Trotter Decomposition Research Articles

Does the full configuration interaction method based on quantum phase estimation with Trotter decomposition satisfy the size consistency condition?

Does the full configuration interaction method based on quantum phase estimation with Trotter decomposition satisfy the size consistency condition?

Pairing Hamiltonians of Nearest-Neighbor Interacting Superconducting Qubits on an IBM Quantum Computer

A quantum simulation experiment pairing Hamiltonians of nearest-neighbor interacting superconducting qubits was performed with a complete set of algorithms on an IBM Quantum Computer-IBMq Lima. The experiment revealed that the fidelity is a function of iteration using Suzuki–Trotter decomposition for four different types of nearest-neighbor Heisenberg, XY, transverse, and longitudinal Ising superconducting qubit couplings of Hamiltonians. The experiment displayed the models of how the experimental density matrices shift from the theoretical density matrices and how their behavior changes with different numbers of iterations. It also demonstrated the reconstruction of quantum states and how the states change as a function of iteration with the IBM Quantum Computer-IBMq Lima. The time evolutions of the states for different models were also shown to predict the dominance of each state.

Optimised Trotter decompositions for classical and quantum computing

Suzuki–Trotter decompositions of exponential operators like are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators can be applied to such generic Suzuki–Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order . Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of 10−4. Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.

Optimal compression of quantum many-body time evolution operators into brickwall circuits

Near term quantum computers suffer from a degree of decoherence which is prohibitive for high fidelity simulations with deep circuits. An economical use of circuit depth is therefore paramount. For digital quantum simulation of quantum many-body systems, real time evolution is typically achieved by a Trotter decomposition of the time evolution operator into circuits consisting only of two qubit gates. To match the geometry of the physical system and the CNOT connectivity of the quantum processor, additional SWAP gates are needed. We show that optimal fidelity, beyond what is achievable by simple Trotter decompositions for a fixed gate count, can be obtained by compiling the evolution operator into optimal brickwall circuits for the S=1/2S=1/2 quantum Heisenberg model on chains and ladders, when mapped to one dimensional quantum processors without the need of additional SWAP gates.

Improved quantum computing with higher-order Trotter decomposition

In designing quantum control, it is generally required to simulate the controlled system evolution with a classical computer. However, computing the time evolution operator can be quite resource-consuming since the total Hamiltonian is often hard to diagonalize. In this paper, we mitigate this issue by substituting the time evolution segments with their Trotter decompositions, which reduces the propagator into a combination of single-qubit operations and fixed-time system evolutions. The resulting procedure can provide substantial speed gain with acceptable costs in the propagator error. As a demonstration, we apply the proposed strategy to improve the efficiency of the gradient ascent pulse engineering algorithm for searching optimal control fields. Furthermore, we show that the higher-order Trotter decompositions can provide efficient Ans\"atze for the variational quantum algorithm, leading to improved performance in solving the ground-state problem. The strategy presented here is also applicable for many other quantum optimization and simulation tasks.

Efficient world-line-based quantum Monte Carlo method without Hubbard–Stratonovich transformation

By precisely writing down the matrix element of the local Boltzmann operator ({mathrm{e}}^{-tau h}, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard–Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.

Digital quantum simulation and pseudoquantum simulation of the Z2 gauge–Higgs model

We present a quantum algorithm for digital quantum simulation of the $\mathbb{Z}_2$ gauge-Higgs model on a $3\times 3$ lattice, which is based on Trotter decomposition, the quantum adiabatic algorithm and its circuit realization. Then we perform a classical demonstration, dubbed a pseudoquantum simulation, on a GPU simulator. We obtain useful results on this model, which suggest the topological properties of the deconfined phase and help to clarify the phase diagram. It is suggested that the tricitical point, where the second-order critical lines of deconfinement-confinement transition and of deconfinement-Higgs transition meet, seems to be on the the first-order critical line of confinement-Higgs transition, at a point other than the end of this critical line.

Higher Order Auxiliary Field Quantum Monte Carlo Methods

The auxiliary field quantum Monte Carlo (AFQMC) method has been a workhorse in the field of strongly correlated electrons for a long time and has found its most recent implementation in the ALF package (alf.physik.uni-wuerzburg.de). The utilization of the Trotter decomposition to decouple the interaction from the non-interacting Hamiltonian makes this method inherently second order in terms of the imaginary time slice. We show that due to the use of the Hubbard-Stratonovich transformation (HST) a semigroup structure on the time evolution is imposed that necessitates the introduction of a new family of complex-hermitian splitting methods for the purpose of reaching higher order. We will give examples of these new methods and study their efficiency, as well as perform comparisons with other established second and higher order methods in the realm of the AFQMC method.

Optimize quantum simulation using a force-gradient integrator

Quantum simulation has shown great potential in many fields due to its powerful computational capabilities. However, the limited fidelity can lead to a severe limitation on the number of gate operations, which requires us to find optimized algorithms. Trotter decomposition and high-order Trotter decompositions are widely used in quantum simulations. We find that they can be significantly improved by force-gradient integrator in lattice QCD. By using two applications as examples, we show that the force-gradient decomposition can reduce the number of gate operations up to about a third of those using high-order Trotter decompositions. Therefore, force-gradient decomposition shows a great prospective in future applications of quantum simulation.

Trotter errors in digital adiabatic quantum simulation of quantum ℤ2 lattice gauge theory

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[Formula: see text]+[Formula: see text]1)-dimensional quantum [Formula: see text] 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 equal to the Trotter error usually defined in terms of the evolution operators. They are much smaller than the order-of-magnitude estimation. The error in the symmetric decomposition is much smaller than that in the asymmetric decomposition. The features of the Trotter errors obtained here are useful in the experimental implementation of digital quantum simulation and its numerical demonstration.

Circuit-based digital adiabatic quantum simulation and pseudoquantum simulation as new approaches to lattice gauge theory

Gauge theory is the framework of the Standard Model of particle physics and is also important in condensed matter physics. As its major non-perturbative approach, lattice gauge theory is traditionally implemented using Monte Carlo simulation, consequently it usually suffers such problems as the Fermion sign problem and the lack of real-time dynamics. Hopefully they can be avoided by using quantum simulation, which simulates quantum systems by using controllable true quantum processes. The field of quantum simulation is under rapid development. Here we present a circuit-based digital scheme of quantum simulation of quantum ℤ2 lattice gauge theory in 2 + 1 and 3 + 1 dimensions, using quantum adiabatic algorithms implemented in terms of universal quantum gates. Our algorithm generalizes the Trotter and symmetric decompositions to the case that the Hamiltonian varies at each step in the decomposition. Furthermore, we carry through a complete demonstration of this scheme in classical GPU simulator, and obtain key features of quantum ℤ2 lattice gauge theory, including quantum phase transitions, topological properties, gauge invariance and duality. Hereby dubbed pseudoquantum simulation, classical demonstration of quantum simulation in state-of-art fast computers not only facilitates the development of schemes and algorithms of real quantum simulation, but also represents a new approach of practical computation.

Phonon-induced dephasing in quantum-dot–cavity QED

We present a semi-analytic and asymptotically exact solution to the problem of phonon-induced decoherence in a quantum dot-microcavity system. Particular emphasis is placed on the linear polarization and optical absorption, but the approach presented herein may be straightforwardly adapted to address any elements of the exciton-cavity density matrix. At its core, the approach combines Trotter's decomposition theorem with the linked cluster expansion. The effects of the exciton-cavity and exciton-phonon couplings are taken into account on equal footing, thereby providing access to regimes of comparable polaron and polariton timescales. We show that the optical decoherence is realized by real phonon-assisted transitions between different polariton states of the quantum dot-cavity system, and that the polariton line broadening is well-described by Fermi's golden rule in the polariton frame. We also provide purely analytic approximations which accurately describe the system dynamics in the limit of longer polariton timescales.

Quantum computing methods for electronic states of the water molecule

ABSTRACTWe compare recently proposed methods to compute the electronic state energies of the water molecule on a quantum computer. The methods include the phase estimation algorithm based on Trotter decomposition, the phase estimation algorithm based on the direct implementation of the Hamiltonian, direct measurement based on the implementation of the Hamiltonian and a specific variational quantum eigensolver, Pairwise VQE. After deriving the Hamiltonian using STO-3G basis, we first explain how each method works and then compare the simulation results in terms of gate complexity and the number of measurements for the ground state of the water molecule with different O–H bond lengths. Moreover, we present the analytical analyses of the error and the gate-complexity for each method. While the required number of qubits for each method is almost the same, the number of gates and the error vary a lot. In conclusion, among methods based on the phase estimation algorithm, the second-order direct method provides the most efficient circuit implementations in terms of the gate complexity. Moreover, Pairwise VQE serves the most practical method for near-term applications on the current available quantum computers. Finally the possibility of extending the calculation to excited states and resonances is discussed.

Floquet topological transition by unpolarized light

We study Floquet topological transition in irradiated graphene when the polarization of incident light changes randomly with time. We numerically confirm that the noise averaged time evolution operator approaches a steady value in the limit of exact Trotter decomposition of the whole period where incident light has different polarization at each interval of the decomposition. This steady limit is found to coincide with time-evolution operator calculated from the noise-averaged Hamiltonian. We observe that at the six corners (Dirac($K$) point) of the hexagonal Brillouin zone of graphene random Gaussian noise strongly modifies the phaseband structure induced by circularly polarized light whereas in zone-center ($\Gamma$ point) even a strong noise isn't able to do the same. This can be understood by analyzing the deterministic noise averaged Hamiltonian which has a different Fourier structure as well as lesser no of symmetries compared to the noise-free one. In 1D systems noise is found to renormalize the drive amplitude only.

Open source matrix product states: exact diagonalization and other entanglement-accurate methods revisited in quantum systems

Tensor network methods as presented in our open source matrix product states software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g. symmetry used for number conservation in the Bose–Hubbard model or discrete symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.

Massively parallel symplectic algorithm for coupled magnetic spin dynamics and molecular dynamics

A parallel implementation of coupled spin–lattice dynamics in the LAMMPS molecular dynamics package is presented. The approach is very general, and can be applied to simple ferromagnets, magnetic alloys, or amorphous magnetic materials. Equations of motion for both spin only and coupled spin–lattice dynamics are first reviewed, including a detailed account of how magneto-mechanical potentials can be used to perform a proper coupling between spin and lattice degrees of freedom. A symplectic numerical integration algorithm is then presented which combines the Suzuki–Trotter decomposition for non-commuting variables and conserves the geometric properties of the equations of motion. The numerical accuracy of the serial implementation was assessed by verifying that it conserves the total energy and the norm of the total magnetization up to second order in the timestep size. Finally, a very general parallel algorithm is proposed that allows large spin–lattice systems to be efficiently simulated on large numbers of processors without degrading its mathematical accuracy. Its correctness as well as scaling efficiency were tested for realistic coupled spin–lattice systems, confirming that the new parallel algorithm is both accurate and efficient.

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.

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.

On the configurational temperature Nosè–Hoover thermostat

Abstract In this paper we reformulate the configurational temperature Nose–Hoover thermostat of Braga and Travis (2005) by means of a quasi-Hamiltonian theory in phase space Sergi and Ferrario (2001). The quasi-Hamiltonian structure is exploited to introduce a hybrid configurational-kinetic temperature Nose–Hoover chain thermostat that can achieve a uniform sampling of phase space (also for stiff harmonic systems), as illustrated by simulating the dynamics of one-dimensional harmonic and quartic oscillators. An integration algorithm, based on the symmetric Trotter decomposition of the propagator, is presented and tested against implicit geometric algorithms with a structure similar to the velocity and position Verlet. In order to obtain an explicit form for the symmetric Trotter propagator algorithm, in the case of non-harmonic and non-linear interaction potentials, a position-dependent harmonically approximated propagator is introduced. Such a propagator approximates the dynamics of the configurational degrees of freedom as if they were locally moving in a harmonic potential. The resulting approximated locally harmonic dynamics is tested with good results in the case of a one-dimensional quartic oscillator: The integration is stable and locally time-reversible. Instead, the implicit geometric integrator is stable and time-reversible globally (when convergence is achieved). We also verify the stability of the approximated explicit integrator for a three-dimensional N-particle system interacting through a soft Weeks–Chandler–Andersen potential.

Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays

We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems, based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method brings considerable advantage when compared to the integration of the time-dependent problem via Trotter decomposition, as it can address arbitrarily long-ranged couplings. Additionally, it ensures numerical stability in the case of weakly dissipative systems thanks to a slow tuning of the dissipation rates along the sweeps. We have tested the method on a driven-dissipative spin chain, under various assumptions for the Hamiltonian, drive, and dissipation parameters, and compared the results to those obtained both by Trotter dynamics and Monte-Carlo wave function. Accurate and numerically stable convergence was always achieved when applying the method to systems with a gapped Liouvillian and a non-degenerate steady-state.