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    $\begingroup$ To make the before/after comparison more effective you can include timing information from AbsoluteTiming. Also, include the output of $Version to your post. What type of entries does emat have? Please include emat and UV. In short, please present a minimal example and include definitions. $\endgroup$ Commented Mar 31, 2023 at 6:11
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    $\begingroup$ @Syed Thanks for pointing out those. I edited my post. $\endgroup$ Commented Mar 31, 2023 at 6:38
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    $\begingroup$ You are recomputing IntegrateUnitaryHaar[UV.emat[[j]].ConjugateTranspose[UV],{u,15},{v,1}] for every value of i instead of memoizing it. $\endgroup$ Commented Mar 31, 2023 at 8:07
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    $\begingroup$ Use sparse Pauli matrices for speed: Clear[σ, emat]; σ[i_] := σ[i] = SparseArray[PauliMatrix[i]]; emat[i_Integer, j_Integer, k_Integer, l_Integer] := emat[i, j, k, l] = KroneckerProduct[σ[i], σ[j], σ[k], σ[l]] will give you memoized, sparse, quick access to the emat values without forcing a pre-compute. $\endgroup$ Commented Mar 31, 2023 at 8:08
  • $\begingroup$ Can you please edit your post so that the code can be copy-pasted into a notebook and will actually run? Please don't expect people to piece it together by themselves. $\endgroup$ Commented Mar 31, 2023 at 8:19