  Non-Hermitian topological phases exhibit a number of exotic features that
have no Hermitian counterparts, including the skin effect and breakdown of the
conventional bulk-boundary correspondence. Here, we implement the non-Hermitian
Su-Schrieffer-Heeger (SSH) Hamiltonian, which is a prototypical model for
studying non-Hermitian topological phases, with a solid-state quantum simulator
consisting of an electron spin and a $^{13}$C nuclear spin in a
nitrogen-vacancy (NV) center in a diamond. By employing a dilation method, we
realize the desired non-unitary dynamics for the electron spin and map out its
spin texture in the momentum space, from which the corresponding topological
invariant can be obtained directly. Our result paves the way for further
exploiting and understanding the intriguing properties of non-Hermitian
topological phases with solid-state spins or other quantum simulation
platforms.
