Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).
Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).
Orús, R. A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014).
Chubb, C. T. General tensor network decoding of 2D Pauli codes. Preprint at https://arxiv.org/abs/2101.04125 (2021).
Pan, F. & Zhang, P. Simulation of quantum circuits using the big-batch tensor network method. Phys. Rev. Lett. 128, 030501 (2022).
Fu, R. et al. Achieving energetic superiority through system-level quantum circuit simulation. In SC ’24: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis 1–20 (ACM, 2024).
Oh, C., Liu, M., Alexeev, Y., Fefferman, B. & Jiang, L. Classical algorithm for simulating experimental Gaussian boson sampling. Nat. Phys. 20, 1461–1468 (2024).
Tindall, J., Fishman, M., Stoudenmire, E. M. & Sels, D. Efficient tensor network simulation of IBM’s Eagle kicked Ising experiment. PRX Quantum 5, 010308 (2024).
Tindall, J., Mello, A. F., Fishman, M., Stoudenmire, E. M. & Sels, D. Dynamics of disordered quantum systems with two- and three-dimensional tensor networks. Preprint at https://arxiv.org/abs/2503.05693 (2025).
Bridgeman, J. C. & Chubb, C. T. Hand-waving and interpretive dance: an introductory course on tensor networks. J. Phys. A 50, 223001 (2017). A comprehensive and pedagogical introduction to tensor networks from the traditional many-body physics perspective.
Eckart, C. & Young, G. The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218 (1936).
Stewart, G. W. Matrix Algorithms, Vol. 1 Basic Decompositions (SIAM, 1998).
Verstraete, F. & Cirac, J. I. Matrix product states represent ground states faithfully. Phys. Rev. B 73, 094423 (2006). Analytical results showing that MPS efficiently represents the ground states of 1D systems.
Bañuls, M. C. Tensor network algorithms: a route map. Annu. Rev. Condens. Matter Phys. 14, 173–191 (2023).
Oseledets, I. V. Tensor-train decomposition. SIAM J. Sci. Comput. 33, 2295–2317 (2011).
Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Computational complexity of projected entangled pair states. Phys. Rev. Lett. 98, 140506 (2007). Shows that contracting PEPS and arbitrary tensor networks is #P-complete.
Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003).
Shi, Y.-Y., Duan, L.-M. & Vidal, G. Classical simulation of quantum many-body systems with a tree tensor network. Phys. Rev. A 74, 022320 (2006).
Wang, H. & Thoss, M. Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys. 119, 1289–1299 (2003).
Markov, I. L. & Shi, Y. Simulating quantum computation by contracting tensor networks. SIAM J. Comput. 38, 963–981 (2008). Proposes exact simulation of quantum circuits with tensor networks and links the complexity to the treewidth of the graph underlying the circuit.
DeCross, M. et al. The computational power of random quantum circuits in arbitrary geometries. Phys. Rev. X. 15, 021052 (2025).
Bayraktar, H. et al. cuQuantum SDK: a high-performance library for accelerating quantum science. Preprint at https://arxiv.org/pdf/2308.01999 (2023).
Gray, J. quimb: A python package for quantum information and many-body calculations. J. Open Source Softw. 3, 819 (2018).
Fishman, M., White, S. & Stoudenmire, E. The ITensor software library for tensor network calculations. SciPost Physics Codebases 004 (2022).
Gray, J. & Kourtis, S. Hyper-optimized tensor network contraction. Quantum 5, 410 (2021).
Hauschild, J. & Pollmann, F. Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy). SciPost Physics Lecture Notes 005 (2018).
Zhai, H. et al. Block2: A comprehensive open source framework to develop and apply state-of-the-art dmrg algorithms in electronic structure and beyond. The Journal of Chemical Physics 159, 234801 (2023).
White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992).
Wang, M. et al. Tensor networks meet neural networks: a survey and future perspectives. Preprint at https://arxiv.org/abs/2302.09019 (2023).
Liao, H.-J., Liu, J.-G., Wang, L. & Xiang, T. Differentiable programming tensor networks. Phys. Rev.X 9, 031041 (2019).
Hauru, M., Van Damme, M. & Haegeman, J. Riemannian optimization of isometric tensor networks. SciPost Phys 10, 040 (2021).
Luchnikov, I., Ryzhov, A., Filippov, S. & Ouerdane, H. QGOpt: Riemannian optimization for quantum technologies. SciPost Physics 10, 079 (2021).
Berezutskii, A., Luchnikov, I. & Fedorov, A. Simulating quantum circuits using the multi-scale entanglement renormalization ansatz. Phys. Rev. Res. 7, 013063 (2025).
Luchnikov, I., Krechetov, M. & Filippov, S. Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies. New J. Phys. (2021).
Sandvik, A. W. & Vidal, G. Variational quantum Monte Carlo simulations with tensor-network states. Phys. Rev. Lett. 99, 220602 (2007).
Wang, L., Pižorn, I. & Verstraete, F. Monte Carlo simulation with tensor network states. Phys. Rev. B—Condensed Matter and Materials Physics 83, 134421 (2011).
Paeckel, S. et al. Time-evolution methods for matrix-product states. Ann. Phys. 411, 167998 (2019). A review of time-evolution methods on MPS.
Feiguin, A. E. & White, S. R. Time-step targeting methods for real-time dynamics using the density matrix renormalization group. Phys. Rev. B 72, 020404 (2005).
Alvarez, G., Dias da Silva, L. G., Ponce, E. & Dagotto, E. Time evolution with the density-matrix renormalization-group algorithm: A generic implementation for strongly correlated electronic systems. Phys. Rev. E 84, 056706 (2011).
Haegeman, J. et al. Time-dependent variational principle for quantum lattices. Phys. Rev. Lett. 107, 070601 (2011).
Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. & Verstraete, F. Unifying time evolution and optimization with matrix product states. Phys. Rev. B 94, 165116 (2016).
Ran, S.-J., Sun, Z.-Z., Fei, S.-M., Su, G. & Lewenstein, M. Tensor network compressed sensing with unsupervised machine learning. Phys. Rev. Res. 2, 033293 (2020).
Arnborg, S., Corneil, D. G. & Proskurowski, A. Complexity of finding embeddings in ak-tree. SIAM Journal on Algebraic Discrete Methods 8, 277–284 (1987).
Pfeifer, R. N., Haegeman, J. & Verstraete, F. Faster identification of optimal contraction sequences for tensor networks. Phys. Rev.E 90, 033315 (2014).
Kourtis, S., Chamon, C., Mucciolo, E. & Ruckenstein, A. Fast counting with tensor networks. SciPost Physics 7, 060 (2019).
Kalachev, G., Panteleev, P., Zhou, P. & Yung, M.-H. Classical sampling of random quantum circuits with bounded fidelity. Preprint at https://arxiv.org/abs/2112.15083 (2021).
Morvan, A. et al. Phase transitions in random circuit sampling. Nature 634, 328–333 (2024).
Meirom, E., Maron, H., Mannor, S. & Chechik, G. Optimizing tensor network contraction using reinforcement learning. (2022).
Liu, X.-Y. & Zhang, Z. Classical simulation of quantum circuits using reinforcement learning: Parallel environments and benchmark (2023).
Kalachev, G., Panteleev, P. & Yung, M.-H. Multi-tensor contraction for xeb verification of quantum circuits. Preprint at https://arxiv.org/abs/2108.05665 (2021).
Huang, C. et al. Classical simulation of quantum supremacy circuits. Preprint at https://arxiv.org/abs/2005.06787 (2020).
Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. In CCC ’17: Proc. 32nd Computational Complexity Conference (CCC, 2017).
Chen, J., Zhang, F., Huang, C., Newman, M. & Shi, Y. Classical simulation of intermediate-size quantum circuits. Preprint at https://arxiv.org/abs/1805.01450 (2018).
Markov, I. L., Fatima, A., Isakov, S. V. & Boixo, S. Quantum supremacy is both closer and farther than it appears. Preprint at https://arxiv.org/abs/1807.10749 (2018).
Villalonga, B. et al. A flexible high-performance simulator for verifying and benchmarking quantum circuits implemented on real hardware. npj Quantum Inf. 5, 86 (2019).
Pednault, E. et al. Pareto-efficient quantum circuit simulation using tensor contraction deferral. Preprint at https://arxiv.org/abs/1710.05867 (2017).
Huang, C. et al. Efficient parallelization of tensor network contraction for simulating quantum computation. Nature Computational Science 1, 578–587 (2021).
Pan, F., Chen, K. & Zhang, P. Solving the Sampling Problem of the Sycamore Quantum Circuits. Phys. Rev. Lett. 129, 090502 (2022).
Liu, Y. et al. Verifying Quantum Advantage Experiments with Multiple Amplitude Tensor Network Contraction. Phys. Rev. Lett. 132, 030601 (2024).
Zhao, X.-H. et al. Leapfrogging sycamore: harnessing 1432 gpus for 7 × faster quantum random circuit sampling. National Science Review 12, nwae317 (2025).
Roth, D. On the hardness of approximate reasoning. Artificial Intelligence 82, 273–302 (1996).
Nishino, T. & Okunishi, K. Corner transfer matrix renormalization group method. J. Phys. Soc. Japan 65, 891–894 (1996).
Levin, M. & Nave, C. P. Tensor renormalization group approach to two-dimensional classical lattice models. Phys. Rev. Lett. 99, 120601 (2007).
Xie, Z.-Y. et al. Coarse-graining renormalization by higher-order singular value decomposition. Phys. Rev. B 86, 045139 (2012).
Evenbly, G. & Vidal, G. Tensor network renormalization. Phys. Rev. Lett. 115, 180405 (2015).
Chen, J., Jiang, J., Hangleiter, D. & Schuch, N. Sign problem in tensor network contraction. PRX Quantum 6, 010312 (2025).
Jiang, J., Chen, J., Schuch, N. & Hangleiter, D. Positive bias makes tensor-network contraction tractable. In STOC ’25: Proc. 57th Annual ACM Symposium on Theory of Computing (ACM, 2025).
Vidal, G. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101, 110501 (2008).
Verstraete, F. & Cirac, J. I. Renormalization algorithms for quantum-many body systems in two and higher dimensions. arXiv preprint cond-mat/0407066 (2004).
Jiang, H.-C., Weng, Z.-Y. & Xiang, T. Accurate determination of tensor network state of quantum lattice models in two dimensions. Phys. Rev. Lett. 101, 090603 (2008).
Corboz, P., Jordan, J. & Vidal, G. Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor hamiltonians. Phys. Rev. B—Condensed Matter and Materials Physics 82, 245119 (2010).
Lubasch, M., Cirac, J. I. & Banuls, M.-C. Unifying projected entangled pair state contractions. New J. Phys. 16, 033014 (2014).
Pan, F., Zhou, P., Li, S. & Zhang, P. Contracting arbitrary tensor networks: general approximate algorithm and applications in graphical models and quantum circuit simulations. Phys. Rev. Lett. 125, 060503 (2020).
Ma, L., Fishman, M., Stoudenmire, M. & Solomonik, E. Approximate contraction of arbitrary tensor networks with a flexible and efficient density matrix algorithm. Quantum 8, 1580 (2024).
Gray, J. & Chan, G. K.-L. Hyperoptimized approximate contraction of tensor networks with arbitrary geometry. Phys. Rev.X 14, 011009 (2024).
Zhou, Y., Stoudenmire, E. M. & Waintal, X. What limits the simulation of quantum computers? Phys. Rev.X 10, 041038 (2020).
Liu, M., Liu, J., Alexeev, Y. & Jiang, L. Estimating the randomness of quantum circuit ensembles up to 50 qubits. npj Quantum Inf. 8, 137 (2022).
Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023).
Zaletel, M. P. & Pollmann, F. Isometric tensor network states in two dimensions. Phys. Rev. Lett. 124, 037201 (2020).
Tindall, J. & Fishman, M. Gauging tensor networks with belief propagation. SciPost Physics 15, 222 (2023).
Anand, S., Temme, K., Kandala, A. & Zaletel, M. Classical benchmarking of zero noise extrapolation beyond the exactly-verifiable regime. Preprint at https://arxiv.org/abs/2306.17839 (2023).
Patra, S., Jahromi, S. S., Singh, S. & Orús, R. Efficient tensor network simulation of ibm’s largest quantum processors. Phys. Rev. Res. 6, 013326 (2024).
Rudolph, M. S., Fontana, E., Holmes, Z. & Cincio, L. Classical surrogate simulation of quantum systems with lowesa. Preprint at https://arxiv.org/abs/2308.09109 (2023).
Bouland, A., Fefferman, B., Landau, Z. & Liu, Y. Noise and the frontier of quantum supremacy (2022).
Krovi, H. Average-case hardness of estimating probabilities of random quantum circuits with a linear scaling in the error exponent. Preprint at https://arxiv.org/abs/2206.05642 (2022).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019). The first experimental realization of random circuit sampling claiming quantum advantage.
Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Zhu, Q. et al. Quantum computational advantage via 60-qubit 24-cycle random circuit sampling. Science bulletin 67, 240–245 (2022).
Aaronson, S. & Hung, S.-H. Certified randomness from quantum supremacy (2023).
Liu, M. et al. Certified randomness with a trapped-ion quantum processor. Nature 640, 343–348 (2025).
Kaleoglu, F. et al. On the equivalence between classical position verification and certified randomness. Preprint at https://arxiv.org/abs/2410.03982 (2024).
Amer, O. et al. Applications of certified randomness. Nat. Rev. Phys. https://doi.org/10.1038/s42254-025-00845-1 (2025).
Pednault, E., Gunnels, J., Maslov, D. & Gambetta, J. On “quantum supremacy”. IBM Research Blog 21 (2019).
Schutski, R., Lykov, D. & Oseledets, I. Adaptive algorithm for quantum circuit simulation. Phys. Rev.A 101, 042335 (2020).
Ayral, T. et al. Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity. PRX Quantum 4, 020304 (2023).
Haghshenas, R. et al. Digital quantum magnetism at the frontier of classical simulations. Preprint at https://arxiv.org/abs/2503.20870 (2025).
Noh, K., Jiang, L. & Fefferman, B. Efficient classical simulation of noisy random quantum circuits in one dimension. Quantum 4, 318 (2020).
Cheng, S. et al. Simulating noisy quantum circuits with matrix product density operators. Phys. Rev. Res. 3, 023005 (2021).
Guo, C. et al. General-purpose quantum circuit simulator with projected entangled-pair states and the quantum supremacy frontier. Phys. Rev. Lett. 123, 190501 (2019).
Ellerbrock, R. & Martinez, T. J. A multilayer multi-configurational approach to efficiently simulate large-scale circuit-based quantum computers on classical machines. J. Chem. Phys. 153, 051101 (2020).
Dumitrescu, E. Tree tensor network approach to simulating shor’s algorithm. Phys. Rev.A 96, 062322 (2017).
Gao, D. et al. Establishing a new benchmark in quantum computational advantage with 105-qubit zuchongzhi 3.0 processor. Phys. Rev. Lett. 134, 090601 (2025).
Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nature Physics 8, 277–284 (2012).
Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nature Physics 16, 132–142 (2020).
Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators. Nature physics 8, 285–291 (2012).
Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Physics 8, 292–299 (2012).
Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse ising model. Phys. Rev.E 58, 5355 (1998).
King, A. D. et al. Beyond-classical computation in quantum simulation. Science 388, 6743 (2025).
Shaw, A. L. et al. Benchmarking highly entangled states on a 60-atom analogue quantum simulator. Nature 628, 71–77 (2024).
King, A. D. et al. Comment on: “Dynamics of disordered quantum systems with two- and three-dimensional tensor networks”. Preprint at https://arxiv.org/abs/2504.06283 (2025).
Aaronson, S. & Arkhipov, A. The computational complexity of linear optics (2011). Proposal of a near-term experiment for quantum advantage based on sampling, specifically boson sampling using linear optics.
Zhong, H.-S. et al. Quantum computational advantage using photons. Science 370, 1460–1463 (2020). The first experimental realization of boson sampling claiming quantum advantage.
Zhong, H.-S. et al. Phase-programmable Gaussian boson sampling using stimulated squeezed light. Phys. Rev. Lett. 127, 180502 (2021).
Madsen, L. S. et al. Quantum computational advantage with a programmable photonic processor. Nature 606, 75–81 (2022).
Deng, Y.-H. et al. Gaussian boson sampling with pseudo-photon-number-resolving detectors and quantum computational advantage. Phys. Rev. Lett. 131, 150601 (2023).
Huang, H.-L., Bao, W.-S. & Guo, C. Simulating the dynamics of single photons in boson sampling devices with matrix product states. Phys. Rev. A 100, 032305 (2019).
Oh, C., Noh, K., Fefferman, B. & Jiang, L. Classical simulation of lossy boson sampling using matrix product operators. Phys. Rev. A 104, 022407 (2021).
Liu, M., Oh, C., Liu, J., Jiang, L. & Alexeev, Y. Simulating lossy Gaussian boson sampling with matrix-product operators. Phys. Rev. A 108, 052604 (2023).
Hamilton, C. S. et al. Gaussian boson sampling. Phys. Rev. Lett. 119, 170501 (2017).
Quesada, N. et al. Quadratic speed-up for simulating Gaussian boson sampling. PRX Quantum 3, 010306 (2022).
Cilluffo, D., Lorenzoni, N. & Plenio, M. B. Simulating Gaussian boson sampling with tensor networks in the Heisenberg picture. Preprint at https://arxiv.org/abs/2305.11215 (2023).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).
Biamonte, J. & Bergholm, V. Tensor networks in a nutshell. Preprint at https://arxiv.org/abs/1708.00006 (2017).
Schön, C., Solano, E., Verstraete, F., Cirac, J. I. & Wolf, M. M. Sequential generation of entangled multiqubit states. Phys. Rev. Lett. 95, 110503 (2005). This paper shows how MPS can be synthesized as quantum circuits.
Perez-Garcia, D., Verstraete, F., Wolf, M. M. & Cirac, J. I. Matrix product state representations. Preprint at https://arxiv.org/abs/quant-ph/0608197 (2006).
Ran, S.-J. Encoding of matrix product states into quantum circuits of one- and two-qubit gates. Phys. Rev. A 101, 032310 (2020).
Wei, Z.-Y., Malz, D. & Cirac, J. I. Efficient adiabatic preparation of tensor network states. Phys. Rev. Res. 5, L022037 (2023).
Malz, D., Styliaris, G., Wei, Z.-Y. & Cirac, J. I. Preparation of matrix product states with log-depth quantum circuits. Phys. Rev. Lett. 132, 040404 (2024).
Haghshenas, R., Gray, J., Potter, A. C. & Chan, G. K.-L. Variational power of quantum circuit tensor networks. Phys. Rev. X 12, 011047 (2022).
Foss-Feig, M. et al. Holographic quantum algorithms for simulating correlated spin systems. Phys. Rev. Res. 3, 033002 (2021).
Barratt, F. et al. Parallel quantum simulation of large systems on small NISQ computers. npj Quantum Inf. 7, 79 (2021).
Smith, A., Jobst, B., Green, A. G. & Pollmann, F. Crossing a topological phase transition with a quantum computer. Phys. Rev. Res. 4, L022020 (2022).
Meth, M. et al. Probing phases of quantum matter with an ion-trap tensor-network quantum eigensolver. Phys. Rev. X 12, 041035 (2022).
Lin, S.-H., Dilip, R., Green, A. G., Smith, A. & Pollmann, F. Real-and imaginary-time evolution with compressed quantum circuits. PRX Quantum 2, 010342 (2021).
Haghshenas, R., O’Rourke, M. J. & Chan, G. K.-L. Conversion of projected entangled pair states into a canonical form. Phys. Rev. B 100, 054404 (2019).
Wei, Z.-Y., Malz, D. & Cirac, J. I. Sequential generation of projected entangled-pair states. Phys. Rev. Lett. 128, 010607 (2022).
Kim, I. H. & Swingle, B. Robust entanglement renormalization on a noisy quantum computer. Preprint at https://arxiv.org/abs/1711.07500 (2017).
Martín, E. C., Plekhanov, K. & Lubasch, M. Barren plateaus in quantum tensor network optimization. Quantum 7, 974 (2023).
Barthel, T. & Miao, Q. Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states. Commun. Math. Phys. 406, 86 (2025).
Sewell, T. J. & Jordan, S. P. Preparing renormalization group fixed points on NISQ hardware. Preprint at https://arxiv.org/abs/2109.09787 (2021).
Anand, S., Hauschild, J., Zhang, Y., Potter, A. C. & Zaletel, M. P. Holographic quantum simulation of entanglement renormalization circuits. PRX Quantum 4, 030334 (2023). This paper uses quantum circuits derived from MERA and uses a holographic encoding to simulate quantum systems larger than the quantum device.
Haghshenas, R. et al. Probing critical states of matter on a digital quantum computer. Phys. Rev. Lett. 133, 266502 (2024).
Miao, Q., Wang, T., Brown, K. R., Barthel, T. & Cetina, M. Probing entanglement scaling across a quantum phase transition on a quantum computer. Preprint at https://arxiv.org/abs/2412.18602 (2024).
Zhang, Y., Gopalakrishnan, S. & Styliaris, G. Characterizing matrix-product states and projected entangled-pair states preparable via measurement and feedback. PRX Quantum 5, 040304 (2024).
Larsen, J. B., Grace, M. D., Baczewski, A. D. & Magann, A. B. Feedback-based quantum algorithms for ground state preparation. Phys. Rev. Res. 6, 033336 (2024).
Stephen, D. T. & Hart, O. Preparing matrix product states via fusion: constraints and extensions. Preprint at https://arxiv.org/abs/2404.16360 (2024).
Smith, K. C., Crane, E., Wiebe, N. & Girvin, S. Deterministic constant-depth preparation of the AKLT state on a quantum processor using fusion measurements. PRX Quantum 4, 020315 (2023).
Sahay, R. & Verresen, R. Classifying one-dimensional quantum states prepared by a single round of measurements. Preprint at https://arxiv.org/abs/2404.16753 (2024).
Ben-Dov, M., Shnaiderov, D., Makmal, A. & Dalla Torre, E. G. Approximate encoding of quantum states using shallow circuits. npj Quantum Inf. 10, 65 (2024).
Rudolph, M. S., Chen, J., Miller, J., Acharya, A. & Perdomo-Ortiz, A. Decomposition of matrix product states into shallow quantum circuits. Quantum Sci. Technol. 9, 015012 (2023).
Melnikov, A. A., Termanova, A. A., Dolgov, S. V., Neukart, F. & Perelshtein, M. Quantum state preparation using tensor networks. Quantum Sci. Technol. 8, 035027 (2023).
Termanova, A. et al. Tensor quantum programming. New J. Phys. 26, 123019 (2024).
Jaderberg, B. et al. Variational preparation of normal matrix product states on quantum computers. Preprint at https://arxiv.org/abs/2503.09683 (2025).
Kukliansky, A., Younis, E., Cincio, L. & Iancu, C. QFactor: a domain-specific optimizer for quantum circuit instantiation. Preprint at https://arxiv.org/abs/2306.08152 (2023).
Lidar, D. A. & Brun, T. A. Quantum Error Correction (Cambridge Univ. Press, 2013).
Ferris, A. J. & Poulin, D. Tensor networks and quantum error correction. Phys. Rev. Lett. 113, 030501 (2014).
Farrelly, T., Harris, R. J., McMahon, N. A. & Stace, T. M. Tensor-network codes. Phys. Rev. Lett. 127, 040507 (2021). Introduces TN codes with a natural TN decoder.
Farrelly, T., Tuckett, D. K. & Stace, T. M. Local tensor-network codes. New J. Phys. 24, 043015 (2022).
Cao, C. & Lackey, B. Quantum lego: Building quantum error correction codes from tensor networks. PRX Quantum 3, 020332 (2022). Generalizes code concatenation with TNs, dubbed ‘quantum Lego’.
Cao, C., Gullans, M. J., Lackey, B. & Wang, Z. Quantum Lego expansion pack: enumerators from tensor networks. PRX Quantum 5, 030313 (2024).
Fan, J. et al. LEGO_HQEC: a software tool for analyzing holographic quantum codes. Preprint at https://arxiv.org/abs/2410.22861 (2024).
Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. J. High Energy Phys. 2015, 149 (2015).
Jahn, A. & Eisert, J. Holographic tensor network models and quantum error correction: a topical review. Quantum Sci. Technol. 6, 033002 (2021).
Steinberg, M., Feld, S. & Jahn, A. Holographic codes from hyperinvariant tensor networks. Nat. Commun. 14, 7314 (2023).
Shen, R., Wang, Y. & Cao, C. Quantum Lego and XP stabilizer codes. Preprint at https://arxiv.org/abs/2310.19538 (2023).
Schotte, A., Zhu, G., Burgelman, L. & Verstraete, F. Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code. Phys. Rev. X 12, 021012 (2022).
Bettaque, V. & Swingle, B. NoRA: a tensor network ansatz for volume-law entangled equilibrium states of highly connected Hamiltonians. Quantum 8, 1362 (2024).
Cao, C. & Lackey, B. Approximate Bacon–Shor code and holography. J. High Energy Phys. 2021, 127 (2021).
Steinberg, M. et al. Far from perfect: quantum error correction with (hyperinvariant) Evenbly codes. Preprint at https://arxiv.org/abs/2407.11926 (2024).
Pastawski, F. & Preskill, J. Code properties from holographic geometries. Phys. Rev. X 7, 021022 (2017).
Su, V. P. et al. Discovery of optimal quantum error correcting codes via reinforcement learning. Phys. Rev. Appl. 23, 034048 (2025).
Mauron, C., Farrelly, T. & Stace, T. M. Optimization of tensor network codes with reinforcement learning. New J. Phys. 26, 023024 (2024).
Bravyi, S., Suchara, M. & Vargo, A. Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A 90, 032326 (2014).
Chubb, C. T. & Flammia, S. T. Statistical mechanical models for quantum codes with correlated noise. Ann. Inst. Henri Poincaré D 8, 269–321 (2021). Introduces a mapping of any stabilizer code to a statistical-mechanical model, connecting the optimal threshold of the QEC code to a phase transition.
Darmawan, A. S., Nakata, Y., Tamiya, S. & Yamasaki, H. Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder. Phys. Rev. Res. 6, 023055 (2025).
Farrelly, T., Milicevic, N., Harris, R. J., McMahon, N. A. & Stace, T. M. Parallel decoding of multiple logical qubits in tensor-network codes. Phys. Rev. A 105, 052446 (2022).
Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).
Piveteau, C., Chubb, C. T. & Renes, J. M. Tensor-network decoding beyond 2D. PRX Quantum 5, 040303 (2024).
Shutty, N., Newman, M. & Villalonga, B. Efficient near-optimal decoding of the surface code through ensembling. Preprint at https://arxiv.org/abs/2401.12434 (2024).
Kukliansky, A. & Lackey, B. Quantum circuit tensors and enumerators with applications to quantum fault tolerance. Preprint at https://arxiv.org/abs/2405.19643 (2024).
Cao, C. & Lackey, B. Quantum weight enumerators and tensor networks. IEEE Trans. Inf. Theory 70, 3512–3528 (2023).
Harris, R. J., McMahon, N. A., Brennen, G. K. & Stace, T. M. Calderbank-shor-steane holographic quantum error-correcting codes. Phys. Rev. A 98, 052301 (2018).
Harris, R. J., Coupe, E., McMahon, N. A., Brennen, G. K. & Stace, T. M. Decoding holographic codes with an integer optimization decoder. Phys. Rev. A 102, 062417 (2020).
Bao, N. & Naskar, J. Code properties of the holographic Sierpinski triangle. Phys. Rev. D 106, 126006 (2022).
Fan, J., Steinberg, M., Jahn, A., Cao, C. & Feld, S. Overcoming the zero-rate hashing bound with holographic quantum error correction. Preprint at https://arxiv.org/abs/2408.06232 (2024).
Tuckett, D. K. et al. Tailoring surface codes for highly biased noise. Phys. Rev. X 9, 041031 (2019).
Xu, Q. et al. Tailored xzzx codes for biased noise. Phys. Rev. Res. 5, 013035 (2023).
Tuckett, D. K., Bartlett, S. D. & Flammia, S. T. Ultrahigh error threshold for surface codes with biased noise. Phys. Rev. Lett. 120, 050505 (2018).
Bonilla Ataides, J. P., Tuckett, D. K., Bartlett, S. D., Flammia, S. T. & Brown, B. J. The XZZX surface code. Nat. Commun. 12, 2172 (2021).
Dua, A., Kubica, A., Jiang, L., Flammia, S. T. & Gullans, M. J. Clifford-deformed surface codes. PRX Quantum 5, 010347 (2024).
Kobayashi, F. et al. Tensor-network decoders for process tensor descriptions of non-Markovian noise. Preprint at https://arxiv.org/abs/2412.13739 (2024).
Battistel, F. et al. Real-time decoding for fault-tolerant quantum computing: progress, challenges and outlook. Nano Futures 7, 032003 (2023).
Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).
Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).
Guo, Y. & Yang, S. Quantum error mitigation via matrix product operators. PRX Quantum 3, 040313 (2022).
Tepaske, M. S. & Luitz, D. J. Compressed quantum error mitigation. Phys. Rev. B 107, L201114 (2023).
Filippov, S., Leahy, M., Rossi, M. A. & García-Pérez, G. Scalable tensor-network error mitigation for near-term quantum computing. Preprint at https://arxiv.org/abs/2307.11740 (2023).
Filippov, S. N., Maniscalco, S. & García-Pérez, G. Scalability of quantum error mitigation techniques: from utility to advantage. Preprint at https://arxiv.org/abs/2403.13542 (2024).
Fischer, L. E. et al. Dynamical simulations of many-body quantum chaos on a quantum computer. Preprint at https://arxiv.org/abs/2411.00765 (2024).
Piveteau, C., Sutter, D., Bravyi, S., Gambetta, J. M. & Temme, K. Error mitigation for universal gates on encoded qubits. Phys. Rev. Lett. 127, 200505 (2021).
Novikov, A., Podoprikhin, D., Osokin, A. & Vetrov, D. P. Tensorizing neural networks. In Advances in Neural Information Processing Systems (eds Cortes, C. et al.) Vol. 28, 442–450 (Curran, 2015).
Novikov, A., Trofimov, M. & Oseledets, I. Exponential machines. Preprint at https://arxiv.org/abs/1605.03795 (2016).
Stoudenmire, E. & Schwab, D. J. Supervised learning with tensor networks. In 30th Conference on Neural Information Processing Systems (NIPS, 2016)
Chen, J., Cheng, S., Xie, H., Wang, L. & Xiang, T. Equivalence of restricted Boltzmann machines and tensor network states. Phys. Rev. B 97, 085104 (2018).
Li, S., Pan, F., Zhou, P. & Zhang, P. Boltzmann machines as two-dimensional tensor networks. Phys. Rev. B 104, 075154 (2021).
Han, Z.-Y., Wang, J., Fan, H., Wang, L. & Zhang, P. Unsupervised generative modeling using matrix product states. Phys. Rev.X 8, 031012 (2018).
Liu, J., Li, S., Zhang, J. & Zhang, P. Tensor networks for unsupervised machine learning. Phys. Rev.E 107, L012103 (2023).
Glasser, I., Sweke, R., Pancotti, N., Eisert, J. & Cirac, I. Expressive power of tensor-network factorizations for probabilistic modeling. In Advances in Neural Information Processing Systems Vol. 32 pp. 1498–1510 (eds Wallach, H. et al.) (Curran, 2019).
Cheng, S., Wang, L., Xiang, T. & Zhang, P. Tree tensor networks for generative modeling. Phys. Rev. B 99, 155131 (2019).
Vieijra, T., Vanderstraeten, L. & Verstraete, F. Generative modeling with projected entangled-pair states. Preprint at https://arxiv.org/abs/2202.08177 (2022).
Reyes, J. A. & Stoudenmire, E. M. Multi-scale tensor network architecture for machine learning. Mach. Learn. Sci. Technol. 2, 035036 (2021).
Huggins, W., Patil, P., Mitchell, B., Whaley, K. B. & Stoudenmire, E. M. Towards quantum machine learning with tensor networks. Quantum Sci. Technol. 4, 024001 (2019).
Wall, M. L., Abernathy, M. R. & Quiroz, G. Generative machine learning with tensor networks: benchmarks on near-term quantum computers. Phys. Rev. Res. 3, 023010 (2021).
Grant, E. et al. Hierarchical quantum classifiers. npj Quantum Inf. 4, 65 (2018).
Lazzarin, M., Galli, D. E. & Prati, E. Multi-class quantum classifiers with tensor network circuits for quantum phase recognition. Phys. Lett. A 434, 128056 (2022).
Rieser, H.-M., Köster, F. & Raulf, A. P. Tensor networks for quantum machine learning. Proc. R. Soc. A 479, 20230218 (2023).
Zhao, C. & Gao, X.-S. Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. Quantum 5, 466 (2021).
Pesah, A. et al. Absence of barren plateaus in quantum convolutional neural networks. Phys. Rev. X 11, 041011 (2021).
Liao, H., Convy, I., Yang, Z. & Whaley, K. B. Decohering tensor network quantum machine learning models. Quant. Mach. Intell. 5, 7 (2023).
Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. 15, 1273–1278 (2019).
Bermejo, P. et al. Quantum convolutional neural networks are (effectively) classically simulable. Preprint at https://arxiv.org/abs/2408.12739 (2024).
Dilip, R., Liu, Y.-J., Smith, A. & Pollmann, F. Data compression for quantum machine learning. Phys. Rev. Res. 4, 043007 (2022).
Iaconis, J. & Johri, S. Tensor network based efficient quantum data loading of images. Preprint at https://arxiv.org/abs/2310.05897 (2023).
Dborin, J., Barratt, F., Wimalaweera, V., Wright, L. & Green, A. G. Matrix product state pre-training for quantum machine learning. Quantum Sci. Technol. 7, 035014 (2022).
Rudolph, M. S. et al. Synergy between quantum circuits and tensor networks: short-cutting the race to practical quantum advantage. Preprint at https://arxiv.org/abs/2208.13673 (2022).
Khan, A., Clark, B. K. & Tubman, N. M. Pre-optimizing variational quantum eigensolvers with tensor networks. Preprint at https://arxiv.org/abs/2310.12965 (2023).
Shin, S., Teo, Y. S. & Jeong, H. Dequantizing quantum machine learning models using tensor networks. Phys. Rev. Res. 6, 023218 (2024).
de Beaudrap, N., Kissinger, A. & Meichanetzidis, K. Tensor network rewriting strategies for satisfiability and counting. Preprint at https://arxiv.org/abs/2004.06455 (2020).
Rakovszky, T., Von Keyserlingk, C. & Pollmann, F. Dissipation-assisted operator evolution method for capturing hydrodynamic transport. Phys. Rev. B 105, 075131 (2022).
Tarabunga, P. S., Tirrito, E., Bañuls, M. C. & Dalmonte, M. Nonstabilizerness via matrix product states in the Pauli basis. Phys. Rev. Lett. 133, 010601 (2024).
Masot-Llima, S. & Garcia-Saez, A. Stabilizer tensor networks: universal quantum simulator on a basis of stabilizer states. Phys. Rev. Lett. 133, 230601 (2024).
Breuckmann, N. P. & Eberhardt, J. N. Quantum low-density parity-check codes. PRX Quantum 2, 040101 (2021).
Panteleev, P. & Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 375–388 (ACM, 2022).
Bausch, J. et al. Learning high-accuracy error decoding for quantum processors. Nature 635, 834–840 (2024).
Zhou, H. H. et al. NVIDIA and QuEra decode quantum errors with AI. NVIDIA Technical Blog https://developer.nvidia.com/blog/nvidia-and-quera-decode-quantum-errors-with-ai/ (2025).
Cerezo, M. et al. Does provable absence of barren plateaus imply classical simulability? or, why we need to rethink variational quantum computing. Preprint at https://arxiv.org/abs/2312.09121 (2023).
Gil-Fuster, E., Gyurik, C., Pérez-Salinas, A. & Dunjko, V. On the relation between trainability and dequantization of variational quantum learning models. Preprint at https://arxiv.org/abs/2406.07072 (2024).