Fast parallel circuits for the quantum Fourier transform

  • Cleve R
  • Watrous J
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We give new bounds on the circuit complexity of the quantum
Fourier transform (QFT). We give an upper bound of O(log n+log
log(1/ε)) on the circuit depth for computing an approximation of
the QFT with respect to the modulus 2n with error bounded by
ε. Thus, even for exponentially small error, our circuits have
depth O(log n). The best previous depth bound was O(n), even for
approximations with constant error. Moreover, our circuits have size O(n
log(n/ε)). As an application of this depth bound, we show that P.
Shor's (1997) factoring algorithm may be based on quantum circuits with
depth only O(log n) and polynomial size, in combination with classical
polynomial-time pre- and postprocessing. Next, we prove an Ω(log
n) 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 ε). We
also give an upper bound of O(n(log n)2 log log n) on the
circuit size of the exact QFT modulo 2n, for which the best
previous bound was O(n2). Finally, based on our circuits for
the QFT with power-of-2 moduli, we show that the QFT with respect to an
arbitrary modulus m can be approximated with accuracy ε with
circuits of depth O((log log m)(log log 1/ε)) and size polynomial
in log m+log(1/ε)

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  • R. Cleve

  • J. Watrous

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