.

Belief propagation has recently emerged as a powerful framework for evaluating tensor networks in higher dimensions, combining computational efficiency with provable analytical guarantees. In this work, we develop the first end-to-end theory of tensor network belief propagation for a class of projected entangled pair states satisfying \emph{strong injectivity}. We show that when the injectivity parameter exceeds a constant threshold, BP fixed points can be found efficiently, and a cluster-corrected BP algorithm computes physical quantities to

1/\mathrm{poly}(N)

error in

\mathrm{poly}(N)

time for an

N

qubit system. We further identify a striking phenomenon we term \emph{algorithmic locality}: local perturbations of the tensor network affect the BP fixed point with an influence decaying rapidly with distance. Moreover, through the cluster expansion, this locality extends to observables, implying that local expectation values can be approximated from local data with controlled accuracy. We exhibit three regimes admitting strong injectivity: random Gaussian PEPS with large physical dimension, high-temperature Gibbs states of commuting two-body Hamiltonians, and fixed-depth quantum circuits subject to sufficiently strong depolarizing noise. Our bounds are worst-case sufficient conditions and do not identify sharp algorithmic transitions.

The variational quantum eigensolver (VQE) combines classical and quantum resources in order simulate classically intractable quantum states. Amongst other variables, successful VQE depends on the choice of variational ansatz for a problem Hamiltonian. We give a detailed description of a quantum circuit version of the 2D isometric tensor network (isoTNS) ansatz which we call qisoTNS. We benchmark the performance of qisoTNS on two different 2D spin 1/2 Hamiltonians. We find that the ansatz has several advantages. It is qubit efficient with the number of qubits allowing for access to some exponentially large bond-dimension tensors at polynomial quantum cost. In addition, the ansatz is robust to the barren plateau problem due emergent layerwise training. We further explore the effect of noise on the efficacy of the ansatz. Overall, we find that qisoTNS is a suitable variational ansatz for 2D Hamiltonians with local interactions.

Recently, a class of tensor networks called isometric tensor network states (isoTNS) was proposed which generalizes the canonical form of matrix product states to tensor networks in higher dimensions. While this ansatz allows for efficient numerical computations, it remained unclear which phases admit an isoTNS representation. In this work, we show that two-dimensional string-net liquids, which represent a wide variety of topological phases including discrete gauge theories, admit an exact isoTNS representation. We further show that the isometric form can be preserved after applying a finite-depth local quantum circuit. Taken together, these results show that long-range entanglement by itself is not an obstruction to isoTNS representation and suggest that all two-dimensional gapped phases with gappable edges admit an isoTNS representation.