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 2 n 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)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s 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 post-processing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with bounded-error probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the

The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then2 extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on ℜ. Finally, we show that this problem of period-finding over ℜ, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, periodfinding

The hardness to solve an unstructured quantum search problem by a standard quantum search algorithm mainly originates from the low efficiency to amplify the amplitude of the marked state in the Hilbert space of an n−qubit pure-state quantum system by the oracle unitary operation associated with other known quantum operations, which usually is inversely proportional to the square root of dimension of the Hilbert space. In order to bypass this square speedup limitation it is necessary to develop other type of quantum search algorithms. It is described in detail in the paper for a quantum dynamical method to solve the quantum search problem in the cyclic group state space. The binary dynamical representation for a quantum state in the Hilbert space of the n−qubit quantum system is generalized to the multi-base dynamical representation for a quantum state in the cyclic group state space. Thus, any quantum state in the cyclic group state space may be described completely in terms of a set of dynamical parameters that are closely related

Quantum search processes in the cyclic group state spaces

Quantum search problems in the cyclic group state spaces