Abstract
Let Xm,ε be the distribution over m bits X1,…,Xm where the Xi are independent and each Xi equals 1 with probability (1−ε)/2 and 0 with probability (1 − ε)/2. We consider the smallest value ε* of ε such that the distributions Xm, ε and Xm, 0 can be distinguished with constant advantage by a function f : {0,1}m → S, which is the product of k functions f1,f2,…, fk on disjoint inputs of n bits, where each fi : {0,1}n → S and m = nk.
We prove that ε* = Θ(1/√n log k) if S = [−1,1], while ε* = Θ(1/√nk) if S is the set of unit-norm complex numbers.
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Index Terms
The Coin Problem for Product Tests
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