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Volume 1 (2005) Article 5 pp. 81-103
Quantum Fan-out is Powerful
Received: October 26, 2004
Published: August 3, 2005
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Keywords: quantum computing, quantum circuits, fan-out, quantum Fourier transform, constant depth circuits, threshold circuits
ACM Classification: F.2.1, F.2.2
AMS Classification: 68Q15, 81P68

Abstract: [Plain Text Version]

We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by $\mathrm{QNC}_{\mathrm{f}}^0$) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth.