Trotter Formula Research Articles

Sets of rich lines in general position

AbstractThe Szemerédi–Trotter theorem implies that the number of lines incident to at least of points in is . J. Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position—no two parallel, no three meeting at a point—then one can get a much tighter bound. We prove a slight variant of his conjecture: for every there exists some such that for sufficiently large values of , every set of lines in general position, each intersecting an grid of points in at least places, has size at most . This implies a conjecture of Gy. Elekes about the existence of a uniform statistical version of Freiman's theorem for linear functions with small image sets.

Linear kernels for separating a graph into components of bounded size

The p-Size Separator problem is to check whether we can delete at most k vertices in a given graph such that each connected component of the remaining graph has at most p vertices. We prove a kernel of O(pk) vertices for this problem, i.e., a linear vertex kernel for each fixed p≥1. Our proofs are based on extremal combinatorial arguments and the main result can be regarded as a generalization of the Nemhauser and Trotter's theorem for the Vertex Cover problem. Our techniques are possible to be used to improve kernel algorithms based on the expansion lemma or crown decompositions.

Bilayer linearized tensor renormalization group approach for thermal tensor networks

In this paper, we perform a comprehensive study of the renormalization group (RG) method on thermal tensor networks (TTN). By Trotter-Suzuki decomposition, one obtains the 1+1D TTN representing the partition function of 1D quantum lattice models, and then employs efficient RG contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN in an efficient and accurate way, is briefly reviewed. In addition, the single-layer LTRG can be generalized to a bilayer form, dubbed as LTRG++, in both finite- and infinite-size systems, with accuracies significantly improved. We provide the details of LTRG++ in finite-size system, comparing its accuracy with single-layer algorithm, and elaborate the infinite-size LTRG++ in the context of fermion chain model. We show that the LTRG++ algorithm in infinite-size system is in essence equivalent to transfer-matrix renormalization group method, while expressed in a tensor network language. LTRG++ is then applied to study an extended Hubbard model, where the phase separation phenomenon, groundstate phase diagram, as well as quantum criticality-enhanced magnetocaloric effects under external fields, are investigated.

On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations

We improve some recent convergence rate estimates for approximations of solution operators of linear non-autonomous evolution equations. The approximation results from the Trotter product formula which is proved to converge in the operator-norm and its convergence can be estimated. The result is applied to a diffusion perturbed by a time-dependent potential.

Log-majorization and Lie–Trotter formula for the Cartan barycenter on probability measure spaces

We extend Ando–Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie–Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.

Optimization of quantum Hamiltonian evolution: From two projection operators to local Hamiltonians

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie–Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretization errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.

Quantum Monte Carlo simulation of a particular class of non-stoquastic Hamiltonians in quantum annealing

Quantum annealing is a generic solver of the optimization problem that uses fictitious quantum fluctuation. Its simulation in classical computing is often performed using the quantum Monte Carlo simulation via the Suzuki–Trotter decomposition. However, the negative sign problem sometimes emerges in the simulation of quantum annealing with an elaborate driver Hamiltonian, since it belongs to a class of non-stoquastic Hamiltonians. In the present study, we propose an alternative way to avoid the negative sign problem involved in a particular class of the non-stoquastic Hamiltonians. To check the validity of the method, we demonstrate our method by applying it to a simple problem that includes the anti-ferromagnetic XX interaction, which is a typical instance of the non-stoquastic Hamiltonians.

Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces

We generalize the theory of gradient flows of semi-convex functions on ${\rm CAT}(0)$-spaces, developed by Mayer and Ambrosio-Gigli-Savar\'e, to ${\rm CAT}(1)$-spaces. The key tool is the so-called ``commutativity'' representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativity with semi-convex squared distance functions. Our approach combining the semi-convexity of the squared distance function with a Riemannian property of the space seems to be of independent interest, and can be compared with Savar\'e's work on the local angle condition under lower curvature bounds. Applications include the convergence of the discrete variational scheme to a unique gradient curve, the contraction property and the evolution variational inequality of the gradient flow, and a Trotter-Kato product formula for pairs of semi-convex functions.

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.

On a generalization of Nemhauser and Trotter's local optimization theorem

Fellows, Guo, Moser and Niedermeier [14] proved a generalization of Nemhauser and Trotter's theorem, which applies to d-Bounded-Degree Vertex Deletion (to delete k vertices of the input graph to make the maximum degree of it ≤d) and gives a linear-vertex kernel for d=0 and 1, and a superlinear-vertex kernel for each d≥2 for the problem parameterized by k. It is still left as an open problem whether this parameterized problem admits a linear-vertex kernel for each d≥3. In this paper, we refine the generalized Nemhauser and Trotter's theorem and give a linear-vertex kernel for each d≥0.

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n × n Hermitian matrices A and B the function fA,B(t) = trace etA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.

Quantum chemistry and charge transport in biomolecules with superconducting circuits.

We propose an efficient protocol for digital quantum simulation of quantum chemistry problems and enhanced digital-analog quantum simulation of transport phenomena in biomolecules with superconducting circuits. Along these lines, we optimally digitize fermionic models of molecular structure with single-qubit and two-qubit gates, by means of Trotter-Suzuki decomposition and Jordan-Wigner transformation. Furthermore, we address the modelling of system-environment interactions of biomolecules involving bosonic degrees of freedom with a digital-analog approach. Finally, we consider gate-truncated quantum algorithms to allow the study of environmental effects.

SPILADY: A parallel CPU and GPU code for spin–lattice magnetic molecular dynamics simulations

Spin–lattice dynamics generalizes molecular dynamics to magnetic materials, where dynamic variables describing an evolving atomic system include not only coordinates and velocities of atoms but also directions and magnitudes of atomic magnetic moments (spins). Spin–lattice dynamics simulates the collective time evolution of spins and atoms, taking into account the effect of non-collinear magnetism on interatomic forces. Applications of the method include atomistic models for defects, dislocations and surfaces in magnetic materials, thermally activated diffusion of defects, magnetic phase transitions, and various magnetic and lattice relaxation phenomena. Spin–lattice dynamics retains all the capabilities of molecular dynamics, adding to them the treatment of non-collinear magnetic degrees of freedom. The spin–lattice dynamics time integration algorithm uses symplectic Suzuki–Trotter decomposition of atomic coordinate, velocity and spin evolution operators, and delivers highly accurate numerical solutions of dynamic evolution equations over extended intervals of time. The code is parallelized in coordinate and spin spaces, and is written in OpenMP C/C++ for CPU and in CUDA C/C++ for Nvidia GPU implementations. Temperatures of atoms and spins are controlled by Langevin thermostats. Conduction electrons are treated by coupling the discrete spin–lattice dynamics equations for atoms and spins to the heat transfer equation for the electrons. Worked examples include simulations of thermalization of ferromagnetic bcc iron, the dynamics of laser pulse demagnetization, and collision cascades. Program summaryProgram title: SPILADY, version 1.0Catalogue identifier: AFAN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Apache License, Version 2.0No. of lines in distributed program, including test data, etc.: 1611165No. of bytes in distributed program, including test data, etc.: 367246683Distribution format: tar.gzProgramming language: OpenMP C/C++, CUDA C/C++.Computer: Any computer with an OpenMP capable C/C++ compiler or computer with CUDA capable GPU card and an nvcc compiler.Operating system: Linux, Unix, Windows.RAM: At least 500 MB, depending on the number of atoms or atomic spins, and on the simulation type.Classification: 7.7.Nature of problem:Excitation of magnetic degrees of freedom affects a broad range of properties of magnetic materials, including their equilibrium crystal structure and response to mechanical deformation. Existing atomistic simulation methods, for example molecular dynamics, do not treat magnetic degrees of freedom and do not describe the effect of magnetism on interatomic forces. This is addressed by the spin–lattice dynamics approach. The integration algorithm satisfies the requirement of phase volume conservation in the multi- dimensional space of atomic coordinates, velocities, and atomic spins, which is achieved through the application of the symplectic Suzuki–Trotter decomposition. Numerical solutions of spin–lattice dynamics equations retain high accuracy over extended intervals of time.Solution method:An atomic scale simulation technique for modelling the coupled dynamics of atomic coordinates and spins. The method generalizes molecular dynamics to the case of magnetic materials, and uses a parallel Suzuki–Trotter decomposition-based time integration algorithm.Restrictions:The current version assumes a single chemical element material composition. The spin Hamiltonian and equations of motion assume isotropic magnetic interactions. Evolution equations assume the validity of localized magnetic moment approximation.Unusual features:An open source spin–lattice dynamics code. The time integration algorithm uses the Suzuki–Trotter decomposition, which is a symplectic integration method retaining high accuracy over extended intervals of time. The code can be executed in parallel on multiple CPUs using OpenMP directives, or on an Nvidia GPU card.Additional comments:Further information available on website:http://spilady.ccfe.ac.ukRunning time:Similar to molecular dynamics, from several minutes to several weeks or months, depending on the number of atoms or spins involved in a simulation, and on the type of the simulation.

Exponentially more precise quantum simulation of fermions in second quantization

We introduce novel algorithms for the quantum simulation of fermionic systems which are dramatically more efficient than those based on the Lie–Trotter–Suzuki decomposition. We present the first application of a general technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The key difficulty in applying algorithms for general sparse Hamiltonian simulation to fermionic simulation is that a query, corresponding to computation of an entry of the Hamiltonian, is costly to compute. This means that the gate complexity would be much higher than quantified by the query complexity. We solve this problem with a novel quantum algorithm for on-the-fly computation of integrals that is exponentially faster than classical sampling. While the approaches presented here are readily applicable to a wide class of fermionic models, we focus on quantum chemistry simulation in second quantization, perhaps the most studied application of Hamiltonian simulation. Our central result is an algorithm for simulating an N spin–orbital system that requires gates. This approach is exponentially faster in the inverse precision and at least cubically faster in N than all previous approaches to chemistry simulation in the literature.

Ultrafast adiabatic quantum algorithm for the NP-complete exact cover problem

An adiabatic quantum algorithm may lose quantumness such as quantum coherence entirely in its long runtime, and consequently the expected quantum speedup of the algorithm does not show up. Here we present a general ultrafast adiabatic quantum algorithm. We show that by applying a sequence of fast random or regular signals during evolution, the runtime can be reduced substantially, whereas advantages of the adiabatic algorithm remain intact. We also propose a randomized Trotter formula and show that the driving Hamiltonian and the proposed sequence of fast signals can be implemented simultaneously. We illustrate the algorithm by solving the NP-complete 3-bit exact cover problem (EC3), where NP stands for nondeterministic polynomial time, and put forward an approach to implementing the problem with trapped ions.

A Lie product type formula in Euclidean Jordan algebras

Abstract In this paper,we state and prove an analog of Lie product formula in the setting of Euclidean Jordan algebras.

On the BMV Conjecture for 2 $$\times $$ × 2 Matrices and the Exponential Convexity of the Function $${\cosh (\sqrt{at^2+b})}$$ cosh ( a t 2 + b )

The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)={{\mathrm{trace\,}}}e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for \(2\times 2\) matrices. This proof is based on the Lie product formula for the exponential of the sum of two matrices. The proof also uses the commutation relations for the Pauli matrices and does not use anything else.

Partly invariant steady state of two interacting open quantum systems

We investigate two interacting open quantum systems whose time evolutions are governed by Markovian master equations. We show a class of coupled systems whose interaction leaves invariant the steady state of one of the systems, i.e., only one of the reduced steady states is sensitive to the interactions. A detailed proof with the help of the Trotter product formula is presented. We apply this general statement to a few models, one of which is the optomechanical coupling model where an optical cavity is coupled to a small mechanical oscillator.

Reciprocal Lie–Trotter formula

Let and be positive semidefinite matrices. The limit of the expression as tends to is given by the well-known Lie–Trotter formula. A similar formula holds for the limit of as tends to , where is the geometric mean of and . In this paper we study the limit of and as tends to instead of , with the ultimate goal of finding an explicit formula, which we call the reciprocal Lie–Trotter formula. We show that the limit of exists and find an explicit formula in a special case. The limit of is shown for matrices only.