We introduce an approach to scattering problems in theories with non-Hermitian Hamiltonian, usually known as PT-symmetric quantum theories, by means of the adiabatic switching of the interaction. The modifications of usual methods needed to employ time-dependent metrics are described. We argue that an analogue of the adiabatic theorem hold for time dependent metrics and that its validity forms a necessary condition for consistency of the procedure.
Exceptional points, the spectral degeneracy points in the complex parameter space, are fundamental to non-Hermitian quantum systems. The dynamics of non-Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here we investigate non-adiabatic transitions in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric systems, in which the exceptional points are driven through at finite speed which are quadratic or cubic functions of time. We identity different transmission dynamics separated by exceptional points, and derive analytical approximate formula…
We give a simple way to detect the geometric phase shift and the conditional geometric phase shift with Josephson junction system. Comparing with the previous work(Falcl G, Fazio R, Palma G.M., Siewert J and Verdal V, {\it Nature} {\bf 407}, 355(2000)), our scheme has two advantages. We use the non-adiabatic operation, thus the detection is less affected by the decoherence. Also, we take the time evolution on zero dynamic phase loop, we need not take any extra operation to cancel the dynamic phase.
In this paper, we discuss time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics. we give the time evolving equation for a class of $PT$-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of $PT$-symmetric Hamiltonians.
We propose a technique for robust and efficient navigation in the Hilbert space of entangled symmetric states of a multiparticle system with externally controllable linear and nonlinear collective interactions. A linearly changing external field applied along the quantization axis creates a network of well separated level crossings in the energy diagram of the collective states. One or more transverse pulsed fields applied at the times of specific level crossings induce adiabatic passage between these states. By choosing the timing of the pulsed field appropriately, one can transfer an initial…
We present a quantum algorithm for implementing $ϕ^4$ lattice scalar field theory on qubit computers. The field is represented in the discretized field amplitude basis. The number of qubits and elementary gates required by the implementation of the evolution operator is proportional to the lattice size. The algorithm allows efficient $ϕ^4$ state preparation for a large range of input parameters in both the normal and broken-symmetry phases. The states are prepared using a combination of variational and adiabatic evolution methods. First, the ground state of a local Hamiltonian, which includes …
We analyze neutral atom Rydberg $C_Z$ gates based on adiabatic pulses applied symmetrically to both atoms. Analysis with smooth pulse shapes and Cs atom parameters predicts the gates can create Bell states with fidelity ${\mathcal F}>0.999$ using adiabatic rapid passage (ARP) pulses. With globally optimized adiabatic pulse shapes, in a two-photon excitation process, we generate Bell states with fidelity ${\mathcal F}=0.997$. The analysis fully accounts for spontaneous emission from intermediate and Rydberg states, including the Rydberg lifetime in a room temperature environment, but does not i…
Quantum annealing (QA) is a promising approach for not only solving combinatorial optimization problems but also simulating quantum many-body systems such as those in condensed matter physics. However, non-adiabatic transitions constitute a key challenge in QA. The choice of the drive Hamiltonian is known to affect the performance of QA because of the possible suppression of non-adiabatic transitions. Here, we propose the use of a drive Hamiltonian that preserves the symmetry of the problem Hamiltonian for more efficient QA. Owing to our choice of the drive Hamiltonian, the solution is searche…
Using Fourier transform on a time series generated by unitary evolution, we extract many-body eigenstates of the Bose-Hubbard model corresponding to low energy excitations, which are generated when the insulator-superfluid phase transition is realized in a typical experiment. The analysis is conducted in a symmetric external potential both without and with and disorder. A simple classification of excitations in the absence disorder is provided. The evolution is performed assuming the presence of the parity symmetry in the system rendering many-body quantum states either symmetric or antisymmet…
Exceptional points, at which two or more eigenfunctions of a Hamiltonian coalesce, occur in non-Hermitian systems and lead to surprising physical effects. In particular, the behaviour of a system under parameter variation can differ significantly from the familiar Hermitian case in the presence of exceptional points. Here we analytically derive the probability of a non-adiabatic transition in a two-level system driven through two consecutive exceptional points at finite speed. The system is Hermitian far away from the exceptional points. In the adiabatic limit an equal redistribution between t…
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming-weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semi-classical potential arising from the spin-coherent path integral formalism. W…
We explain why quantum adiabatic evolution and simulated annealing perform similarly in certain examples of searching for the minimum of a cost function of n bits. In these examples each bit is treated symmetrically so the cost function depends only on the Hamming weight of the n bits. We also give two examples, closely related to these, where the similarity breaks down in that the quantum adiabatic algorithm succeeds in polynomial time whereas simulated annealing requires exponential time.
Much research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians with Hamming symmetric potentials, such as the well studied "spike" example. Due to the large amount of symmetry in these potentials such problems are readily open to analysis both analytically and computationally. However, more realistic potentials do not have such a high degree of symmetry and may have many local minima. Here we present a somewhat more realistic class of problems consisting of many individually Hamming symmetric potential wells. For two or three such wells we demonstrate that such a…
$PT$ symmetric quantum mechanics for a particle trapped by the generalized non-Hermitian harmonic oscillator potential is studied. It is shown that energy and the expectation value of the position operator $x$ can not be real simultaneously, if the particle is trapped. Non-vanishing boundary conditions for the trapped particle in $PT$ symmetric theory are also discussed.
We study Landau-Zener interband transitions for a non-symmetric optical lattice in the presence of an external force. We show that gain and losses of the light beam, as well as the relative occupation probabilities of the bands involved in the transitions can be accurately managed upon tuning the amplitude of the non-Hermitian component of the lattice. Exact expressions for the transition and non-transition probabilities for a non-symmetric system obtained within a two-mode approximation are provided. These equations successfully account for the main features of the transitions in the optical …
We explore to what extent path-integral quantum Monte Carlo methods can efficiently simulate the tunneling behavior of quantum adiabatic optimization algorithms. Specifically we look at symmetric cost functions defined over n bits with a single potential barrier that a successful optimization algorithm will have to tunnel through. The height and width of this barrier depend on n, and by tuning these dependencies, we can make the optimization algorithm succeed or fail in polynomial time. In this article we compare the strength of quantum adiabatic tunneling with that of path-integral quantum Mo…
We provide a theoretical study of the quantum adiabatic evolution algorithm with different evolution paths proposed in [E. Farhi, et al., arXiv:quant-ph/0208135]. The algorithm is applied to a random binary optimization problem (a version of the 3-Satisfiability problem) where the n-bit cost function is symmetric with respect to the permutation of individual bits. The evolution paths are produced, using the generic control Hamiltonians H(τ) that preserve the bit symmetry of the underlying optimization problem. In the case where the ground state of H(0) coincides with the totally-symmetric stat…
Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a non-perturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized …
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adia…
Most research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians, whose ground states can be expressed with only real, nonnegative amplitudes. This raises the question of whether classical Monte Carlo algorithms can efficiently simulate quantum adiabatic optimization with stoquastic Hamiltonians. Recent results have given counterexamples in which path integral and diffusion Monte Carlo fail to do so. However, most adiabatic optimization algorithms, such as for solving MAX-k-SAT problems, use k-local Hamiltonians, whereas our previous counterexample for diffusion Mo…
Observation of Restored Adiabatic State Transfer in Time-Modulated Non-Hermitian Systems
Beijing Computational Science Research Center（北京计算科学研究中心）；Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University（帕尔卡大学与捷克科学院物理研究所联合光学实验室，帕尔卡大学理学院）；School of Physics, Southeast University（东南大学物理学院）；Quantum Information Physics Theory Research Team, Quantum Computing Center, RIKEN（理研量子计算中心量子信息物理理论研究组）
We present a general unified approach for the study of quantum thermal machines, including both heat engines and refrigerators, operating under periodic adiabatic driving and in contact with thermal reservoirs kept at different temperatures. We show that many observables characterizing this operating mode and the performance of the machine are of geometric nature. Heat-work conversion mechanisms and dissipation of energy can be described, respectively, by the antisymmetric and symmetric components of a thermal geometric tensor defined in the space of time-dependent parameters generalized to in…
Trotter decomposition is the basis of the digital quantum simulation. Asymmetric and symmetric decompositions are used in our GPU demonstration of the digital adiabatic quantum simulations of $2+1$ dimensional quantum $\mathbb{Z}_2$ lattice gauge theory. The actual errors in Trotter decompositions are investigated as functions of the coupling parameter and the number of Trotter substeps in each step of the variation of coupling parameter. The relative error of energy is shown to be closely related to the Trotter error usually defined defined in terms of the evolution operators. They are much s…
We consider a generalized Dicke model with U(1) symmetry, which can undergo a transition to a superradiant state that spontaneously breaks this symmetry. By exploiting the difference in timescale between atomic and cavity dynamics, one may eliminate the cavity dynamics, providing an atom-only theory. We show that the standard Redfield theory cannot describe the transition to the superradiant state, but including higher-order corrections does recover the transition. Our work reveals how the forms of effective theories must vary for models with continuous symmetry, and provides a template to dev…
We introduce an improved semiclassical dynamics approach to quantum vibrational spectroscopy. In this method, a harmonic-based phase space sampling is preliminarily driven toward non-harmonic quantization by slowly switching on the actual potential. The new coordinates and momenta serve as initial conditions for the semiclassical dynamics calculation, leading to a substantial decrease in the number of chaotic trajectories to deal with. Applications are presented for model and molecular systems of increasing dimensionality characterized by moderate or high chaoticity. They include a bidimension…
We describe recent progress in understanding the continuous symmetry properties of non-Hermitian, PT-symmetric quantum field theories. Focussing on a simple non-Hermitian theory composed of one complex scalar and one complex pseudoscalar, we revisit the derivation of Noether's theorem to show that the conserved currents of non-Hermitian theories correspond to transformations that do not leave the Lagrangian invariant. We illustrate the impact that this has on the consistent formulation of (Abelian) gauge theories by studying a non-Hermitian extension of scalar quantum electrodynamics. We consi…
We use the Higgs mechanism to investigate connections between higher-rank symmetric $U(1)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A…
Some quantum field theories described by non-Hermitian Hamiltonians are investigated. It is shown that for the case of a free fermion field theory with a $γ_5$ mass term the Hamiltonian is $\cal PT$-symmetric. Depending on the mass parameter this symmetry may be either broken or unbroken. When the $\cal PT$ symmetry is unbroken, the spectrum of the quantum field theory is real. For the $\cal PT$-symmetric version of the massive Thirring model in two-dimensional space-time, which is dual to the $\cal PT$-symmetric scalar Sine-Gordon model, an exact construction of the $\cal C$ operator is given…
This paper provides the first ab-initio on-the-fly example of using the Quasi-Diabatic (QD) scheme for non-adiabatic simulations with diabatic dynamics approaches. The QD scheme provides a seamless interface between diabatic quantum dynamics approaches and {\it adiabatic} electronic structure calculations. It completely avoids additional theoretical efforts to reformulate the equation of motion from diabatic to adiabatic representation, or construct global diabatic surfaces. This scheme enables many recently developed diabatic quantum dynamics approaches for ab-inito on-the-fly simulations, pr…
