In parity-time-symmetric ( -symmetric) Hamiltonian theory, the optimal evolution time can be reduced drastically and can even be zero. In this article, we report our experimental simulation of the fast evolution of a -symmetric Hamiltonian in a nuclear magnetic resonance quantum system. The experimental results demonstrate that the -symmetric Hamiltonian system can indeed evolve much faster than the quantum system, and the evolution time can be arbitrarily close to zero.

The recently theoretical and experimental researches related to PT -symmetric system have attracted unprecedented attention because of various novel features and potentials in extending canonical quantum mechanics. However, as the counterpart of PT -symmetry, there are only a few researches on anti-PT -symmetry. Here, we propose an algorithm for simulating the universal anti-PT -symmetric system with quantum circuit. Utilizing the protocols, an oscillation of information flow is observed for the first time in our Nuclear Magnetic Resonance quantum simulator. We will show that information will recover from the environment completely when the anti-PT -symmetry is broken, whereas no information can be retrieved in the symmetry-unbroken phase. Our work opens the gate for practical quantum simulation and experimental investigation of universal anti-PT -symmetric system in quantum computer. Traditional quantum mechanics requires HermitianHamiltonians to describe closed physical systems, while the dynamic evolution of open systems is typically described by non-Hermitian Hamiltonians [1,2]. The nonhermitian Hamiltonian of open systems has attracted extensive attention and research because of the discovery by Bender and Boettcher in 1998 [3]. It was found that Hamiltonians satisfying parity P (spatial reflection) and T (time reversal) symmetry instead of hermiticity can still have real energy spectra and orthogonal eigenstates in the symmetry-unbroken phase, in which the eigenfunction of system Hamiltonian is at the same time an eigenfunction of the PT operator [4,5]. When the Hamiltonian parameters cross the exceptional point, PT -symmetry will be broken and lead to a symmetrybreaking transition [6][7][8]. This work has inspired numerous theoretical and experimental studies [9-15] of the non-hermitian systems, including demonstrating novel properties of open systems [16,17] and extending fundamental quantum mechanics [20,21]. However, there are limited investigations on another important counterpart anti-PT -symmetry, which means the system Hamiltonian is anti-commutative with the joint PT operator {H, PT } = 0. Some relevant experimental demonstrations have been realized in atoms [29][30][31], optical [32][33][34][35][36][37], electrical circuit resonators [38] and diffusive systems [39]. Quantum processes such as symmetry breaking transition, observation of exceptional point and simulation of anti-PT -symmetric Lorentz dynamics have been presented in these experiments [29,33,37,38], whereas the novel characteristics of entanglement [17][18][19] and information flow [22][23][24] in the anti-PT -symmetric system, which would present various phenomena different from Hermitian quantum mechanics and reveal the relationship between system and environment, have not been fully thorough investigated in the experiment.In this work, we propose an algorithm for the simulation of universal anti-PT -symmetric evolution with quantum circuit model and report the first experimental observation of information flow oscillation in anti-PT...

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The quantum Fourier transform and quantum phase estimation are the key components for many quantum algorithms, such as order-finding, factoring, and etc. In this article, the general procedure of quantum Fourier transform and phase estimation are investigated for high dimensional case. They can be seen as subroutines in a main program run in a qudit quantum computer, and the quantum circuits are given.