Non-Hermitian Computers Need No Complex Numbers

Abstract

In traditional quantum computing, it has been established that real quantum computation augmented with non-Clifford gates is as powerful as universal quantum computation. Here we investigate this phenomenon in the non-Hermitian setting. We show that a non-Hermitian quantum computer equipped with the real gate set $H,\text{CCNOT},G$, where $G=\mathrm{diag}(g^{-1},g)$ with $g>0$ and $g\ne 1$, can solve problems in ${\text{P}}^{♯\text{P}}$ in polynomial time, matching the capability of its universal non-Hermitian counterpart $H,T,\text{CNOT},G$. This demonstrates that non-unitarity, rather than universality, is the essential resource, and that complex numbers are unnecessary.

10 pages, 4 figures