Preparing low-variance states using a distributed quantum algorithm

Xiaoyu Liu; Benjamin F. Schiffer; Jordi Tura

doi:10.22331/q-2025-08-28-1838

Preparing low-variance states using a distributed quantum algorithm

Xiaoyu Liu^1,2, Benjamin F. Schiffer³, and Jordi Tura^1,2

¹$\langle aQa ^L\rangle $ Applied Quantum Algorithms, Universiteit Leiden
²Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
³Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

Published:	2025-08-28, volume 9, page 1838
Editor:	Daniel Malz
Eprint:	arXiv:2501.13097v2
Doi:	https://doi.org/10.22331/q-2025-08-28-1838
Citation:	Quantum 9, 1838 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum computers are a highly promising tool for efficiently simulating quantum many-body systems. The preparation of their eigenstates is of particular interest and can be addressed, e.g., by quantum phase estimation algorithms. The routine then acts as an effective filtering operation, reducing the energy variance of the initial state. In this work, we present a distributed quantum algorithm inspired by iterative phase estimation to prepare low-variance states. Our method uses a single auxiliary qubit per quantum device, which controls its dynamics, and a postselection strategy for a joint quantum measurement on such auxiliary qubits. In the multi-device case, the result of this measurement heralds the successful runs of the protocol. This allows us to demonstrate that our distributed algorithm reduces the energy variance faster compared to single-device implementations, thereby highlighting the potential of distributed algorithms for near-term and early fault-tolerant devices.

Popular summary

Quantum computers are a powerful tool for simulating quantum-mechanical systems. A key ingredient is the ability to prepare the lowest-energy configuration (the ground state) of such systems. Quantum algorithms typically start from a rough guess and iteratively refine it toward the target configuration. The quantum phase estimation algorithm is a standard approach: it acts like a narrow energy filter, concentrating the state around a single energy and suppressing everything else. The drawback is that very sharp filtering usually demands deep and noise-sensitive circuits.

In this work, we introduce a distributed version of the phase estimation algorithm that runs across several quantum processors linked by quantum connections. The devices share only a limited number of entangled qubit pairs (Bell pairs) to coordinate the steps of the algorithm. This design achieves the same energy-filter effect with shallower circuits than a single-processor approach. By keeping only runs with the desired measurement outcomes (post-selection), the protocol can produce identical copies of the target eigenstate on different processors. These results show that networking quantum processors yields a concrete performance advantage for a core quantum simulation task.

► BibTeX data

@article{Liu2025preparinglow,
  doi = {10.22331/q-2025-08-28-1838},
  url = {https://doi.org/10.22331/q-2025-08-28-1838},
  title = {Preparing low-variance states using a distributed quantum algorithm},
  author = {Liu, Xiaoyu and Schiffer, Benjamin F. and Tura, Jordi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1838},
  month = aug,
  year = {2025}
}

► References

[1] Michael A. Nielsen and Isaac L. Chuang. ``Quantum Computation and Quantum Information: 10th Anniversary Edition''. Cambridge University Press. (2010). url: https://doi.org/10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667

[2] A. Yu Kitaev. ``Quantum measurements and the Abelian Stabilizer Problem'' (1995). arXiv:quant-ph/9511026.
arXiv:quant-ph/9511026

[3] Thomas E. O'Brien, Brian Tarasinski, and Barbara M. Terhal. ``Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments''. New Journal of Physics 21, 023022 (2019).
https://doi.org/10.1088/1367-2630/aafb8e

[4] Alicja Dutkiewicz, Barbara M. Terhal, and Thomas E. O'Brien. ``Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits''. Quantum 6, 830 (2022).
https://doi.org/10.22331/q-2022-10-06-830

[5] Michael Hartmann, Günter Mahler, and Ortwin Hess. ``Spectral Densities and Partition Functions of Modular Quantum Systems as Derived from a Central Limit Theorem''. Journal of Statistical Physics 119, 1139–1151 (2005).
https://doi.org/10.1007/s10955-004-4298-5

[6] Daniel S. Abrams and Seth Lloyd. ``Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors''. Physical Review Letters 83, 5162–5165 (1999).
https://doi.org/10.1103/PhysRevLett.83.5162

[7] David Poulin and Pawel Wocjan. ``Preparing Ground States of Quantum Many-Body Systems on a Quantum Computer''. Physical Review Letters 102, 130503 (2009).
https://doi.org/10.1103/PhysRevLett.102.130503

[8] Yimin Ge, Jordi Tura, and J. Ignacio Cirac. ``Faster ground state preparation and high-precision ground energy estimation with fewer qubits''. Journal of Mathematical Physics 60, 022202 (2019).
https://doi.org/10.1063/1.5027484

[9] Sirui Lu, Mari Carmen Bañuls, and J. Ignacio Cirac. ``Algorithms for Quantum Simulation at Finite Energies''. PRX Quantum 2, 020321 (2021).
https://doi.org/10.1103/PRXQuantum.2.020321

[10] Reinis Irmejs, Mari Carmen Bañuls, and J. Ignacio Cirac. ``Efficient Quantum Algorithm for Filtering Product States''. Quantum 8, 1389 (2024).
https://doi.org/10.22331/q-2024-06-27-1389

[11] Kshiti Sneh Rai, J. Ignacio Cirac, and Álvaro M. Alhambra. ``Matrix product state approximations to quantum states of low energy variance''. Quantum 8, 1401 (2024).
https://doi.org/10.22331/q-2024-07-10-1401

[12] Miroslav Dobšíček, Göran Johansson, Vitaly Shumeiko, and Göran Wendin. ``Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark''. Physical Review A 76, 030306 (2007).
https://doi.org/10.1103/PhysRevA.76.030306

[13] Jin-Shi Xu, Man-Hong Yung, Xiao-Ye Xu, Sergio Boixo, Zheng-Wei Zhou, Chuan-Feng Li, Alán Aspuru-Guzik, and Guang-Can Guo. ``Demon-like algorithmic quantum cooling and its realization with quantum optics''. Nature Photonics 8, 113–118 (2014).
https://doi.org/10.1038/nphoton.2013.354

[14] Richard Meister and Simon C. Benjamin. ``Resource-frugal Hamiltonian eigenstate preparation via repeated quantum phase estimation measurements'' (2022). arXiv:2212.00846.
arXiv:2212.00846

[15] Yanzhu Chen and Tzu-Chieh Wei. ``Quantum algorithm for spectral projection by measuring an ancilla iteratively''. Physical Review A 101, 032339 (2020).
https://doi.org/10.1103/PhysRevA.101.032339

[16] Benjamin F. Schiffer and Jordi Tura. ``Quantum eigenstate preparation assisted by a coherent link''. Physical Review A 111, 012445 (2025).
https://doi.org/10.1103/PhysRevA.111.012445

[17] Zhengrong Qian, Jacob Watkins, Gabriel Given, Joey Bonitati, Kenneth Choi, and Dean Lee. ``Demonstration of the rodeo algorithm on a quantum computer''. The European Physical Journal A 60, 151 (2024).
https://doi.org/10.1140/epja/s10050-024-01373-9

[18] Kenneth Choi, Dean Lee, Joey Bonitati, Zhengrong Qian, and Jacob Watkins. ``Rodeo Algorithm for Quantum Computing''. Physical Review Letters 127, 040505 (2021).
https://doi.org/10.1103/PhysRevLett.127.040505

[19] Dorit Aharonov, Vaughan Jones, and Zeph Landau. ``A polynomial quantum algorithm for approximating the Jones polynomial''. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing. Pages 427–436. STOC '06New York, NY, USA (2006). Association for Computing Machinery.
https://doi.org/10.1145/1132516.1132579

[20] P. Magnard, S. Storz, P. Kurpiers, J. Schär, F. Marxer, J. Lütolf, T. Walter, J.-C. Besse, M. Gabureac, K. Reuer, A. Akin, B. Royer, A. Blais, and A. Wallraff. ``Microwave Quantum Link between Superconducting Circuits Housed in Spatially Separated Cryogenic Systems''. Physical Review Letters 125, 260502 (2020).
https://doi.org/10.1103/PhysRevLett.125.260502

[21] Patrick Harvey-Collard, Jurgen Dijkema, Guoji Zheng, Amir Sammak, Giordano Scappucci, and Lieven M. K. Vandersypen. ``Coherent Spin-Spin Coupling Mediated by Virtual Microwave Photons''. Physical Review X 12, 021026 (2022).
https://doi.org/10.1103/PhysRevX.12.021026

[22] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello. ``Distributed quantum computation over noisy channels''. Physical Review A 59, 4249–4254 (1999).
https://doi.org/10.1103/PhysRevA.59.4249

[23] Harry Buhrman and Hein Röhrig. ``Distributed Quantum Computing''. In Branislav Rovan and Peter Vojtáš, editors, Mathematical Foundations of Computer Science 2003. Pages 1–20. Berlin, Heidelberg (2003). Springer.
https://doi.org/10.1007/978-3-540-45138-9_1

[24] Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. ``Stabilization of Quantum Computations by Symmetrization''. SIAM Journal on Computing 26, 1541–1557 (1997).
https://doi.org/10.1137/S0097539796302452

[25] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. ``Quantum Fingerprinting''. Physical Review Letters 87, 167902 (2001).
https://doi.org/10.1103/PhysRevLett.87.167902

[26] Max Born. ``Zur Quantenmechanik der Stoßvorgänge''. Zeitschrift für Physik 37, 863–867 (1926).
https://doi.org/10.1007/BF01397477

[27] Masaru Kada, Harumichi Nishimura, and Tomoyuki Yamakami. ``The efficiency of quantum identity testing of multiple states''. Journal of Physics A: Mathematical and Theoretical 41, 395309 (2008).
https://doi.org/10.1088/1751-8113/41/39/395309

[28] Harry Buhrman, Dmitry Grinko, Philip Verduyn Lunel, and Jordi Weggemans. ``Permutation tests for quantum state identity'' (2024). arXiv:2405.09626.
arXiv:2405.09626

[29] Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, and Jordi Tura. ``Generalized concentratable entanglement via parallelized permutation tests''. Physical Review Research 7, L032022 (2025).
https://doi.org/10.1103/jtlj-qs3y

[30] Hermann Weyl. ``Über die Gleichverteilung von Zahlen mod. Eins''. Mathematische Annalen 77, 313–352 (1916).
https://doi.org/10.1007/BF01475864

Cited by

[1] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera, "Symmetric quantum states: a review of recent progress", Reports on Progress in Physics 89 2, 024001 (2026).

[2] J Knörzer, X Liu, B F Schiffer, and J Tura, "Distributed quantum information processing: a review of recent progress", Reports on Progress in Physics 89 7, 074401 (2026).

[3] Benjamin F. Schiffer and Jordi Tura, "Quantum eigenstate preparation assisted by a coherent link", Physical Review A 111 1, 012445 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-19 13:43:26) and SAO/NASA ADS (last updated successfully 2026-08-19 13:43:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.