Fast parallel circuits for the quantum Fourier transform

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of Ç´ÐÓ Ò · ÐÓ ÐÓ ´½ µµ on the circuit depth for computing an approximation of the QFT with respect to the modulus ¾Ò with error bounded by . Thus, even for exponentially small error, our circuits have depth Ç´ÐÓ Òµ. The best previous depth bound was Ç´Òµ, even for approximations with constant error. Moreover, our circuits have size Ç´ÒÐÓ ´Ò µµ. As an application of this depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only Ç´ÐÓ Òµ and polynomial size, in combination with classical polynomial-time pre- and postprocessing. Next, we prove an ª´ÐÓ Òµ lower bound on the depth complexity of approximations of the QFT with constant error. This implies that the above upper bound is asymptotically tight (for a reasonable range of values of ). We also give an upper bound of Ç´Ò´ÐÓ Òµ¾...

• Type: Conference
• Year: 2000
• Where: FOCS
• Authors: Richard Cleve, John Watrous
• Keywords: Constant Error, Depth Bound, FOCS 2000, Theoretical Computer Science, Upper Bound