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Tensor networks for quantum computing

Aleksandr Berezutskii, Minzhao Liu, Atithi Acharya, Roman Ellerbrock, Johnnie Gray, Reza Haghshenas, Zichang He, Abid Khan, Viacheslav Kuzmin, Dmitry Lyakh, Danylo Lykov, Salvatore Mandrà, Christopher Mansell, Alexey Melnikov, Artem Melnikov, Vladimir Mironov, Dmitry Morozov, Florian Neukart, Alberto Nocera, Michael A. Perlin, Michael Perelshtein, Matthew Steinberg, Ruslan Shaydulin, Benjamin Villalonga, Markus Pflitsch, Marco Pistoia, Valerii Vinokur, Yuri Alexeev

arXiv:2503.08626v3quant-ph

TL;DR

Quantum computing needs methods that address difficult simulation, synthesis, error-management, and learning problems. This review surveys tensor-network applications across these domains and concludes that tensor networks remain valuable for simulation, benchmarking, error mitigation, and future hybrid quantum-classical systems.

  • Problem

    Quantum computing faces challenges across simulation, circuit synthesis, error correction and mitigation, and quantum machine learning that motivate tensor-network methods.

  • Method

    The paper reviews tensor-network techniques and their applications across quantum-computing subdomains, including contraction methods, synthesis, error mitigation, and tensor-network quantum machine learning.

  • Results

    The review reports that tensor networks support quantum-circuit simulation, circuit analysis, error mitigation, quantum machine learning, and validation of quantum-advantage claims.

  • Takeaways & Limitations

    Tensor networks remain important tools for quantum computing and may become especially significant in hybrid quantum-classical and fault-tolerant systems.

Abstract

from arXiv · show

In the rapidly evolving field of quantum computing, tensor networks serve as an important tool due to their multifaceted utility. In this paper, we review the diverse applications of tensor networks and show that they are an important instrument for quantum computing. Specifically, we summarize the application of tensor networks in various domains of quantum computing, including simulation of quantum computation, quantum circuit synthesis, quantum error correction and mitigation, and quantum machine learning. Finally, we provide an outlook on the opportunities and the challenges of the tensor-network techniques.

I. INTRODUCTION

Tensor networks represent and manipulate quantum states and processes, helping quantum computing address exponentially large state spaces. This section introduces their mathematical operations and previews applications across simulation, synthesis, error correction and mitigation, and machine learning.

  • Scope: Tensor networks represent and manipulate quantum states and processes, supporting applications across quantum computing.The review organizes these applications into simulation, quantum circuit synthesis, quantum error correction and mitigation, and quantum machine learning.
  • Motivation: They help address the curse of dimensionality, whose state-space size grows exponentially with the number of degrees of freedom.Tensor networks have enabled classical simulation of quantum experiments previously believed beyond classical computational capability.
  • Tensor-network operations: Tensor contraction composes tensors by summing over shared indices, with matrix multiplication as a special case.More general contractions can combine multiple tensors into a resulting tensor with selected remaining indices.
  • Tensor-network operations: Tensor networks use connected-node diagrams in which nodes represent tensors and edges represent indices.Joining shared edges denotes contraction, so connected mathematical expressions form tensor networks.
  • Tensor decompositions: SVD decomposes a matrix into isometries and singular values, enabling approximate representations by truncating singular values.For fixed-rank approximations, the Eckart–Young theorem states that the SVD solution is optimal.

A. Common ansatzes

Common tensor-network ansatzes trade structure, expressiveness, and computational cost to represent quantum states across different geometries and correlation patterns.

  • Matrix product states: MPS is the simplest and most widely used ansatz, representing many-body wavefunctions through contracted site tensors.Its bond dimension χ controls compression accuracy, and the related MPO represents operators by adding a dual index.
  • Matrix product states: MPS is especially effective for one-dimensional systems because its contraction cost is polynomial in system size and local observables can be evaluated efficiently.Its structure follows the area law of entanglement, although it has also been used beyond one dimension.
  • Higher-dimensional ansatzes: PEPS generalizes MPS locality to higher dimensions and can reproduce larger entanglement, but its algorithms are typically much more computationally demanding.The passage describes PEPS for two- and higher-dimensional quantum systems.
  • Tree tensor networks: TTNS generalizes MPS to nonlocal correlations, while balanced trees capture long-range interactions with logarithmic worst-case correlation length.Its scaling is comparable to MPS, although accurately describing states may require a higher rank.
  • Multiscale ansatzes: MERA captures entanglement across different length scales and can reproduce logarithmic violations of the area law.Tensor networks can also use periodic boundaries, represent operators and channels, and mirror circuit connectivity for simulation.

B. Manipulation of TNs

Tensor-network techniques are used to obtain interpretable quantities from quantum-information objects, supported by software libraries for tensor manipulation.

  • Tensor networks can produce expectation values, reduced density matrices, amplitudes, samples, and other interpretable quantities.
  • The field has many software libraries suited to different tensor-network manipulations.The passage names cuQuantum, quimb, and ITensor among available libraries.
  • Tensor-network software supports practical manipulation of diverse quantum-information representations.The passage connects library availability with the broad range of quantities and objects handled by tensor networks.

1. Update

Tensor-network algorithms optimize represented quantum states and operators through variational, gradient-based, Monte Carlo, time-evolution, and contraction methods. Exact simulation also relies on slicing and multi-contraction, while larger networks require approximate contraction whose tractability remains unresolved for some circuit classes.

  • Optimization: DMRG variationally optimizes an MPS approximation of a many-body ground state for Hamiltonian-defined cost functions.Gradient-based and Riemannian optimization provide alternatives when gradients are accessible or isometric constraints must be maintained.
  • Time evolution: TEBD evolves tensor-network states or operators by applying local gates or Kraus operators and restoring the network form with SVD.Continuous Hamiltonian evolution can be approximated by discrete steps when interactions are sufficiently short ranged.
  • Contraction: Contraction trees replace naive contraction with pairwise intermediate tensors, reducing cost substantially while still generally requiring exponential resources.Some tree- and fractal-like geometries admit polynomial-cost contraction.
  • Contraction: Finding an optimal contraction scheme is NP-complete, motivating heuristic methods based on graph partitioning, simulated annealing, and reinforcement learning.The quality of the contraction tree can change total cost by many orders of magnitude.
  • Exact simulation: Exact circuit simulation uses slicing to parallelize smaller contractions and multi-contraction to reuse work across related tensor networks.The largest exact simulations combine both techniques.
  • Approximate simulation: Approximate contraction extends simulations beyond exactly treatable network sizes but is limited by entanglement buildup and lacks known tractability classes for some quantum circuits.Recent work has pursued automatic approximation and contraction-sequence optimization.

III. SIMULATION OF QUANTUM COMPUTATION

Tensor networks simulate gate-based quantum algorithms by representing states, operators, and circuits, enabling amplitude and expectation-value calculations. These methods have reproduced or challenged claims about quantum computational difficulty, although the newest high-fidelity, large-volume experiments remain hard to simulate.

  • General methods: Tensor networks represent quantum states and circuits, allowing amplitudes, expectation values, and circuit traces to be computed by contracting suitable networks.They can evolve an ansatz such as an MPS or PEPS using TEBD-like techniques.
  • Examples: Tensor-network methods solved kicked-Ising dynamics more efficiently and sometimes more accurately than the quantum processor itself.The problem was initially considered intractable using state-vector MPS and isometric-TN approaches.
  • Random circuit sampling: Random circuit sampling is classically hard under standard complexity-theoretic assumptions, so circuit-network contraction is used to simulate it at controlled fidelity.Slicing, approximation, and simplification trade computational cost against fidelity.
  • Random circuit sampling: MPS-based DMRG decomposes a sample amplitude into beginning, middle, and end circuit segments, contracting MPS representations around the middle segment.This approach was used to establish quantum utility in simulating quantum Ising-model dynamics.
  • Scope: Earlier random-circuit-sampling experiments can now be simulated, whereas newer experiments with higher fidelity and larger circuit volume remain difficult.MPO, MPDO, PEPS, converted 3D-to-2D networks, and TTNS have also been applied to gate-based simulations.

B. Analog evolution

Analog quantum simulations use tensor-network methods to study continuously controlled quantum dynamics, including annealing and optical sampling. Reported performance ranges from comparable accuracy on low-dimensional annealers to unresolved difficulty for larger or more complex settings.

  • Analog quantum computing: Analog quantum devices implement model-specific dynamics through simultaneous, time-dependent control of pairwise interactions across available qubits.Examples include neutral atoms, trapped ions, Rydberg tweezers, photonic waveguides, and superconducting circuits.
  • Analog evolution: For a studied 3D and biclique transverse-field Ising spin glass, approximate MPS and PEPS methods reportedly failed to match the quantum simulator’s solution quality.The comparison used GPU-accelerated TDVP time evolution on unfolded lattice geometries.
  • Analog evolution: Tensor-network simulations were reported to match sampled QPU-distribution quality against converged MPS simulations at simulatable scales.This evidence concerns the comparison described for the analog-evolution setting.
  • Analog evolution: A belief-propagation simulation achieved comparable accuracy to a quantum annealer for two- and three-dimensional systems, but was not performed for the studied infinite-dimensional biclique lattice.The method used belief propagation for time evolution and more sophisticated variants for expectation values.
  • Boson sampling: Boson sampling is non-universal yet difficult to simulate classically under plausible complexity-theoretic assumptions, and MPS methods exploit photon-number conservation without loss.With photon loss, MPO simulation can remain efficient when output-photon scaling keeps MPO entanglement entropy manageable.
  • Boson sampling: Lossy Gaussian boson-sampling states can be simulated using MPS representations after random classical operations and photon-number reduction.The largest such experiments were simulated on up to 288 GPUs in under two hours, leaving no experiment as clear evidence of beyond-classical hardness.
  • Boson sampling: Heisenberg-picture boson-sampling simulation evolves a Fock-state projector rather than the quantum state, which can reduce bond dimension relative to a Gaussian Schrödinger-picture state.The contrast is especially relevant because Gaussian states are infinite superpositions of Fock states.

IV. QUANTUM CIRCUIT SYNTHESIS

Quantum circuit synthesis must respect hardware connectivity and noise-limited depth, but no universally scalable or precise solution is currently available. Tensor-network representations provide a geometry-aware route to native gate realization and simpler circuit design.

  • Problem: Quantum circuit synthesis decomposes a target operation into executable gates compatible with a specific quantum architecture.Modern processors impose native-connectivity and noise-limited-depth constraints, and no universal solution currently achieves both scalability and precision.
  • Tensor-network approach: MPS and PEPS layouts align naturally with adjacent-qubit connectivity, providing a partial solution to hardware-aware circuit synthesis.Their prescribed geometrical decomposition supports circuit descriptions that are themselves tensor networks.
  • Tensor-network approach: Representing states and operations as tensor networks and recasting them as circuits can simplify design and compile arbitrary unitaries into natively realizable gates.The paper states that this can reduce circuit complexity and enable more efficient synthesis.

A. Promoting TNs to quantum gates

Tensor networks can be mapped to quantum circuits by converting tensors into isometries, embedding time direction, promoting them to unitaries, and decomposing those unitaries into quantum gates. MPS is especially suitable because it has an exact canonical form, whereas PEPS generally lacks one and may require exponential postselection.

  • General procedure: Mapping a tensor network to a quantum circuit requires isometric conversion, temporal ordering, unitary promotion, and gate decomposition.Gauge freedom enables isometric tensors, while circuit directionality is introduced by specifying incoming and outgoing wires.
  • MPS preparation: MPS architectures can be converted into quantum circuits by promoting their canonical isometric tensors into unitaries.The procedure has been theoretically explored and experimentally realized across multiple quantum-computer architectures.
  • MPS preparation: Exact MPS preparation requires operations acting on ⌊log(m)⌋ + 1 qubits, which can conflict with limited connectivity and restricted native-gate sets.Variational local circuits and native-gate disentanglers are proposed as approximate alternatives.
  • PEPS preparation: PEPS generally lacks a canonical exact isometric form, preventing exact circuit mapping without an exponential amount of postselection.The isoPEPS subclass restores a circuit connection by requiring tensors or tensor groups to satisfy isometric conditions.
  • Other architectures: MERA is naturally related to quantum circuits because its isometries are embedded into unitaries, and it is reported to be robust to noise without barren plateaus.The paper also reports an ion-trapped digital-quantum-computer demonstration for probing many-body criticality.

C. Implementation techniques

Implementation techniques use tensor-network structure to prepare states and construct error-correcting codes, including sequential, adaptive, variational, and contraction-based approaches. These methods support broad code constructions and decoding, but practical real-time decoding remains constrained by computational cost.

  • State preparation: Holographic preparation sequentially converts a spatial dimension of a tensor-network state into a temporal preparation procedure.Tensor-network time-ordering freedom permits this sequential implementation.
  • State preparation: Adaptive circuits use mid-circuit measurements to determine subsequent gates when preparing tensor-network states.They are explored alongside holographic implementations for states with long-range correlations.
  • State preparation: Variational methods optimize circuit ansätze to approximate target tensor-network states, typically producing shallower circuits.Tensor-network formulations can also instantiate generic quantum circuits through efficient optimization.
  • Tensor-network codes: Tensor-network representations formally connect convolutional, concatenated block, and topological codes with MPS, tree networks, and PEPS, respectively.For convolutional and concatenated block codes, efficient contraction enables efficient optimal decoders.
  • Tensor-network codes: The quantum LEGO formalism builds large codes from small seed tensors, with universality using three elementary seed tensors.These tensor-network codes naturally include optimal decoders evaluated by tensor-network contraction.
  • Tensor-network codes: Tensor-network code constructions generalize code concatenation and include topological, holographic, non-Abelian stabilizer, non-additive topological, and approximate codes.These constructions are used across several code families and physical settings.

B. Syndrome decoding

Tensor-network decoding reformulates maximum-likelihood inference as tensor-network contraction over error configurations compatible with syndrome data. The approach accommodates diverse noise models and large networks, but current decoders are accurate yet generally not practical for real-time correction.

  • Decoding formulation: Maximum-likelihood decoding finds the most likely physical-error configuration from syndrome measurements and an error model.The task is formulated as an inference problem over compatible error configurations.
  • Decoding approaches: One decoding approach enumerates coset errors by identifying symmetries that preserve logical information and syndrome measurements.A representative error configuration is combined with all symmetry operations to generate the coset.
  • Decoding approaches: A second approach builds a tensor network directly from error-model connectivity and zeroes entries for configurations outside the target coset.This handles more general error models without requiring computationally useful symmetry sets.
  • Noise models: Tensor-network decoding has been applied to erasure, fractal, depolarizing, biased, and non-Markovian noise models, while circuit-level noise is decoded approximately.The two decoding approaches support a broad range of noise assumptions.
  • Practical limitations: Tensor-network decoders are highly accurate but not currently practical for real-time decoding, making heuristic decoders more suitable for experiments.They remain useful for benchmarking experiments and less accurate decoders.
  • Error mitigation: Tensor-network error-mitigation methods provide a universal quadratic cost reduction and were experimentally demonstrated on a 91-qubit circuit with 4095 two-qubit gates.The procedure contracts a circuit-level tensor network after randomized local measurements and classical noise inversion.
  • Error mitigation: Whether error mitigation alone or combined with quantum error correction yields near-term computational utility remains an open question.The unresolved issue concerns practical advantage in near-term quantum computation and algorithms.

VI. TENSOR NETWORKS FOR QUANTUM MACHINE LEARNING

Tensor-network quantum machine learning compresses models and uses quantum states to replace costly classical contractions or sampling. Its architectures support multiple learning tasks, but proposed quantum advantages remain contested for some models such as QCNNs.

  • Motivation: Tensor networks reduce memory usage and training parameters by compressing high-dimensional neural-network linear layers.They also provide analytical interpretability and allow inductive biases through structural constraints.
  • Quantum architectures: TN-QML uses parameterized quantum-circuit ansätze based on MPS, TTN, and MERA architectures to address costly classical contraction complexity.Quantum computers may implement these architectures directly rather than relying on classical tensor contractions.
  • Learning tasks: Regression and classification models obtain tensor-network neural-layer outputs by measuring observables on corresponding quantum states.Generative models instead sample directly from quantum states encoding the tensor network.
  • Optimization: Quantum-gate parameterizations can require exponentially fewer parameters than classical tensor networks for similar results and are reported to avoid barren plateaus.These properties are presented as potential advantages for quantum machine learning.
  • Quantum applications: TN-QML supports quantum phase classification on quantum devices through QCNNs based on TTN and MERA architectures.Recent work argues that QCNNs can be dequantized using classical-shadow techniques, challenging claims of exponential quantum advantage.
  • Workflow and validation: Tensor networks can encode classical data into quantum circuits, pre-train TN-PQCs classically, and test QML models for dequantization through surrogate modeling.These roles connect classical preprocessing, quantum ansätze, and evaluation of purported quantum advantages.

VII. DISCUSSION AND OUTLOOK

The discussion presents tensor networks as useful for quantum-computing simulation, complexity analysis, quantum error correction, and future hybrid systems, while identifying open challenges in decoding and quantum machine learning.

  • Tensor networks can benchmark whether finite-size quantum experiments are beyond classical reach and provide evidence about computational complexity.Polynomial-time tensor-network algorithms can establish membership in P, while observed exponential scaling can support empirical hardness arguments.
  • Recent tensor-network developments have improved simulation of quantum many-body systems and near-Clifford circuits, including representations in Pauli and computational bases.The classes of circuits that can be efficiently simulated remain an open question.
  • Quantum LEGO formalisms may enable systematic construction of quantum low-density parity-check codes and further exploration of non-Abelian stabilizer and non-additive codes.These directions are motivated by the promise of quantum low-density parity-check codes in quantum error correction.
  • Highly accurate tensor-network decoding may be too slow for real-time hardware, motivating approximate tensor-network decoders or AI-based alternatives.Tensor-network simulations could provide noisy training data for AI decoders.
  • Tensor networks are expected to remain significant in hybrid quantum-classical systems, alongside new algorithms, AI integration, and fault-tolerant quantum computing techniques.The review also notes that whether tensor-network quantum machine learning offers a polynomial advantage remains unresolved.
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