本文推广了Levy-Leblond方法，将薛定谔方程线性化得到单粒子Pauli方程，得到了一个带自旋的$N$粒子方程。这是通过使用更通用的矩阵分解$G\tilde{G} = |G| I = (-K)^l I$来实现的。这里方阵$G$在总能量E和所有动量中是线性的，$\tilde G$是$G$的伴随矩阵，$I$是单位矩阵，$|G|$是$G$的行列式，$l$是正整数，$K=H-E$是Lanczos的扩展哈密顿量，其中$H$是机电系统的经典哈密顿量。对于所有这样的系统，$K$都是零，因此矩阵$G$是奇异的。因此，总是存在具有属性$G\underlineθ=0$的向量函数$\underlineθ$。这种因式分解，以获得矩阵$G$和矢量函数$\underlineθ$示出的一维粒子在一个简单的势阱。同样的技术，当应用到经典的非相对论性哈密顿量的$N$相互作用的粒子在电磁场中，示出产生N=1的泡利波方程的自旋和其推广到$N$粒子。最后，本文用泡利方程的非相对论性推广，把类氢原子的塞曼效应当作一个带自旋的两粒子问题来处理。
英文摘要：
In this paper the Levy-Leblond procedure for linearizing the Schrödinger equation to obtain the Pauli equation for one particle is generalized to obtain an $N$-particle equation with spin. This is achieved by using the more universal matrix factorization, $G\tilde{G} = |G| I = (-K)^l I$. Here the square matrix $G$ is linear in the total energy E and all momenta, $\tilde G$ is the matrix adjoint of $G$, $I$ is the identity matrix, $|G|$ is the determinant of $G$, $l$ is a positive integer and $K=H-E$ is Lanczos' extended Hamiltonian where $H$ is the classical Hamiltonian of the electro-mechanical system. $K$ is identically zero for all such systems, so that matrix $G$ is singular. As a consequence there always exists a vector function $\underlineθ$ with the property $G\underlineθ=0$. This factorization to obtain the matrix $G$ and vector function $\underlineθ$ is illustrated first for a one-dimensional particle in a simple potential well. This same technique, when applied to the classical nonrelativistic Hamiltonian for $N$ interacting particles in an electromagnetic field, is shown to yield for N=1 the Pauli wave equation with spin and its generalization to $N$ particles. Finally this nonrelativistic generalization of the Pauli equation is used to treat the simple Zeeman effect of a hydrogen-like atom as a two-particle problem with spin.
Noncontextual Pauli Hamiltonians decompose into sets of Pauli terms to which joint values may be assigned without contradiction. We construct a quasi-quantized model for noncontextual Pauli Hamiltonians. Using this model, we give an algorithm to classically simulate noncontextual VQE. We also use the model to show that the noncontextual Hamiltonian problem is NP-complete. Finally, we explore the applicability of our quasi-quantized model as an approximate simulation tool for contextual Hamiltonians. These results support the notion of noncontextuality as classicality in near-term quantum algor…
Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitio…
Efficient Quantum Circuit Construction of Controlled Time-Evolution for Arbitrary Pauli-Sum Hamiltonians
Faculty of Engineering, Yokohama National University（横滨国立大学工学部）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理·物理信息系）
We apply the method of direct perturbation theory for the Foldy-Wouthuysen (FW) transformation upon the Dirac-Pauli Hamiltonian subject to external electromagnetic fields. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, by mathematical induction on the orders of $1/c$ in the power series, we rigorously prove the long-held speculation: the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital H…
Planted-Solution Pauli Hamiltonians as a Quantum Benchmarking Primitive
Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）；Department of Physics and Astronomy, and Center for Quantum Information Science & Technology, University of Southern California（南加州大学物理与天文系及量子信息技术中心）
A simple protocol based on low-weight Pauli Hamiltonians is introduced for performing quantum gates that are robust to control noise. Gates are implemented by an adiabatic sequence of single-qubit fields and two-qubit interactions with a single ancillary qubit, whereas related techniques require three-qubit interactions, perturbation gadgets, higher dimensional subsystems, and/or more ancilla qubits. Low-weight interactions and low qubit overhead open a viable path to experimental investigation, while operation in a degenerate ground space allows for physical qubit designs that are immune to e…
Schroedinger's equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in spe…
It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of $1/c$ in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreemen…
A High-Performance Pauli-Algebra Framework for Large-Scale Quantum Simulations
State Key Laboratory of Precision and Intelligent Chemistry, University of Science and Technology of China（精密与智能化学国家重点实验室，中国科学技术大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（合肥微尺度物质科学国家研究中心，中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）
We show that there is a manifestly covariant version of the Pauli Hamiltonian with equations of motion quadratic on spin and field strength. Relativistic covariance inevitably leads to noncommutative positions: classical brackets of the position variables are proportional to the spin. It is the spin-induced noncommutativity that is responsible for transforming the covariant Hamiltonian into the Pauli Hamiltonian, without any appeal to the Thomas precession formula. The Pauli theory can be thought to be $1/c^2$ approximation of the covariant theory written in special variables. These observatio…
In previous implementations of adiabatic quantum algorithms using spin systems, the average Hamiltonian method with Trotter's formula was conventionally adopted to generate an effective instantaneous Hamiltonian that simulates an adiabatic passage. However, this approach had issues with the precision of the effective Hamiltonian and with the adiabaticity of the evolution. In order to address these, we here propose and experimentally demonstrate a novel scheme for adiabatic quantum computation by using the intrinsic Hamiltonian of a realistic spin system to represent the problem Hamiltonian whi…
Quantitative semidefinite certificates for ground-state energies of Pauli Hamiltonians
University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学与物理学院）；University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies（沿海大学数学、自然科学和信息技术学院）；Université de Toulouse（图卢兹大学；法国国家科学研究中心劳动力自动化研究与应用实验室）；LAAS-CNRS
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum (NISQ) computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We pro…
Pauli's Theorem and Quantum Canonical Pairs: The Consistency Of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum
Pauli Weight Hamiltonian Term Selection for Optimized Machine Learning Based Quantum Error Mitigation
Bandung Institute of Technology（万隆理工学院）；National Research and Innovation Agency (BRIN)（国家研究与创新署）；National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）
Fault-tolerant quantum simulation of the Pauli-Breit Hamiltonian for ab initio hybrid quantum-classical molecular design with applications to photodynamic therapy
We report numerical results on the diagonalization of 1D transverse field Ising model. Numerical simulations using the Pauli product representation yield diagonalization from 3 spins to 22 spins in the transverse field Ising model with the number of global Jacobi unitary transformations and number of final terms in diagonalized spin z representation both grew polynomial with the number of spins. These results computed on a classical computer show promise in constructing a quantum circuit to simulate diagonalized generic many-particle Hamiltonians using polynomial number of gates.
For the variational quantum eigensolver we propose to generate trial wavefunctions from a small amount of selected Pauli terms of the problem Hamiltonian. Two different approaches, one inspired by the quantum approximate optimization algorithm and the other by imaginary-time evolution, are proposed and studied in detail. Using numerical calculations, we study the efficiency of these trial wavefunctions for finding the ground-state energy of three molecules: H2, LiH and H2O. We find that only a small number of Pauli terms are needed to reach chemical accuracy, leading to short-depth quantum cir…
The modified Dirac-Pauli equations, which is entered by means of ${γ_5}$-mass extension of Hamiltonian operators, are considered. We also take into account the interaction of fermions with the intensive homogenous magnetic field focusing attention on (g-2) gyromagnetic factor of particles with spin $1/2$. Without the use of perturbation theory in the external field the exact energy spectra are deduced with regard to spin effects of fermions. We discuss the possible proposals of experimental measuring of properties of new particles which arising in this model.
We generalize Schroedinger's factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach, is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach…
Affine spin-pencil factorization of generalized driven anisotropic Rabi--Stark Hamiltonians in an inhomogeneous orthosymplectic superalgebra
Universidad Politécnica Metropolitana de Hidalgo（伊达尔戈大都会理工大学）；SECIHTI-Instituto Nacional de Astrofísica, Óptica y Electrónica（国家天体物理学、光学和电子学研究所）；Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional（国立理工学院高级研究中心）
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found,there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.
The factorization method was introduced by Schroedinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain -- instead all energies are solved using a "single-shot factorization," enabled b…
