Heisenberg models and Schur–Weyl duality

Björnberg, Jakob E.; Rosengren, Hjalmar; Ryan, Kevin

doi:10.1016/j.aam.2023.102572

Advances in Applied Mathematics

2023

DOI: 10.1016/j.aam.2023.102572

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Heisenberg models and Schur–Weyl duality

Jakob E. Björnberg

Hjalmar Rosengren

Kevin Ryan

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Linear Programming with Unitary-Equivariant Constraints

Grinko,

Ozols

2024

Commun. Math. Phys.

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Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $$d^{p+q}$$ d p + q -dimensional matrix variable that commutes with $$U^{\otimes p} \otimes {\bar{U}}^{\otimes q}$$ U ⊗ p ⊗ U ¯ ⊗ q , for all $$U \in \textrm{U}(d)$$ U ∈ U ( d ) . Solving such problems naively can be prohibitively expensive even if $$p+q$$ p + q is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

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Linear Programming with Unitary-Equivariant Constraints

Grinko,

Ozols

2024

Commun. Math. Phys.

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The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems

Manai,

Warzel

2023

Forum of Mathematics, Sigma

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A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.

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Heisenberg models and Schur–Weyl duality

Cited by 2 publications

References 24 publications

Linear Programming with Unitary-Equivariant Constraints

Linear Programming with Unitary-Equivariant Constraints

The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems

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