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<jats:p>Nuclear physics, whose underling theory is described by quantum gauge field coupled with matter, is fundamentally important and yet is formidably challenge for simulation with classical computers. Quantum computing provides a perhaps transformative approach for studying and understanding nuclear physics. With rapid scaling-up of quantum processors as well as advances on quantum algorithms, the digital quantum simulation approach for simulating quantum gauge fields and nuclear physics has gained lots of attention. In this review, we aim to summarize recent efforts on solving nuclear physics with quantum computers. We first discuss a formulation of nuclear physics in the language of quantum computing. In particular, we review how quantum gauge fields (both Abelian and non-Abelian) and their coupling to matter field can be mapped and studied on a quantum computer. We then introduce related quantum algorithms for solving static properties and real-time evolution for quantum systems, and show their applications for a broad range of problems in nuclear physics, including simulation of lattice gauge field, solving nucleon and nuclear structures, quantum advantage for simulating scattering in quantum field theory, non-equilibrium dynamics, and so on. Finally, a short outlook on future work is given.</jats:p>

<jats:p>Quantum simulation has been developed extensively over the past decades, widely applied to different models to explore dynamics in the quantum regime. Rydberg atoms have strong dipole–dipole interactions and interact with each other over a long distance, which makes it straightforward to build many-body interacting quantum systems to simulate specific models. Additionally, neutral atoms are easily manipulated due to their weak interactions. These advantages make Rydberg many-body system an ideal platform to implement quantum simulations. This paper reviews several quantum simulations for different models based on Rydberg many-body systems, including quantum Ising models in one dimension and two dimensions mainly for quantum magnetism, <jats:italic>XY</jats:italic> model for excitation transport, SSH model for symmetry-protected topological phases, and critical self-organized behaviors in many-body systems. Besides, some challenges and promising directions of quantum simulations based on Rydberg many-body system are discussed in this paper.</jats:p>

<jats:p>We present a quantum adiabatic algorithm for a set of quantum 2-satisfiability (Q2SAT) problem, which is a generalization of 2-satisfiability (2SAT) problem. For a Q2SAT problem, we construct the Hamiltonian which is similar to that of a Heisenberg chain. All the solutions of the given Q2SAT problem span the subspace of the degenerate ground states. The Hamiltonian is adiabatically evolved so that the system stays in the degenerate subspace. Our numerical results suggest that the time complexity of our algorithm is <jats:italic>O</jats:italic>(<jats:italic>n</jats:italic> <jats:sup>3.9</jats:sup>) for yielding non-trivial solutions for problems with the number of clauses <jats:italic>m</jats:italic> = <jats:italic>dn</jats:italic>(<jats:italic>n</jats:italic> – 1)/2 (<jats:italic>d</jats:italic> ≲ 0.1). We discuss the advantages of our algorithm over the known quantum and classical algorithms.</jats:p>

<jats:p>To improve the complexity of chaotic signals, in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system, then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders, as well as the Lyaponov exponent spectrum, bifurcation diagram, and SE complexity of the 0.99-order system. In the process of analyzing the system, we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations. Next, we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time, and obtain the basins of attraction under different order conditions. Finally, we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators, thus realizing the 0.9-order multi-scroll hidden chaotic attractors.</jats:p>

<jats:p>We consider the Hamiltonian of <jats:italic>α</jats:italic>, <jats:italic>β</jats:italic>-Fermi Pasta Ulam lattice and explore the Hamilton–Jacobi formalism to obtain the discrete equation of motion. By using the continuum limit approximations and incorporating some normalized parameters, the extended Korteweg–de Vries equation is obtained, with solutions that elucidate on the Fermi Pasta Ulam paradox. We further derive the nonlinear Schrödinger amplitude equation from the extended Korteweg–de Vries equation, by exploring the reductive perturbative technique. The dispersion and nonlinear coefficients of this amplitude equation are functions of the <jats:italic>α</jats:italic> and <jats:italic>β</jats:italic> parameters, with the <jats:italic>β</jats:italic> parameter playing a crucial role in the modulational instability analysis of the system. For <jats:italic>β</jats:italic> greater than or equal to zero, no modulational instability is observed and only dark solitons are identified in the lattice. However for <jats:italic>β</jats:italic> less than zero, bright solitons are traced in the lattice for some large values of the wavenumber. Results of numerical simulations of both the Korteweg–de Vries and nonlinear Schrödinger amplitude equations with periodic boundary conditions clearly show that the bright solitons conserve their amplitude and shape after collisions.</jats:p>

<jats:p>New two-component vector breather solution of the modified Benjamin–Bona–Mahony (MBBM) equation is considered. Using the generalized perturbation reduction method, the MBBM equation is reduced to the coupled nonlinear Schrödinger equations for auxiliary functions. Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented. The two-component vector breather and single-component scalar breather of the MBBM equation is compared.</jats:p>

<jats:p>The dynamic phase transition properties for ferroelectric nanotube under a spin-1/2 transverse Ising model are studied under the effective field theory (EFT) with correlations. The temperature effects on the pseudo-spin systems are unveiled in three-dimensional (3-D) and two-dimensional (2-D) phase diagrams. Moreover, the dynamic behaviors of exchange interactions on the 3-D and 2-D phase transitions under high temperature are exhibited. The results present that it is hard to obtain pure ferroelectric phase under high temperature; that is, the vibration of orderly pseudo-spins cannot be eliminated completely.</jats:p>

<jats:p>Combing with the generalized Hamiltonian system theory, by introducing a special form of sinusoidal function, a class of <jats:italic>n</jats:italic>-dimensional (<jats:italic>n</jats:italic> = 1,2,3) controllable multi-scroll conservative chaos with complicated dynamics is constructed. The dynamics characteristics including bifurcation behavior and coexistence of the system are analyzed in detail, the latter reveals abundant coexisting flows. Furthermore, the proposed system passes the NIST tests and has been implemented physically by FPGA. Compared to the multi-scroll dissipative chaos, the experimental portraits of the proposed system show better ergodicity, which have potential application value in secure communication and image encryption.</jats:p>

<jats:p>We investigate quantum dynamics of a quantum walker on a finite bipartite non-Hermitian lattice, in which the particle can leak out with certain rate whenever it visits one of the two sublattices. Quantum walker initially located on one of the non-leaky sites will finally totally disappear after a length of evolution time and the distribution of decay probability on each unit cell is obtained. In one regime, the resultant distribution shows an expected decreasing behavior as the distance from the initial site increases. However, in the other regime, we find that the resultant distribution of local decay probability is very counterintuitive, in which a relatively high population of decay probability appears on the edge unit cell which is the farthest from the starting point of the quantum walker. We then analyze the energy spectrum of the non-Hermitian lattice with pure loss, and find that the intriguing behavior of the resultant decay probability distribution is intimately related to the existence and specific property of the edge states, which are topologically protected and can be well predicted by the non-Bloch winding number. The exotic dynamics may be observed experimentally with arrays of coupled resonator optical waveguides.</jats:p>

<jats:p>Based on a quadratic transformation and a switching function, a novel multi-wing chaotic switched system is proposed. First, a 4-wing chaotic system is constructed from a 2-wing chaotic system on the basis of a quadratic transformation. Then, a switching function is designed and by adjusting the switching function, the number and the distribution of the saddle-focus equilibrium points of the switched system can be regulated. Thus, a set of chaotic switched systems, which can produce 6-to-8-12-16-wing attractors, are generated. The Lyapunov exponent spectra, bifurcation diagrams, and Poincaré maps are given to verify the existence of the chaotic attractors. Besides, the digital circuit of the multi-wing chaotic switched system is designed by using the Verilog HDL fixed-point algorithm and the state machine control. Finally, the multi-wing chaotic attractors are demonstrated via FPGA platform. The experimental results show that the number of the wings of the chaotic attractors can be expanded more effectively with the combination of the quadratic transformation and the switching function methods.</jats:p>