Abstract
Quantum simulation algorithms often require numerous ancilla qubits and deep circuits, prohibitive for near-term hardware. We introduce a framework for simulating quantum channels using ensembles of low-depth circuits in place of many-qubit dilations. This naturally enables simulations of open systems, which we demonstrate by preparing damped many-qubit GHZ states on ibm_hanoi. The technique further inspires two Hamiltonian simulation algorithms with gate counts that are asymptotically independent of the spectral precision target, reducing resource requirements by several orders of magnitude for a benchmark system.
Similar content being viewed by others
Data availability
Data for the GHZ state preparation results run on ibm_hanoi can be provided upon reasonable request.
Code availability
The code used to generate Figs. 2 and 4 can be provided upon reasonable request.
References
Childs, A. M. & Wiebe, N. Hamiltonian simulation using linear combinations of unitary operations. Quantum Inf. Comput. 12, 901–924 (2012).
Childs, A. M., Kothari, R. & Somma, R. D. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM J. Comput. 46, 1920–1950 (2017).
Low, G. H. & Chuang, I. L. Hamiltonian simulation by qubitization. Quantum 3, 163 (2019).
Chakraborty, S., Gilyén, A. & Jeffery, S. The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. In Proc. 46th International Colloquium on Automata, Languages, and Programming (ICALP), Vol. 132, 1–33 (2019)..
Wan, L.-C. et al. Block-encoding-based quantum algorithm for linear systems with displacement structures. Phys. Rev. A 104, 062414 (2021).
Wang, S. et al. Fast black-box quantum state preparation based on linear combination of unitaries. Quantum Inf. Process. 20, 270 (2021).
Nguyen, Q. T., Kiani, B. T. & Lloyd, S. Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra. Quantum 6, 876 (2022).
An, D., Liu, J.-P. & Lin, L. Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost. Phys. Rev. Lett. 131, 150603 (2023).
Stinespring, W. F. Positive functions on c*-algebras (1955).
Paulsen, V. Completely Bounded Maps and Operator Algebras Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, Cambridge, 2003).
Hu, Z., Xia, R. & Kais, S. A quantum algorithm for evolving open quantum dynamics on quantum computing devices. Sci. Rep. 10, 3301 (2020).
Langer, H. B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert Space. VIII + 387 S. Budapest/Amsterdam/London 1970. Akadémiai Kiadó/North-Holland Publishing Company. Z. Angew. Math. Mech. 52, 501–501 (1972).
Faehrmann, P. K., Steudtner, M., Kueng, R., Kieferova, M. & Eisert, J. Randomizing multi-product formulas for Hamiltonian simulation. Quantum 6, 806 (2022).
Chakraborty, S. Implementing any linear combination of unitaries on intermediate-term quantum computers. Quantum 8, 1496 (2024).
Breuer, H.-P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford Univ. Press, Oxford, 2007).
Head-Marsden, K., Flick, J., Ciccarino, C. J. & Narang, P. Quantum information and algorithms for correlated quantum matter. Chem. Rev. 121, 3061–3120 (2021).
Schlimgen, A. W., Head-Marsden, K., Sager, L. M., Narang, P. & Mazziotti, D. A. Quantum simulation of open quantum systems using a unitary decomposition of operators. Phys. Rev. Lett. 127, 270503 (2021).
Kamakari, H., Sun, S.-N., Motta, M. & Minnich, A. J. Digital quantum simulation of open quantum systems using quantum imaginary–time evolution. PRX Quantum 3, 010320 (2022).
Egorova, D., Kühl, A. & Domcke, W. Modeling of ultrafast electron-transfer dynamics: multi-level Redfield theory and validity of approximations. Chem. Phys 268, 105–120 (2001).
Okolowicz, J., Ploszajczak, M., & Rotter, I. Dynamics of quantum systems embedded in a continuum. Phys. Rep. 374, 271–383 (2003).
Rotter, I. & Bird, J. P. A review of progress in the physics of open quantum systems: theory and experiment. Rep. Prog. Phys. 78, 114001 (2015).
Delle Site, L. & Praprotnik, M. Molecular systems with open boundaries: theory and simulation. Phys. Rep. 693, 1–56 (2017).
Moueddene, A. A., Khammassi, N., Bertels, K. & Almudever, C. G. Realistic simulation of quantum computation using unitary and measurement channels. Phys. Rev. A 102, 052608 (2020).
Wiebe, N., Berry, D. W., Hoyer, P. & Sanders, B. C. Simulating quantum dynamics on a quantum computer. J. Phys. A Math. Theor. 44, 445308 (2011).
Miessen, A., Ollitrault, P. J., Tacchino, F. & Tavernelli, I. Quantum algorithms for quantum dynamics. Nat. Comput. Sci. 3, 25–37 (2023).
Ollitrault, P. J., Miessen, A. & Tavernelli, I. Molecular quantum dynamics: a quantum computing perspective. Acc. Chem. Res. 54, 4229–4238 (2021).
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).
Dong, Y., Lin, L. & Tong, Y. Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices. PRX Quantum 3, 040305 (2022).
Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023).
Nam, Y. et al. Ground-state energy estimation of the water molecule on a trapped-ion quantum computer. npj Quantum Inf. 6, 1–6 (2020).
Bauer, B., Bravyi, S., Motta, M. & Chan, G. K.-L. Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev. 120, 12685–12717 (2020).
Lordi, V. & Nichol, J. M. Advances and opportunities in materials science for scalable quantum computing. MRS Bull. 46, 589–595 (2021).
De Leon, N. P. et al. Materials challenges and opportunities for quantum computing hardware. Science 372, eabb2823 (2021).
Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arXiv.org/abs/1411.4028 (2014).
Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).
Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).
Suzuki, M. General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991).
Childs, A. M., Su, Y., Tran, M. C., Wiebe, N. & Zhu, S. Theory of Trotter error with commutator scaling. Phys. Rev. X 11, 011020 (2021).
Childs, A. M. & Su, Y. Nearly optimal lattice simulation by product formulas. Phys. Rev. Lett. 123, 050503 (2019).
Low, G. H. & Chuang, I. L. Optimal Hamiltonian simulation by quantum signal processing. Phys. Rev. Lett. 118, 010501 (2017).
Campbell, E. Random compiler for fast Hamiltonian simulation. Phys. Rev. Lett. 123, 070503 (2019).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge Univ. Press, 2010).
Audretsch, J. Entangled Systems: New Directions in Quantum Physics 1st edn (Wiley, 2007).
Peetz, J., Smart, S. E., Tserkis, S. & Narang, P. Simulation of open quantum systems via low-depth convex unitary evolutions. Phys. Rev. Res. 6, 023263 (2024).
Arrasmith, A., Cincio, L., Somma, R. D. & Coles, P. J. Operator sampling for shot-frugal optimization in variational algorithms. Preprint at https://arXiv.org/abs/2004.06252 (2020).
Cleve, R. & Wang, C. Efficient quantum algorithms for simulating Lindblad evolution. Proc. 44th ICALP 80, 1–17 (2017).
Brassard, G., Høyer, P., Mosca, M. & Tapp, A. Quantum amplitude amplification and estimation. Contemp. Math. 305, 53–74 (2002).
Berry, D. W., Childs, A. M., Cleve, R., Kothari, R. & Somma, R. D. Exponential improvement in precision for simulating sparse Hamiltonians. In Proc. 46th ACM Symposium on Theory of Computing (STOC), 283–292 (ACM, 2014).
Berry, D. W., Childs, A. M., Cleve, R., Kothari, R. & Somma, R. D. Simulating Hamiltonian dynamics with a truncated Taylor series. Phys. Rev. Lett. 114, 090502 (2015).
Avis, G., Rozpȩdek, F. & Wehner, S. Analysis of multipartite entanglement distribution using a central quantum-network node. Phys. Rev. A 107, 012609 (2023).
Meignant, C., Markham, D. & Grosshans, F. Distributing graph states over arbitrary quantum networks. Phys. Rev. A 100, 052333 (2019).
Patil, A., Pant, M., Englund, D., Towsley, D. & Guha, S. Entanglement generation in a quantum network at distance-independent rate. npj Quantum Inf. 8, 1–9 (2022).
Covey, J. P., Weinfurter, H. & Bernien, H. Quantum networks with neutral atom processing nodes. npj Quantum Inf. 9, 1–12 (2023).
Gärttner, M., Hauke, P. & Rey, A. M. Relating out-of-time-order correlations to entanglement via multiple-quantum coherences. Phys. Rev. Lett. 120, 040402 (2018).
Rost, B. et al. Simulation of thermal relaxation in spin chemistry systems on a quantum computer using inherent qubit decoherence. Preprint at https://arXiv.org/abs/2001.00794 (2020).
Wan, K., Berta, M. & Campbell, E. T. Randomized quantum algorithm for statistical phase estimation. Phys. Rev. Lett. 129, 030503 (2022).
Childs, A. M., Maslov, D., Nam, Y., Ross, N. J. & Su, Y. Toward the first quantum simulation with quantum speedup. Proc. Natl. Acad. Sci. USA 115, 9456–9461 (2018).
Hagan, M. & Wiebe, N. Composite quantum simulations. Quantum 7, 1181 (2023).
Yen, T.-C., Verteletskyi, V. & Izmaylov, A. F. Measuring all compatible operators in one series of single-qubit measurements using unitary transformations. J. Chem. Theory Comput. 16, 2400–2409 (2020).
Loaiza, I., Khah, A. M., Wiebe, N. & Izmaylov, A. F. Reducing molecular electronic Hamiltonian simulation cost for linear combination of unitaries approaches. Quantum Sci. Technol. 8, 035019 (2023).
Wu, B., Yan, Y., Wei, F. & Liu, Z. Optimization for expectation value estimation with shallow quantum circuits. Preprint at https://arXiv.org/abs/2407.19499 (2024).
Wei, K. X. et al. Verifying multipartite entangled Greenberger-Horne-Zeilinger states via multiple quantum coherences. Phys. Rev. A 101, 032343 (2020).
Mooney, G. J., White, G. A. L., Hill, C. D. & Hollenberg, L. C. L. Generation and verification of 27-qubit Greenberger-Horne-Zeilinger states in a superconducting quantum computer. J. Phys. Commun. 5, 095004 (2021).
Chen, C.-F., Huang, H.-Y., Kueng, R. & Tropp, J. A. Concentration for random product formulas. PRX Quantum 2, 040305 (2021).
Kiss, O., Grossi, M. & Roggero, A. Importance sampling for stochastic quantum simulations. Quantum 7, 977 (2023).
Acknowledgements
This work is supported by an NSF CAREER Award under Grant No. NSF-ECCS-1944085 and the NSF CNS program under Grant No. 2247007. The authors acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.
Author information
Authors and Affiliations
Contributions
J.P. and S.E.S. developed the framework with guidance from P.N. J.P. designed the Hamiltonian simulation approaches with feedback from S.E.S. and P.N. J.P. implemented the GHZ experiment and computed the TFIM resource estimates with support from S.E.S. All authors contributed to the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Peetz, J., Smart, S.E. & Narang, P. Quantum simulation via stochastic combination of unitaries. npj Quantum Inf (2026). https://doi.org/10.1038/s41534-025-01168-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41534-025-01168-w
