Successive randomized compression: A randomized algorithm for the compressed MPO-MPS product
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AbstractTensor networks like matrix product states (MPSs) and matrix product operators (MPOs) are powerful tools for representing exponentially large states and operators, with applications in quantum many-body physics, machine learning, numerical analysis, and other areas. In these applications, computing a compressed representation of the MPO-MPS product is a fundamental computational primitive. For this operation, this paper introduces a new single-pass, randomized algorithm, called successive randomized compression (SRC), that improves on existing approaches in speed or in accuracy. The performance of the new algorithm is evaluated on synthetic problems and unitary time evolution problems for quantum spin systems.Popular summaryTensor networks have proven to be a powerful computational framework for the classical simulation of large quantum systems, including circuit simulation, quantum chemistry, and many-body dynamics. A core primitive in many tensor network workflows is applying an operator $\hat{O}$ to a state $|{\psi}\rangle$. In one dimension, this operation usually amounts to applying a matrix product operator (MPO) to a matrix product states (MPS). Directly applying an MPO with bond dimension ${D}$ to an MPS with bond dimension ${\chi}$ produces an intermediate state with bond dimension up to ${D\chi}$. This multiplicative growth often makes the resulting MPS impractical to use unless it is compressed, often by dense truncated singular value decompositions, to a more manageable bond dimension ${\bar{\chi}} < {D\chi}$. This compression bottleneck has motivated the development of various fast MPO–MPS multiplication strategies. In this work, we introduce successive randomized compression (SRC), a fast single-pass randomized algorithm for computing the compressed MPO–MPS product. SRC leverages ideas in randomized low rank approximation to directly produce $|{\eta}\rangle \approx \hat{O}|{\psi}\rangle$ without ever forming the costly intermediate state $\hat{O}|{\psi}\rangle$ of bond dimension ${D\chi}$. Across a suite of numerical benchmarks, SRC is as fast as or faster than existing MPO–MPS multiplication algorithms, while delivering accuracy comparable to exact MPO–MPS multiplication. We support this claim using several tests and an end-to-end real-time tensor-network simulation of a long-range XY spin chain with $n=101$ spins time evolved using GSE-TDVP1. In this workflow, SRC is the fastest method and achieves speedups up to $181\times$ over direct MPO–MPS multiplication.► BibTeX data@article{Camano2026successive, doi = {10.22331/q-2026-03-10-2022}, url = {https://doi.org/10.22331/q-2026-03-10-2022}, title = {Successive randomized compression: {A} randomized algorithm for the compressed {MPO}-{MPS} product}, author = {Cama{\~{n}}o, Chris and Epperly, Ethan N. and Tropp, Joel A.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2022}, month = mar, year = {2026} }► References [1] M. Fannes, B. Nachtergaele, and R. F. Werner. ``Finitely correlated states on quantum spin chains''. Communications in Mathematical Physics 144, 443–490 (1992). https://doi.org/10.1007/BF02099178 [2] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. ``Matrix product density operators: Simulation of finite-temperature and dissipative systems''. Phys. Rev. Lett. 93, 207204 (2004). https://doi.org/10.1103/PhysRevLett.93.207204 [3] Y.-Y. Shi, L.-M. Duan, and G. Vidal. ``Classical simulation of quantum many-body systems with a tree tensor network''. Physical Review A 74 (2006). https://doi.org/10.1103/physreva.74.022320 [4] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac. ``Criticality, the area law, and the computational power of projected entangled pair states''.
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Epperly and Joel A. Tropp. ``Efficient Error and Variance Estimation for Randomized Matrix Computations''. SIAM Journal on Scientific Computing 46, A508–A528 (2024). https://doi.org/10.1137/23M1558537Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-10 15:03:38: Could not fetch cited-by data for 10.22331/q-2026-03-10-2022 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-10 15:03:39: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractTensor networks like matrix product states (MPSs) and matrix product operators (MPOs) are powerful tools for representing exponentially large states and operators, with applications in quantum many-body physics, machine learning, numerical analysis, and other areas. In these applications, computing a compressed representation of the MPO-MPS product is a fundamental computational primitive. For this operation, this paper introduces a new single-pass, randomized algorithm, called successive randomized compression (SRC), that improves on existing approaches in speed or in accuracy. The performance of the new algorithm is evaluated on synthetic problems and unitary time evolution problems for quantum spin systems.Popular summaryTensor networks have proven to be a powerful computational framework for the classical simulation of large quantum systems, including circuit simulation, quantum chemistry, and many-body dynamics. A core primitive in many tensor network workflows is applying an operator $\hat{O}$ to a state $|{\psi}\rangle$. In one dimension, this operation usually amounts to applying a matrix product operator (MPO) to a matrix product states (MPS). Directly applying an MPO with bond dimension ${D}$ to an MPS with bond dimension ${\chi}$ produces an intermediate state with bond dimension up to ${D\chi}$. This multiplicative growth often makes the resulting MPS impractical to use unless it is compressed, often by dense truncated singular value decompositions, to a more manageable bond dimension ${\bar{\chi}} < {D\chi}$. This compression bottleneck has motivated the development of various fast MPO–MPS multiplication strategies. In this work, we introduce successive randomized compression (SRC), a fast single-pass randomized algorithm for computing the compressed MPO–MPS product. SRC leverages ideas in randomized low rank approximation to directly produce $|{\eta}\rangle \approx \hat{O}|{\psi}\rangle$ without ever forming the costly intermediate state $\hat{O}|{\psi}\rangle$ of bond dimension ${D\chi}$. Across a suite of numerical benchmarks, SRC is as fast as or faster than existing MPO–MPS multiplication algorithms, while delivering accuracy comparable to exact MPO–MPS multiplication. We support this claim using several tests and an end-to-end real-time tensor-network simulation of a long-range XY spin chain with $n=101$ spins time evolved using GSE-TDVP1. In this workflow, SRC is the fastest method and achieves speedups up to $181\times$ over direct MPO–MPS multiplication.► BibTeX data@article{Camano2026successive, doi = {10.22331/q-2026-03-10-2022}, url = {https://doi.org/10.22331/q-2026-03-10-2022}, title = {Successive randomized compression: {A} randomized algorithm for the compressed {MPO}-{MPS} product}, author = {Cama{\~{n}}o, Chris and Epperly, Ethan N. and Tropp, Joel A.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2022}, month = mar, year = {2026} }► References [1] M. Fannes, B. Nachtergaele, and R. F. Werner. ``Finitely correlated states on quantum spin chains''. Communications in Mathematical Physics 144, 443–490 (1992). https://doi.org/10.1007/BF02099178 [2] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. ``Matrix product density operators: Simulation of finite-temperature and dissipative systems''. Phys. Rev. Lett. 93, 207204 (2004). https://doi.org/10.1103/PhysRevLett.93.207204 [3] Y.-Y. Shi, L.-M. Duan, and G. Vidal. ``Classical simulation of quantum many-body systems with a tree tensor network''. Physical Review A 74 (2006). https://doi.org/10.1103/physreva.74.022320 [4] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac. ``Criticality, the area law, and the computational power of projected entangled pair states''.
Physical Review Letters 96 (2006). https://doi.org/10.1103/physrevlett.96.220601 [5] Guifre Vidal. ``Entanglement renormalization: an introduction'' (2010). url: https://arxiv.org/abs/0912.1651v2. arXiv:0912.1651v2 [6] Jacob C. Bridgeman and Christopher T. Chubb. ``Hand-waving and interpretive dance: An introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017). https://doi.org/10.1088/1751-8121/aa6dc3 [7] G. Evenbly and G. Vidal. ``Tensor network states and geometry''. Journal of Statistical Physics 145, 891918 (2011). https://doi.org/10.1007/s10955-011-0237-4 [8] Román Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Annals of Physics 349, 117–158 (2014). https://doi.org/10.1016/j.aop.2014.06.013 [9] Román Orús. ``Tensor networks for complex quantum systems''.
Nature Reviews Physics 1, 538–550 (2019). https://doi.org/10.1038/s42254-019-0086-7 [10] I. V. Oseledets. ``Approximation of $2^d \times 2^d$ matrices using tensor decomposition''. SIAM Journal on Matrix Analysis and Applications 31, 2130–2145 (2010). https://doi.org/10.1137/090757861 [11] Boris N. Khoromskij. ``$O(d \log N)$-quantics approximation of $N$-d tensors in high-dimensional numerical modeling''. Constructive Approximation 34, 257–280 (2011). https://doi.org/10.1007/s00365-011-9131-1 [12] Michael Lindsey. ``Multiscale interpolative construction of quantized tensor trains'' (2024). url: https://arxiv.org/abs/2311.12554v3. arXiv:2311.12554v3 [13] Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, and Pan Zhang. ``Unsupervised generative modeling using matrix product states''. Physical Review X 8 (2018). https://doi.org/10.1103/physrevx.8.031012 [14] YoonHaeng Hur, Jeremy G. Hoskins, Michael Lindsey, E. M. 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