Document Open Access Logo

Quantum Catalytic Space

Authors Harry Buhrman, Marten Folkertsma , Ian Mertz , Florian Speelman , Sergii Strelchuk , Sathyawageeswar Subramanian , Quinten Tupker

PDF

File

Thumbnail PDF
PDF
LIPIcs.TQC.2025.11.pdf
  • Filesize: 0.82 MB
  • 24 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity classes
Keywords
  • quantum computing
  • quantum complexity
  • space-bounded algorithms
  • catalytic computation
  • one clean qubit

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource.
Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of "dirty" qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. 
We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.

Cite As Get BibTex

Harry Buhrman, Marten Folkertsma, Ian Mertz, Florian Speelman, Sergii Strelchuk, Sathyawageeswar Subramanian, and Quinten Tupker. Quantum Catalytic Space. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 11:1-11:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.TQC.2025.11

Author Details

Harry Buhrman
  • Quantinuum London, UK
  • QuSoft, Amsterdam, The Netherlands
Marten Folkertsma
  • CWI, Amsterdam, The Netherlands
  • QuSoft, Amsterdam, The Netherlands
Ian Mertz
  • Charles University, Prague, Czech Republic
Florian Speelman
  • University of Amsterdam, The Netherlands
  • QuSoft, Amsterdam, The Netherlands
Sergii Strelchuk
  • University of Oxford, UK
Sathyawageeswar Subramanian
  • University of Cambridge, UK
Quinten Tupker
  • CWI, Amsterdam, The Netherlands
  • QuSoft, Amsterdam, The Netherlands

Funding

  • Folkertsma, Marten: Supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL programme.
  • Mertz, Ian: Supported by the Grant Agency of the Czech Republic under the grant agreement no. 24-10306S and by the Center for Foundations of Contemporary Computer Science (Charles Univ. project UNCE 24/SCI/008).
  • Speelman, Florian: Supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL program, and the project Divide and Quantum "D&Q" NWA.1389.20.241 of the program "NWA-ORC", which is partly funded by the Dutch Research Council (NWO).
  • Strelchuk, Sergii: Supported by a Royal Society University Research Fellowship and the EPSRC (RoaRQ), Investigation 005 [grantreference EP/W032635/1].
  • Subramanian, Sathyawageeswar: Supported by a Royal Commission for the Exhibition of 1851 Research Fellowship.
  • Tupker, Quinten: Supported by the Dutch National Growth Fund (NGF), as part of the Quantum Delta NL program.

Acknowledgements

We thank [Dustin G. Mixon, 2023] for pointing us to [Noga Alon, 2003]. Part of this work was done while HB was at CWI, IM was at University of Warwick, SS (1) was at Cambridge University and University of Warwick, and SS (2) was at University of Warwick. SS (2) acknowledges support from the Royal Society University Research Fellowship.

References

  1. Aryan Agarwala and Ian Mertz. Bipartite matching is in catalytic logspace. Electron. Colloquium Comput. Complex., TR25-048, 2025. URL: https://eccc.weizmann.ac.il/report/2025/048/.
  2. Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic computing and register programs beyond log-depth. Electron. Colloquium Comput. Complex., TR25-055, 2025. URL: https://eccc.weizmann.ac.il/report/2025/055/.
  3. Noga Alon. Problems and results in extremal combinatorics—i. Discrete Mathematics, 273(1):31-53, 2003. EuroComb'01. URL: https://doi.org/10.1016/S0012-365X(03)00227-9.
  4. Matthew Amy, Matthew Crawford, Andrew N Glaudell, Melissa L Macasieb, Samuel S Mendelson, and Neil J Ross. Catalytic embeddings of quantum circuits. arXiv preprint arXiv:2305.07720, 2023. Google Scholar
  5. David A. Mix Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC¹. Journal of Computer and System Sciences (J.CSS), 38(1):150-164, 1989. URL: https://doi.org/10.1016/0022-0000(89)90037-8.
  6. Michael Ben-Or and Richard Cleve. Computing algebraic formulas using a constant number of registers. SIAM Journal on Computing (SICOMP), 21(1):54-58, 1992. URL: https://doi.org/10.1137/0221006.
  7. Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. SIAM Journal on Computing, 26(5):1411-1473, 1997. URL: https://doi.org/10.1137/s0097539796300921.
  8. Sagar Bisoyi, Krishnamoorthy Dinesh, Bhabya Rai, and Jayalal Sarma. Almost-catalytic computation. CoRR, abs/2409.07208, 2024. URL: https://doi.org/10.48550/arXiv.2409.07208.
  9. Sagar Bisoyi, Krishnamoorthy Dinesh, and Jayalal Sarma. On pure space vs catalytic space. Theor. Comput. Sci., 921:112-126, 2022. URL: https://doi.org/10.1016/J.TCS.2022.04.005.
  10. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: Catalytic space. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 857-866, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591874.
  11. Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman. Catalytic space: Non-determinism and hierarchy. Theory Comput. Syst., 62(1):116-135, 2018. URL: https://doi.org/10.1007/S00224-017-9784-7.
  12. James Cook, Jiatu Li, Ian Mertz, and Edward Pyne. The structure of catalytic space: Capturing randomness and time via compression. In ACM Symposium on Theory of Computing (STOC), 2025. Google Scholar
  13. James Cook and Ian Mertz. Trading time and space in catalytic branching programs. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 8:1-8:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CCC.2022.8.
  14. James Cook and Ian Mertz. Tree evaluation is in space O(log n ⋅ log log n). In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1268-1278, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649664.
  15. Samir Datta, Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Randomized and symmetric catalytic computation. Electron. Colloquium Comput. Complex., TR20-024, 2020. URL: https://eccc.weizmann.ac.il/report/2020/024.
  16. Christopher M. Dawson and Michael A. Nielsen. The solovay-kitaev algorithm. Quantum Info. Comput., 6(1):81-95, January 2006. Google Scholar
  17. David Deutsch. Quantum theory, the church-turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97-117, 1985. Google Scholar
  18. Yfke Dulek. Catalytic space: on reversibility and multiple-access randomness. 2015. Google Scholar
  19. Bill Fefferman and Zachary Remscrim. Eliminating intermediate measurements in space-bounded quantum computation. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1343-1356, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451051.
  20. Marten Folkertsma, Ian Mertz, Florian Speelman, and Quinten Tupker. Fully characterizing lossy catalytic computation. In Raghu Meka, editor, 16th Innovations in Theoretical Computer Science Conference, ITCS 2025, January 7-10, 2025, Columbia University, New York, NY, USA, volume 325 of LIPIcs, pages 50:1-50:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPICS.ITCS.2025.50.
  21. Uma Girish, Ran Raz, and Wei Zhan. Quantum logspace algorithm for powering matrices with bounded norm. arXiv preprint arXiv:2006.04880, 2020. Google Scholar
  22. Uma Girish, Ran Raz, and Wei Zhan. Quantum logspace computations are verifiable. In 2024 Symposium on Simplicity in Algorithms (SOSA), pages 144-150. SIAM, 2024. Google Scholar
  23. Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Lossy catalytic computation. Computing Research Repository (CoRR), abs/2408.14670, 2024. Google Scholar
  24. Dustin G. Mixon (https://mathoverflow.net/users/29873/dustin-g mixon). How many non-orthogonal vectors fit into a complex vector space? MathOverflow. URL:https://mathoverflow.net/q/458508 (version: 2023-11-16). URL: https://arxiv.org/abs/https://mathoverflow.net/q/458508.
  25. Neil Immerman. Nondeterministic space is closed under complementation. SIAM Journal on computing, 17(5):935-938, 1988. Google Scholar
  26. Richard Jozsa, Barbara Kraus, Akimasa Miyake, and John Watrous. Matchgate and space-bounded quantum computations are equivalent. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2115):809-830, 2010. Google Scholar
  27. A Yu Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6):1191-1249, December 1997. URL: https://doi.org/10.1070/rm1997v052n06abeh002155.
  28. E. Knill and R. Laflamme. Power of one bit of quantum information. Physical Review Letters, 81(25):5672-5675, December 1998. URL: https://doi.org/10.1103/physrevlett.81.5672.
  29. Michal Koucký, Ian Mertz, Ted Pyne, and Sasha Sami. Collapsing catalytic classes. Electronic Colloquium on Computational Complexity (ECCC), TR25-018, 2025. URL: https://eccc.weizmann.ac.il/report/2025/018.
  30. Patryk Lipka-Bartosik, Henrik Wilming, and Nelly HY Ng. Catalysis in quantum information theory. Reviews of Modern Physics, 96(2):025005, 2024. Google Scholar
  31. Ian Mertz. Reusing space: Techniques and open problems. Bulletin of the EATCS (B.EATCS), 141:57-106, 2023. Google Scholar
  32. 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.
  33. Harumichi Nishimura and Masanao Ozawa. Computational complexity of uniform quantum circuit families and quantum turing machines. Theor. Comput. Sci., 276(1–2):147-181, April 2002. URL: https://doi.org/10.1016/S0304-3975(01)00111-6.
  34. Harumichi Nishimura and Masanao Ozawa. Perfect computational equivalence between quantum turing machines and finitely generated uniform quantum circuit families. Quantum Information Processing, 8(1):13-24, January 2009. URL: https://doi.org/10.1007/s11128-008-0091-8.
  35. Tetsuro Nishino. Mathematical models of quantum computation. New Generation Computing, 20(4):317-337, December 2002. URL: https://doi.org/10.1007/bf03037370.
  36. Aaron Potechin. A note on amortized branching program complexity. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 4:1-4:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.CCC.2017.4.
  37. Edward Pyne, Nathan S. Sheffield, and William Wang. Catalytic communication. In Raghu Meka, editor, 16th Innovations in Theoretical Computer Science Conference, ITCS 2025, January 7-10, 2025, Columbia University, New York, NY, USA, volume 325 of LIPIcs, pages 79:1-79:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPICS.ITCS.2025.79.
  38. Walter J Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. Google Scholar
  39. Dan Shepherd. Computation with unitaries and one pure qubit, 2006. URL: https://arxiv.org/abs/quant-ph/0608132.
  40. Peter W. Shor and Stephen P. Jordan. Estimating jones polynomials is a complete problem for one clean qubit, 2008. URL: https://arxiv.org/abs/0707.2831.
  41. Róbert Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta informatica, 26:279-284, 1988. Google Scholar
  42. Amnon Ta-Shma. Inverting well conditioned matrices in quantum logspace. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 881-890, 2013. Google Scholar
  43. John Watrous. Quantum algorithms for solvable groups. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 60-67, 2001. Google Scholar
  44. John Harrison Watrous. Space-bounded quantum computation. The University of Wisconsin-Madison, 1998. Google Scholar
  45. Ryan Williams. Simulating time in square-root space. Electron. Colloquium Comput. Complex., TR25-017, 2025. URL: https://eccc.weizmann.ac.il/report/2025/017/.
  46. Andrew Chi-Chih Yao. Quantum circuit complexity. In 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, USA, 3-5 November 1993, pages 352-361. IEEE Computer Society, 1993. URL: https://doi.org/10.1109/SFCS.1993.366852.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact