Any matrix-vector query algorithm that estimates tr(W^{-p}) for a d×d Wishart matrix within a constant factor needs Ω(d) queries when p>1/2.
Query lower bounds for log-concave sampling
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Lower bounds for trace estimation via Block Krylov and other methods
Any matrix-vector query algorithm that estimates tr(W^{-p}) for a d×d Wishart matrix within a constant factor needs Ω(d) queries when p>1/2.