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A non-Hermitian quantum optimization algorithm is created and used to findthe ground state of an antiferromagnetic Ising chain. We demonstrateanalytically and numerically (for up to N=1024 spins) that our approach leadsto a significant reduction of the annealing time that is proportional to $\lnN$, which is much less than the time (proportional to $N^2$) required for thequantum annealing based on the corresponding Hermitian algorithm. We propose touse this approach to achieve similar speed-up for NP-complete problems by usingclassical computers in combination with quantum algorithms.

Non-Hermitian Quantum Annealing in the Antiferromagnetic Ising Chain