Published: August 3, 2005

**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.