Non-Hermitian quantum annealing in the ferromagnetic Ising model

Abstract

We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to ln N , where N is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NPcomplete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

Non-Hermitian Quantum Annealing and Superradiance

“…Since for the non-Hermitian systems the norm of the wave function is not conserved, we define the partial survival probability as [16,17],…”

“…Recently, we have applied the NQA algorithm to Grover's problem of finding a marked item in an unsorted database, and to study the transition to the ground state in a 1-dimensional ferromagnetic (and anti-ferromagnetic) Ising chain [15,16,17]. Analytical and numerical results demonstrate that, even for a moderate value of the decay parameter, the NQA has a complexity of order ln N, where N is the number of qubits.…”

Non-Hermitian quantum annealing in the antiferromagnetic Ising chain

“…The Hamiltonian in Eq. (1) can be diagonalized using the standard Jordan-Wigner transformation, following procedures outlined in [14,15,16,17,18]. The Jordan-Wigner transformation, which is mapping of a spin-1/2 system to a system of spinless fermions, is given by…”

“…Using the asymptotic formulas for Weber functions with the large argument values, and taking into account that δ ≪ g, we obtain, as in [14],…”

“…In [13,14], we applied this non-Hermitian quantum annealing (NQA) algorithm to Grover's problem of finding a marked item in an unsorted database, and to study the transition to the ground state in a 1-dimensional ferromagnetic Ising chain. We demonstrated that, for a suitable choice of relaxation and other parameters, the search time is proportional to the logarithm of the number of qubits (spins).…”