Plethysm is in #BQP

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. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, 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 our approach as well as the approaches of prior 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.

💡 Research Summary

The paper “Plethysm is in #BQP” addresses a long‑standing open problem in algebraic combinatorics and computational complexity: determining the complexity class of representation‑theoretic multiplicities such as Kostka numbers, Littlewood‑Richardson (LR) coefficients, Kronecker coefficients, and especially plethysm coefficients. While Kostka and LR coefficients are known to be in #P (they admit positive combinatorial formulas), Kronecker and plethysm coefficients are #P‑hard but it is unknown whether they belong to #P. The authors approach the problem from a quantum‑computational perspective, aiming to place these multiplicities in the quantum counting class #BQP, the quantum analogue of #P.

The central contribution is a quantum algorithm that computes any plethysm coefficient a⁽λ⁾{μ,ν} in #BQP. The input consists of three integer partitions λ, μ, ν encoded in unary (standard in the literature). The algorithm proceeds by exploiting Schur‑Weyl duality and multiple applications of the quantum Schur transform (both forward and inverse). The key idea is to embed the GL(V)‑module {μ}↓{G}H (the restriction of the GL({ν}H)‑module {μ} to H = GL(V)) into a high‑dimensional tensor product space V^{⊗|ν|·|μ|}. This embedding is realized by first mapping {μ} into ( {ν}H )^{⊗|μ|} via an inverse Schur transform, then mapping each {ν}H into V^{⊗|ν|} again via an inverse Schur transform. After these embeddings, a forward Schur transform is applied to all |ν|·|μ| registers, and a measurement in the Schur basis is performed. The algorithm accepts precisely when the measurement yields the partition λ together with a fixed basis state of the Weyl module {λ}. The acceptance probability is proportional to the desired multiplicity, allowing exact sampling or estimation of a⁽λ⁾_{μ,ν}.

The authors rely on recent advances that improve the dependence of the Schur transform on the local dimension d, making the overall circuit depth polynomial in the input size (the total length of the three partitions) and polylogarithmic in d and the number of tensor copies. Under the assumption that the gate set permits exact implementation of the high‑dimensional Schur transform, the algorithm is perfectly sound and complete, yielding a #BQP algorithm for plethysm coefficients. As a corollary, the decision problem “is a⁽λ⁾_{μ,ν} > 0?” lies in QMA.

Beyond plethysm, the paper introduces a general framework called “branching multiplicities” (or branching coefficients). Given groups G and H (products of general linear groups) and a homomorphism φ: H → G, together with a succinct description of a G‑representation π, the multiplicities of irreducible H‑representations in the restriction π ∘ φ are called branching multiplicities. The authors prove that computing any such branching multiplicity is also in #BQP (Theorem 1.3) and that the corresponding positivity problem is in QMA. This result subsumes earlier quantum algorithms for Kronecker coefficients and special plethysm cases, providing a unified perspective.

In addition to the quantum results, the authors show that the same multiplicities belong to GapP (Theorem 1.4). By expressing the acceptance probability of the quantum circuit as a difference of two #P functions, they obtain a classical counting characterization. Moreover, they present a specialized classical algorithm for plethysm coefficients that runs in polynomial time when the number of parts of λ and the size of μ are fixed, extending prior work that handled only very special parameter regimes.

The paper discusses the relevance of these results to Geometric Complexity Theory (GCT) and quantum information theory. In GCT, plethysm coefficients appear as multiplicity obstructions when comparing the orbit closures of the permanent (VNP‑complete) and determinant (VP) polynomials; knowing that these coefficients are in #BQP suggests that quantum algorithms could play a role in proving lower bounds such as VP ≠ VNP. In quantum information, Kronecker and plethysm coefficients govern the compatibility of marginal spectra (the quantum marginal problem) and the N‑representability problem in quantum chemistry; the quantum‑complexity classification thus informs the feasibility of quantum verification procedures for many‑body states.

Overall, the paper achieves three major milestones: (1) a general quantum algorithm placing a broad class of representation‑theoretic multiplicities, including all plethysm coefficients, in #BQP; (2) a classical GapP characterization and polynomial‑time algorithms under fixed‑parameter conditions; (3) a unifying framework that captures and extends all prior quantum complexity results for Kronecker, LR, and related coefficients. These contributions deepen our understanding of the computational landscape of algebraic multiplicities and open new avenues for both quantum algorithm design and complexity‑theoretic investigations in algebraic geometry and quantum physics.