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Plethysm is in #BQP

Authors Matthias Christandl , Aram W. Harrow , Greta Panova , Pietro M. Posta , Michael Walter

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum witness counting
  • #BQP
  • algebraic combinatorics
  • representation theory
  • representation-theoretic multiplicities

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Abstract

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #𝖯. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients.
Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. This includes the result that plethysm coefficients are in #BQP, which was only known in certain cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, thus unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform; recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures the approaches of our and previous work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

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Matthias Christandl, Aram W. Harrow, Greta Panova, Pietro M. Posta, and Michael Walter. Plethysm is in #BQP. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 28:1-28:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.28

Author Details

Matthias Christandl
  • University of Copenhagen, Denmark
Aram W. Harrow
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Greta Panova
  • University of Southern California, Los Angeles, USA
Pietro M. Posta
  • University of Copenhagen, Denmark
Michael Walter
  • Ludwig-Maximilians-Universität München, Germany
  • Munich Center for Quantum Science and Technology (MCQST), Germany
  • Korteweg-de Vries Institute for Mathematics, Amsterdam, The Netherlands
  • QuSoft, University of Amsterdam, The Netherlands

Funding

  • Christandl, Matthias: MC acknowledges support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Novo Nordisk Foundation (grant NNF20OC0059939 "Quantum for Life").
  • Harrow, Aram W.: AWH acknowledges support from a grant from the Simons Foundation (MP-SIP-00001553, AWH) and from a Fulbright Scholar Grant.
  • Panova, Greta: GP acknowledges partial support from NSF grant CCF:2302174 and from an AMS Birman fellowship, and a residence at ICERM (NSF grant DMS-1929284) in Fall 2025.
  • Posta, Pietro M.: PMP acknowledges support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059), the Novo Nordisk Foundation (grant NNF20OC0059939 "Quantum for Life") and from a PhD scholarship on Quantum Algorithms from the Danish e-infrastructure Consortium (DeiC), and partial support from Istituto Nazionale di Alta Matematica "Francesco Severi"(INdAM).
  • Walter, Michael: MW acknowledges support from the European Research Council (ERC Grant SYMOPTIC, 101040907) by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, 556164098), from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy– EXC-2111– 390814868, and from the German Federal Ministry of Research, Technology and Space (QuSol, 13N17173). Part of this work was conducted while MW was visiting Q-FARM and the Leinweber Institute for Theoretical Physics at Stanford University and the Simons Institute for the Theory of Computing at UC Berkeley.

Acknowledgements

We are grateful to Christian Ikenmeyer, Vojtech Havlicek and Martin Larocca for fruitful discussions.

References

  1. Matthias Christandl, Aram W. Harrow, Greta Panova, Pietro M. Posta, and Michael Walter. Plethysm is in #BQP, full version, 2026. URL: https://arxiv.org/abs/2602.08441.
  2. M. Altunbulak and A. Klyachko. The Pauli principle revisited. Communications in Mathematical Physics, 282(1):287-322, 2008. URL: https://doi.org/10.1007/s00220-008-0552-z.
  3. Robert Beals. Quantum computation of Fourier transforms over symmetric groups. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC '97), pages 48-53. ACM, 1997. URL: https://doi.org/10.1145/258533.258548.
  4. R. E. Borland and K. Dennis. The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. Journal of Physics B: Atomic and Molecular Physics, 5(1):7-15, 1972. URL: https://doi.org/10.1088/0022-3700/5/1/009.
  5. Sergey Bravyi, Anirban Chowdhury, David Gosset, Vojtěch Havlíček, and Guanyu Zhu. Quantum complexity of the Kronecker coefficients. PRX Quantum, 5(1):010329, 2024. URL: https://doi.org/10.1103/PRXQuantum.5.010329.
  6. Michel Brion. Stable properties of plethysm: on two conjectures of Foulkes. Manuscripta Mathematica, 80:347-371, 1993. URL: https://doi.org/10.1007/BF03026558.
  7. Adam Burchardt, Jiani Fei, Dmitry Grinko, Martin Larocca, Maris Ozols, Sydney Timmerman, and Vladyslav Visnevskyi. High-dimensional quantum Schur transforms, 2025. URL: https://doi.org/10.48550/arXiv.2509.22640.
  8. Peter Bürgisser, Matthias Christandl, and Christian Ikenmeyer. Even partitions in plethysms. Journal of Algebra, 328(1):322-329, 2011. URL: https://doi.org/10.1016/j.jalgebra.2010.10.031.
  9. Peter Bürgisser, Matthias Christandl, Ketan Mulmuley, and Michael Walter. Membership in moment polytopes is in NP and coNP. SIAM Journal on Computing, 46(3):972-991, 2017. URL: https://doi.org/10.1137/15M1048859.
  10. Peter Bürgisser and Christian Ikenmeyer. The complexity of computing Kronecker coefficients. In FPSAC'08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics, 2008. URL: https://doi.org/10.46298/dmtcs.3622.
  11. Peter Bürgisser and Christian Ikenmeyer. Geometric complexity theory and tensor rank. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (STOC '11), pages 509-518. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993704.
  12. Peter Bürgisser and Christian Ikenmeyer. Deciding positivity of Littlewood-Richardson coefficients. SIAM Journal on Discrete Mathematics, 27(4):1639-1681, 2013. URL: https://doi.org/10.1137/120892532.
  13. Peter Bürgisser, Christian Ikenmeyer, and Jesko Hüttenhain. Permanent versus determinant: not via saturations. Proceedings of the American Mathematical Society, 145(3):1247-1258, 2017. URL: https://doi.org/10.1090/proc/13310.
  14. Peter Bürgisser, Christian Ikenmeyer, and Greta Panova. No occurrence obstructions in geometric complexity theory. Journal of the American Mathematical Society, 32(1):163-193, 2019. URL: https://doi.org/10.1090/jams/908.
  15. Matthias Christandl, Brent Doran, and Michael Walter. Computing multiplicities of Lie group representations. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 639-648. IEEE, 2012. URL: https://doi.org/10.1109/FOCS.2012.43.
  16. Matthias Christandl, Aram W. Harrow, and Graeme Mitchison. On nonzero Kronecker coefficients and what they tell us about spectra. Communications in Mathematical Physics, 270(3):575-585, 2007. URL: https://doi.org/10.1007/s00220-006-0157-3.
  17. Matthias Christandl, Aram W. Harrow, and Michael Walter. Presentation at the algorithms and complexity in algebraic geometry reunion workshop. Presented by M. Walter at the Simons Institute for the Theory of Computing, Berkeley, 2015. Google Scholar
  18. Matthias Christandl and Graeme Mitchison. The spectra of quantum states and the Kronecker coefficients of the symmetric group. Communications in Mathematical Physics, 261(3):789-797, 2006. URL: https://doi.org/10.1007/s00220-005-1435-1.
  19. A. J. Coleman and V. I. Yukalov. Reduced Density Matrices: Coulson’s Challenge. Lecture Notes in Chemistry, Vol. 72. Springer Berlin Heidelberg, 2000. URL: https://doi.org/10.1063/1.1359715.
  20. Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki. The mystery of plethysm coefficients. Open Problems in Algebraic Combinatorics (Proc. Sympos. Pure Math.), 110:275-292, 2024. URL: https://doi.org/10.48550/arXiv.2208.07258.
  21. Melanie de Boeck, Rowena Paget, and Mark Wildon. Plethysms of symmetric functions and highest weight representations. Transactions of the American Mathematical Society, 374(11):8013-8043, 2021. URL: https://doi.org/10.1090/tran/8481.
  22. Julian Dörfler, Christian Ikenmeyer, and Greta Panova. On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions. SIAM Journal on Applied Algebra and Geometry, 4(2):354-376, 2020. URL: https://doi.org/10.1137/19M1287638.
  23. Nick Fischer and Christian Ikenmeyer. The computational complexity of plethysm coefficients. Computational Complexity, 29:1-43, 2020. URL: https://doi.org/10.1007/s00037-020-00198-4.
  24. H. O. Foulkes. Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society, 1(3):205-209, 1950. URL: https://doi.org/10.1112/jlms/s1-25.3.205.
  25. Aram W. Harrow. Applications of coherent classical communication and the Schur transform to quantum information theory. PhD thesis, Massachusetts Institute of Technology, 2005. URL: https://arxiv.org/abs/quant-ph/0512255.
  26. Christian Ikenmeyer, Ketan Mulmuley, and Michael Walter. On vanishing of Kronecker coefficients. Computational Complexity, 26:949-992, 2017. URL: https://doi.org/10.1007/s00037-017-0158-y.
  27. Christian Ikenmeyer and Greta Panova. Rectangular Kronecker coefficients and plethysms in geometric complexity theory. Advances in Mathematics, 319:40-66, 2017. URL: https://doi.org/10.1016/j.aim.2017.08.024.
  28. Christian Ikenmeyer and Sathyawageeswar Subramanian. A remark on the quantum complexity of the Kronecker coefficients, 2023. URL: https://arxiv.org/abs/2307.02389.
  29. Thomas Kahle and Mateusz Michałek. Plethysm and lattice point counting. Foundations of Computational Mathematics, 16(5):1241-1261, 2016. URL: https://doi.org/10.1007/s10208-015-9275-7.
  30. Alexander A. Klyachko. Quantum marginal problem and representations of the symmetric group, 2004. URL: https://arxiv.org/abs/quant-ph/0409113.
  31. Alexander A. Klyachko. Quantum marginal problem and N-representability, 2005. URL: https://arxiv.org/abs/quant-ph/0511102.
  32. Allen Knutson and Terence Tao. The honeycomb model of GL_n(ℂ) tensor products I: Proof of the saturation conjecture. Journal of the American Mathematical Society, 12(4):1055-1090, 1999. URL: https://doi.org/10.1090/s0894-0347-99-00299-4.
  33. Martin Larocca and Vojtech Havlicek. Quantum algorithms for representation-theoretic multiplicities, 2024. URL: https://arxiv.org/abs/2407.17649.
  34. D. E. Littlewood. Polynomial concomitants and invariant matrices. Journal of the London Mathematical Society, s1-11(1):49-55, 1936. URL: https://doi.org/10.1112/jlms/s1-11.1.49.
  35. D. E. Littlewood. Products and plethysms of characters with orthogonal, symplectic and symmetric groups. Canadian Journal of Mathematics, 10:17-32, 1958. URL: https://doi.org/10.4153/CJM-1958-002-7.
  36. Dudley Ernest Littlewood and Archibald Read Richardson. Group characters and algebra. Philosophical Transactions of the Royal Society of London. Series A, 233:99-141, 1934. URL: https://doi.org/10.1098/rsta.1934.0015.
  37. Ketan Mulmuley, Hariharan Narayanan, and Milind Sohoni. Geometric complexity theory III: on deciding nonvanishing of a Littlewood-Richardson coefficient. Journal of Algebraic Combinatorics, 36(1):103-110, 2012. URL: https://doi.org/10.1007/s10801-011-0325-1.
  38. Francis D. Murnaghan. On the analysis of the Kronecker product of irreducible representations of the symmetric group. American Journal of Mathematics, 60(3):761-784, 1938. URL: https://doi.org/10.1073/pnas.41.7.515.
  39. Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki. Plethysm and the algebra of uniform block permutations. Algebraic Combinatorics, 5(5):1165-1203, 2022. URL: https://doi.org/10.5802/alco.243.
  40. Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki. From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm. The Electronic Journal of Combinatorics, 31(4):P4.23, 2024. URL: https://doi.org/10.37236/12950.
  41. Igor Pak. What is a combinatorial interpretation? In Open Problems in Algebraic Combinatorics, volume 110 of Proc. Sympos. Pure Math., pages 191-260. Amer. Math. Soc., 2024. URL: https://doi.org/10.1090/pspum/110/02007.
  42. Igor Pak and Greta Panova. On the complexity of computing Kronecker coefficients. Computational Complexity, 26:1-36, 2017. URL: https://doi.org/10.1007/s00037-015-0109-4.
  43. Igor Pak, Greta Panova, and Joshua P. Swanson. A combinatorial interpretation for certain plethysm and Kronecker coefficients, 2025. Unpublished manuscript. Google Scholar
  44. Igor Pak and Colleen Robichaux. Signed combinatorial interpretations in algebraic combinatorics, 2024. URL: https://doi.org/10.48550/arXiv.2406.13902.
  45. Greta Panova. Computational complexity in algebraic combinatorics. In Current Developments in Mathematics. International Press Boston, 2023. URL: https://doi.org/10.4310/CDM.250429173831.
  46. Greta Panova. Complexity and asymptotics of structure constants. In Open Problems in Algebraic Combinatorics, volume 110 of Proc. Sympos. Pure Math., pages 61-86. Amer. Math. Soc., 2024. URL: https://doi.org/10.1090/pspum/110/02016.
  47. Greta Panova. Polynomial time classical versus quantum algorithms for representation theoretic multiplicities, 2025. URL: https://doi.org/10.48550/arXiv.2502.20253.
  48. Richard P. Stanley. Positivity problems and conjectures in algebraic combinatorics. Mathematics: Frontiers and Perspectives, pages 295-319, 2000. URL: https://doi.org/10.5281/zenodo.18768172.
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