Jeffrey Yepez
An efficient quantum algorithm for the many-body three-dimensional Dirac equation is presented. Its computational complexity is dominantly linear in the number of qubits used to spatially resolve the 4-spinor wave function.
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We analyse a system in which, due to entanglement between the spin and spatial degrees of freedom, the reduced transmitted state has the shape of the freely propagating pulse translated in the complex co-ordinate plane. In the case an apparently "superluminal" advancement of the pulse the delay amplitude distribution is found to be a peculiar approximation to the Dirac delta-function, and the transmission coefficient exhibits a well-defined super-oscillatory window. Analogies with potential tunnelling and the Wheeler's delayed choice experiment are highlighted.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, …
We present the exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.
A de Broglie-Bohm like model of Dirac equation, that leads to the correct Pauli equations for electrons and positrons in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm theory--that the local momentum of particles is given by the gradient of the phase of the wave function--wont be accurate. Also, the number of particles wont be locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
A non-relativistic system such as an ultracold trapped ion may perform a quantum simulation of a Dirac equation dynamics under specific conditions. The resulting Hamiltonian and dynamics are highly controllable, but the coupling between momentum and internal levels poses some difficulties to manipulate the internal states accurately in wave packets. We use invariants of motion to inverse engineer robust population inversion processes with a homogeneous, time-dependent simulated electric field. This exemplifies the usefulness of inverse-engineering techniques to improve the performance of quant…
We present an exact quantization condition for the time independent solutions (energy eigenstates) of the one-dimensional Dirac equation with a scalar potential well that gives only two `effective' turning points (defined by the roots of $V(x)+mc^2=\pm E$) for a given energy $E$ and satisfies $\min V(x)+mc^2\geq 0$. This result generalizes the previously known non-relativistic quantization formula and preserves many physically desirable symmetries, besides, attaining the correct non-relativistic limit. Numerical calculations demonstrate the utility of the formula for computing accurate energy …
The quantum walk formalism is a widely used and highly successful framework for modeling quantum systems, such as simulations of the Dirac equation, different dynamics in both the low and high energy regime, and for developing a wide range of quantum algorithms. Here we present the circuit-based implementation of a discrete-time quantum walk in position space on a five-qubit trapped-ion quantum processor. We encode the space of walker positions in particular multi-qubit states and program the system to operate with different quantum walk parameters, experimentally realizing a Dirac cellular au…
Qubit Lattice Algorithms based on the Schrodinger-Dirac representation of Maxwell Equations and their Extensions
针对非均匀介质下麦克斯韦方程薛定谔-狄拉克表示的难题，研究补充非幺正势算符改进量子比特格点算法，验证了折射率导数陡度对二维散射的影响，并将算法扩展到含耗散的开放量子系统，给出了量子实现的门数估计。
The bound state (energy spectrum and two-spinor wave functions) solutions of the Dirac equation with the Hulthen potential for all angular momenta based on the spin and pseudospin symmetry are obtained. The parametric generalization of the Nikiforov-Uvarov method is used in the calculations. The orbital dependency (spin-orbit and pseudospin-orbit dependent coupling too singular 1/r^{2}) of the Dirac equation are included to the solution by introducing a more accurate approximation scheme to deal with the centrifugal (pseudo-centrifugal) term. The approximation is also made for the less singula…
Presented is a quantum computing representation of Dirac particle dynamics. The approach employs an operator splitting method that is an analytically closed-form product decomposition of the unitary evolution operator. This allows the Dirac equation to be cast as a unitary finite-difference equation in a high-energy limit. The split evolution operator (with separate kinetic and interaction terms) is useful for efficient quantum simulation. For pedagogical purposes, here we restrict the treatment to Dirac particle dynamics in 1+1 spacetime dimensions. Independent derivations of the quantum algo…
We consider the problem of searching a d-dimensional lattice of N sites for a single marked location. We present a Hamiltonian that solves this problem in time of order sqrt(N) for d>2 and of order sqrt(N) log(N) in the critical dimension d=2. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of d=4), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom.
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent par…
Digital quantum simulator for the time-dependent Dirac equation using discrete-time quantum walks
本文提出利用离散时间量子行走在3+1维中模拟含时狄拉克方程的量子算法，通过量子谱方法实现指数收敛并降低门复杂度，证明相对论动力学可在量子计算机上高效模拟。
An algorithm for exact analytical solutions for tilted anisotropic Dirac materials
本文提出了一种算法，通过解耦旋量分量微分方程，求解倾斜各向异性 Dirac 材料在外加电磁场下有效 Hamiltonian 的本征值方程，从而获得其束缚态的精确解析解。
We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form $H_{g}=H+gδ(x)$, where $δ(x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction.
We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results.
Presented is a quantum lattice gas algorithm to efficiently model a system of Dirac particles interacting through an intermediary gauge field. The algorithm uses a fixed qubit array to represent both the spacetime and the particles contained in the spacetime. Despite being a lattice based algorithm, Lorentz invariance is preserved down to the grid scale, with the continuum Dirac Hamiltonian generating the local unitary evolution even at that scale: there is nonlinear scaling between the smallest observable time and that time measured in the quantum field theory limit, a kind of time dilation e…
Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm
提出分裂变分量子算法Dirac-sVQA，通过线性狄拉克子步和非线性变分修正分解非线性狄拉克方程，利用自旋-傅里叶传播器和测量基变分更新实现数值模拟。
In this paper, we consider the Dirac-Hartree-Fock equations for system has many-particles. The difficulties associated with Gaussians model are likely to be more complex in relativistic Dirac-Hartree-Fock calculations. To processing these problem, we use accurate techniques. The four-component spinors will be expanded into a finite basis-set, using Gaussian basis-set type dyall.2zp to describe 4-component wave functions, in order to describe the upper and lower two components of the 4-spinors, respectively. The small component Gaussian basis functions have been generated from large component G…
High-precision minmax solution of the two-center Dirac equation
本文提出用极小极大有限元法高精度求解双中心狄拉克方程，在H₂⁺和Th₂¹⁷⁹⁺上获得相对不确定度低至10⁻²³和10⁻²¹的基准结果，为方法应用与扩展奠定基础。
We formulate the Einstein-Cartan-Dirac equations in the Newman-Penrose (NP) formalism, thereby presenting a more accurate and explicit analysis of previous such studies. The equations show in a transparent way how the Einstein-Dirac equations are modified by the inclusion of torsion. In particular, the Hehl-Datta equation is presented in NP notation. We then describe a few solutions of the Hehl-Datta equation on Minkowski space-time, and in particular report a solitonic solution which removes the unphysical behavioiur of the corresponding Dirac solution. The present work serves as a prelude to…
Dirac-Bergmann algorithm and canonical quantization of $k$-essence cosmology
本文利用Dirac-Bergmann算法对$k$-本质宇宙学进行规范量子化，推导出无势能项的哈密顿约束，并探讨了超光速场在量子隧穿效应下的暗能量交叉条件。
On Hamiltonian formulations of the Dirac system
该研究通过狄拉克-伯格曼算法将约束薛定谔系统的哈密顿表述扩展到狄拉克场，引入旋量变量、修改的泊松括号等，恢复狄拉克方程，并提出正则二次量子化方案得到正确反对易关系。
Two-Body Kapitza-Dirac Scattering of One-Dimensional Ultracold Atoms
研究一维谐振子中两原子在光格子中的双体Kapitza-Dirac散射，分析相互作用强度、晶格深度等参数对衍射图案的影响，提供精确的少体基准。
Remarks on Dirac-Bergmann algorithm, Dirac's conjecture and the extended Hamiltonian
本文通过教学方式探讨了Dirac-伯格曼算法、Dirac猜想及扩展哈密顿量在约束系统中的应用，指出规范不变量的重新定义与Stueckelberg技巧的等价性。
Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
本文提出狄拉克方程信号处理框架，利用拓扑狄拉克算子的谱特性，提升信号处理性能，适用于非平滑或非谐振的真实信号。
Gaussian Beam Methods for the Dirac Equation in the Semi-classical Regime
针对半经典区域中狄拉克方程直接模拟成本过高的问题，提出基于特征分解的高斯光束方法，通过独立演化各特征空间的高斯光束并求和构造近似解，数值实验验证了方法的效率和精度。
We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case of unit nuclear charges is relevant for the accurate description of the ${\rm H}_2^+$ molecular ion and its isotopologues, systems that are an active experimental topic. The computation utilizes the 2-spinor minmax approach and the finite-element method. The computed total energies have estimated fractional uncertainties of a few times $10^{-20}$ for unit charges and a bond length of 2 atomic units. The fractional uncertainty of the purely relativistic contribution is $1\times10^{-17}…
The Dirac--Bergmann approach to optimal control theory
本文提出利用Dirac-Bergmann算法解决经典和量子系统最优控制问题，通过闭合Poisson括号代数自动获得最优解，并通过两个实例验证其有效性。
Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger (Nature, \textbf{169}, 538 (1952)), the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger's conclusion that charged fields do not necessarily require complex repr…
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