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arXiv:2602.02245 · Journal of Computational Science

Sampling two-dimensional isometric tensor network states

Alec Dektor, Eugene Dumitrescu, Chao Yang

quant-phphysics.comp-phSubmitted Feb 2, 2026• Announced Feb 3, 2026• 0 citations• Revised Jun 16, 2026 (v2)
v1 · Feb 2, 2026v2 · Jun 16, 2026
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Abstract

Sampling a quantum system's underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that generalize existing 1D tensor networks sampling algorithms. Our first proposed algorithm performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify $K$ high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.

DOI
10.1016/j.jocs.2026.102939
Comments
26 pages, 5 figures

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Simulating the dynamics of finite two-dimensional (2D) quantum many-body systems poses challenges to numerical methods due to the exponential growth of the many-body Hilbert space. In this work, the authors explore the variational power of recently introduced isometric tensor network states (isoTNS) --- a subset of general tensor-network states that allow an efficient numerical simulation. As a efficient tool to find ground states, the variational density matrix renormalization group (DMRG2{}_{2}2​) is introduced. It is then demonstrated that dynamical correlation functions for the 2D transverse field Ising model and the Kitaev Honeycomb model can be obtained.
Quantum optimal control (QOC) provides a systematic framework for achieving high-fidelity operations in quantum systems and plays a central role in tasks such as gate synthesis, state transfer, and pulse design. Existing QOC methods broadly fall into two categories: gradient-based and gradient-free algorithms. The associated optimization landscape is often high-dimensional, non-convex, and populated by numerous local minima, making efficient gradient-free search strategies essential. To address this, we introduce a gradient-free matrix product state/tensor train (MPS/TT) sampling heuristic for discrete quantum optimal control. In our approach, the MPS defines a score function over the space of discrete control parameters, which in turn induces a sampling distribution over candidate control sequences. This distribution is iteratively refined through selection of better performing sequences and local tensor updates to bias the search toward high-performing sequences. We evaluate the method on a range of benchmark problems, including single-qubit state transfer, Bell-pair preparation, qutrit gate implementation, and open-system population transfer. Across these tasks, the method exhibits stable convergence behavior and competitive empirical performance relative to established gradient-free baselines. These results suggest that tensor network sampling offers a viable heuristic framework for discrete quantum control.
We introduce an optimal strategy to sample quantum outcomes of local measurement strings for isometric tensor network states. Our method generates samples based on an exact cumulative bounding function, without prior knowledge, in the minimal amount of tensor network contractions. The algorithm avoids sample repetition and, thus, is efficient at sampling distribution with exponentially decaying tails. We illustrate the computational advantage provided by our optimal sampling method through various numerical examples, involving condensed matter, optimization problems, and quantum circuit scenarios. Theory predicts up to an exponential speedup reducing the scaling for sampling the space up to an accumulated unknown probability ϵ from O(ϵ−1) to O(log⁡(ϵ−1)) for a decaying probability distribution. We confirm this in practice with over one order of magnitude speedup or multiple orders improvement in the error depending on the application. Our sampling strategy extends beyond local observables, e.g., to quantum magic.
Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.
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.