  We investigate the advantage of coherent superposition of two different coded
channels in quantum metrology. In a continuous variable system, we show that
the Heisenberg limit $1/N$ can be beaten by the coherent superposition without
the help of indefinite causal order. And in parameter estimation, we
demonstrate that the strategy with the coherent superposition can perform
better than the strategy with quantum \textsc{switch} which can generate
indefinite causal order. We analytically obtain the general form of estimation
precision in terms of the quantum Fisher information and further prove that the
nonlinear Hamiltonian can improve the estimation precision and make the
measurement uncertainty scale as $1/N^m$ for $m\geq2$. Our results can help to
construct a high-precision measurement equipment, which can be applied to the
detection of coupling strength and the test of time dilation and the
modification of the canonical commutation relation.

Quantum Physics