Physics and computation: An insight from non-Hermitian quantum computing

We elucidate the profound connection between physics and computation by proposing and examining the model of the non-Hermitian quantum computer (NQC). In addition to conventional quantum gates such as the Hadamard, phase, and CNOT gates, this model incorporates a non-unitary quantum gate $G$. We show that NQC is extraordinarily powerful, capable of solving not only all NP problems but also all problems within the complexity class $\text{P}^{\sharp\text{P}}$ in polynomial time. We investigate two physical schemes for implementing the non-unitary gate $G$ and find that the remarkable computational power of NQC originates from the exponentially large physical resources required.