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 nonunitary quantum gate $G$. We show that the NQC is extraordinarily powerful, capable of solving not only all NP problems but also all problems within the complexity class ${\text{P}}^{♯\text{P}}$ in polynomial time. We investigate two physical schemes for implementing the nonunitary gate $G$ and find that the remarkable computational power of NQC originates from the exponentially large physical resources required.

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• Open quantum systems & decoherence
• Quantum algorithms & computation
• Quantum circuits
• Quantum computation