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You have plenty of redundant operations here:

The equivalent code is:

Abs[MatrixA . VectorB]^2

I have:

a = RandomReal[1, {1500, 1500}];
b = RandomReal[1, 1500];

Table[(Norm[Sum[a[[i, j]] b[[j]], {j, 1500}]])^2, {i, 
    1500}]; // AbsoluteTiming
(* {2.90726, Null} *)

Abs[a.b]^2; // AbsoluteTiming
(* {0.000943871, Null} *)

Thus, using Dot almost 3000 times faster than manual loops.

You have plenty of redundant operations here:

The equivalent code is (where a is a matrix and b is a list of 3-vectors):

Map[Norm[#]^2 &, a . b];

I have:

n = 500;
a = RandomReal[1, {n, n}];
b = RandomReal[1, {n, 3}];

oneDimArray = 
   Table[(Norm[Sum[a[[i, j]] b[[j]], {j, n}]])^2, {i, 
     n}]; // AbsoluteTiming
(* {46.5618, Null} *)

oneDimArrayMap = Map[Norm[#]^2 &, a.b]; // AbsoluteTiming
(* {0.000621823, Null} *)

oneDimArray == oneDimArrayMap
(* True *)

Thus, using Dot almost 70000 times faster than manual loops.

You have plenty of redundant operations here:

The equivalent code is:

Abs[MatrixA . VectorB]^2

You have plenty of redundant operations here:

The equivalent code is:

Abs[MatrixA . VectorB]^2

I have:

a = RandomReal[1, {1500, 1500}];
b = RandomReal[1, 1500];

Table[(Norm[Sum[a[[i, j]] b[[j]], {j, 1500}]])^2, {i, 
    1500}]; // AbsoluteTiming
(* {2.90726, Null} *)

Abs[a.b]^2; // AbsoluteTiming
(* {0.000943871, Null} *)

Thus, using Dot almost 3000 times faster than manual loops.

You have plenty of redundant operations here:

The equivalent code is (taking in account that Norm squared is just Abs by default):

Abs[MatrixA . VectorB]^2

You have plenty of redundant operations here:

The equivalent code is:

Abs[MatrixA . VectorB]^2