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Abstract:

Abstract. We exhibit a quantum circuit that performs the Quantum Fourier Transform on n qubits in O(n) depth. Thus, a parallel quantum computer can carry out the QFT in linear time. Griffiths and Niu have already shown this, so this paper is little more than an exercise in quantum circuit design; but perhaps it illustrates a worthwhile idea. We also speculate as to whether the QFT might be in the class QNC 1 of problems solvable in logarithmic parallel time. Shor's factoring algorithm [6] suggests that quantum computers can do things in polynomial time that classical computers cannot. However, since decoherence due to storage errors is a function of time, we should also ask to what extent we can parallelize quantum algorithms; if we can do many quantum operations at once, rather than serially, we can solve larger problems before our computer decoheres. Consider a quantum circuit operating on a set of qubits, containing onequbit gates (2 \Theta 2 unitary matrices) and the two-qubit controlled-not gate; these are universal for quantum computation [1, 4]. We can define the depth of this circuit as the number of layers, where each layer consists of gates operating on mutually disjoint sets of qubits; that is, each qubit interacts with at most one other qubit at a time. (In a model of quantum computation where one qubit can simultaneously interact with several others, we could allow gates operating on the same qubit in the same level, as long as these gates all mutually commute.) The heart of Shor's algorithm is the Quantum Fourier Transform. If we represent n-digit numbers jai with n qubits, the QFT maps jai to 2

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