Randomized decision tree complexity of Deutsch–Jozsa problem and a generalization

Deutsch–Jozsa problem ( DJ_n ) showed for the first time that quantum computation can achieve exponential advantages over classical computers, which encouraged and laid the foundation for further research on quantum algorithms. A generalization of Deutsch–Jozsa problem ( DJ^k_n , proposed by Phys Rev A 97:062331, 2018) maintains the exponential advantage when the parameter k is a constant. However, these achieved exponential advantages are just in the case that all outputs are required to be accurate. In contrast, this paper studies classical randomized decision tree complexities of DJ_n and DJ^k_n . It is proved that the first complexity R_2(DJ_n)≤ 3 in all cases and the second complexity R_2(DJ^k_n) is constant when k/n is constant for n being a large even number. As a result, for these cases, optimal bounded-error quantum algorithms of DJ_n and DJ^k_n can only slightly accelerate the classical computation.