Partitioned Quantum Subspace Expansion

Tom O'Leary; Lewis W. Anderson; Dieter Jaksch; Martin Kiffner

doi:10.22331/q-2025-05-05-1726

Partitioned Quantum Subspace Expansion

Tom O'Leary^1,2, Lewis W. Anderson¹, Dieter Jaksch^1,3, and Martin Kiffner^1,4

¹Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
²Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
³Institut für Quantenphysik, Universität Hamburg, Hamburg, Germany
⁴PlanQC GmbH, Lichtenbergstr. 8, 85748 Garching, Germany

Published:	2025-05-05, volume 9, page 1726
Editor:	Álvaro Alhambra
Eprint:	arXiv:2403.08868v3
Doi:	https://doi.org/10.22331/q-2025-05-05-1726
Citation:	Quantum 9, 1726 (2025).

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Abstract

We present an iterative generalisation of the quantum subspace expansion algorithm used with a Krylov basis. The iterative construction connects a sequence of subspaces via their lowest energy states. Diagonalising a Hamiltonian in a given Krylov subspace requires the same quantum resources in both the single step and sequential cases. We propose a variance-based criterion for determining a good iterative sequence and provide numerical evidence that these good sequences display improved numerical stability over a single step in the presence of finite sampling noise. Implementing the generalisation requires additional classical processing with a polynomial overhead in the subspace dimension. By exchanging quantum circuit depth for additional measurements the quantum subspace expansion algorithm appears to be an approach suited to near term or early error-corrected quantum hardware. Our work suggests that the numerical instability limiting the accuracy of this approach can be substantially alleviated in a parameter-free way.

Featured image: Illustration of the Partitioned Quantum Subspace Expansion algorithm for 2 partitions.

Popular summary

Quantum subspace methods are a class of algorithms for learning properties of low-energy quantum states using a quantum computer. These methods are promising approaches to implement on current and early-fault-tolerant quantum hardware. A subspace method of particular importance is the quantum subspace expansion algorithm with a Krylov basis (QSE), as this approach has known convergence properties and builds on the broad success of classical-Krylov methods.

An obstacle to the implementation of QSE is a sensitivity to statistical noise in data extracted from a quantum computer. Existing proposals for overcoming this obstacle have been successfully deployed in experiment and come with rigorous theoretical guarantees, however, they rely on being correctly parameterized. We show that the challenge of finding a good parameter can be avoided by proposing a generalisation of QSE, called the Partitioned Quantum Subspace Expansion algorithm (PQSE). In PQSE, a single Krylov basis is broken down and recombined in a way that is designed to ensure stability to statistical noise. Our work provides a new technique for handling noise in quantum subspace methods, which can immediately be applied to experiments on quantum hardware and lays a foundation for future optimizations as the field progresses.

► BibTeX data

@article{OLeary2025partitionedquantum,
  doi = {10.22331/q-2025-05-05-1726},
  url = {https://doi.org/10.22331/q-2025-05-05-1726},
  title = {Partitioned {Q}uantum {S}ubspace {E}xpansion},
  author = {O'Leary, Tom and Anderson, Lewis W. and Jaksch, Dieter and Kiffner, Martin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1726},
  month = may,
  year = {2025}
}

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[5] Lewis W. Anderson, Martin Kiffner, Tom O'Leary, Jason Crain, and Dieter Jaksch, "Solving lattice gauge theories using the quantum Krylov algorithm and qubitization", arXiv:2403.08859, (2024).

[6] Bo Yang, Nobuyuki Yoshioka, Hiroyuki Harada, Shigeo Hakkaku, Yuuki Tokunaga, Hideaki Hakoshima, Kaoru Yamamoto, and Suguru Endo, "Resource-efficient generalized quantum subspace expansion", Physical Review Applied 23 5, 054021 (2025).

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[8] Cedric Gaberle and Manpreet Singh Jattana, "A Critical Assessment of the Sample-Based Quantum Diagonalization for Heisenberg and Hubbard Models", arXiv:2605.02494, (2026).

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