Physics and Computation: A Perspective From Non-Hermitian Quantum Computer

We elucidate the profound connection between physics and computation by proposing and examining the model of non-Hermitian quantum computer (NQC). In addition to conventional quantum gates such as the Hadamard gate, phase gate, and CNOT gate, this computing model incorporates a non-unitary quantum gate $G$.We show that NQC is extraordinarily powerful, not only capable of solving 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 amount of physical resources required in these implementations.