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Sparse inverse problems in the presence of Poisson noise with physicalÌýconstraints arise in a variety of applications, includingÌýphoton-limited imaging systems based on compressed sensing. TheÌýperformance of compressed sensing in these settings can be markedlyÌýdifferent from classical settings. Prior results on Poisson compressedÌýsensing provided upper bounds on mean squared error performance;Ìýhowever, it was unknown whether those bounds were tight or if otherÌýestimators could achieve significantly better performance. This workÌýprovides minimax lower bounds on mean-squared error for sparse PoissonÌýinverse problems under physical constraints. The lower bounds areÌýcomplemented by minimax upper bounds which match the lower bounds forÌýcertain problem sizes and noise levels. The upper and lower boundsÌýreveal several distinctions from the classical compressed sensingÌýsetup due to the interplay between the Poisson noise model, theÌýsparsity of the signal, and the physical constraints. For example,Ìýerror decay rates depend heavily upon the sparsifying basis of theÌýsignal. In addition, in many application-relevant scenarios signalÌýacquisition via simple downsampling can significantly outperformÌýcompressed sensing. This is joint work with Xin Jiang and GarveshÌýRaskutti.