Quantum Complexity of the Kronecker Coefficients

Journal Article · 20 February 2024 · PRX Quantum

DOI: https://doi.org/10.1103/PRXQuantum.5.010329 · OSTI ID:2310410

Bravyi, Sergey; Chowdhury, Anirban ; Gosset, David; Havlíček, Vojtěch; Zhu, Guanyu

Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector that can be measured efficiently using a quantum computer. In other words a Kronecker coefficient counts the dimension of the vector space spanned by the accepting witnesses of a $\mathrm{QMA}$ verifier, where $\mathrm{QMA}$ is the quantum analogue of $\mathrm{NP}$ . This implies that approximating the Kronecker coefficients to within a given relative error is not harder than a certain natural class of problems that captures the complexity of estimating thermal properties of quantum many-body systems. A second consequence is that deciding positivity of Kronecker coefficients is contained in $\mathrm{QMA}$ , complementing a recent $\mathrm{NP}$ -hardness result of Ikenmeyer, Mulmuley, and Walter. We obtain similar results for the related problem of approximating row sums of the character table of the symmetric group. Finally, we discuss an efficient quantum algorithm that approximates normalized Kronecker coefficients to inverse-polynomial additive error. Published by the American Physical Society 2024

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0012704
• OSTI ID: 2310410
• Journal Information: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 1 Vol. 5; ISSN 2691-3399; ISSN PQRUAG
• Publisher: American Physical SocietyCopyright Statement
• Country of Publication: United States
• Language: English