Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with N<105 fermionic modes can be performed with roughly O(1) and O(N2) T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.

High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates

Since the introduction of a series of methods for solving the time-dependent Schrödinger equation (TDSE) in the 80s of the last centry, such as the Fourier transform, the split operator (SO), the Chebyshev polynomial propagator, and complex absorbing potential, investigation of the molecular dynamics within quantum mechanics principle have become popular. In this paper, the application of the time-dependent wave packet (TDWP) method using high-order SO propagators in hyperspherical coordinates for solving triatomic reactive scattering was investigated. The fast sine transform was applied to calculate the derivatives of the wave function of the radial degree of freedom. These high-order SO propagators are examined in different forms, i.e., TVT (Kinetic–Potential–Kinetic) and VTV (Potential–Kinetic–Potential) forms with three typical triatomic reactions, H + H 2 , O + O 2 and F + HD. A little difference has been observed among the performances of high-order SO propagators in the TVT and VTV representations in the hyperspherical coordinate. For obtaining total reaction probabilities with 1% error, some of the S class high-order SO propagators, which have symmetric forms, are more efficient than second order SO for reactions involving long lived intermediate states. High order SO propagators are very efficient for obtaining total reaction probabilities.

The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

Numerical Simulation of Ship Motions in Regular and Irregular Waves

A numerical simulation method is presented in this study to predict ship resistance and motion responses in regular and irregular waves. The unsteady RANS (Reynolds Average Navier-Stokes) method is selected as the governing equation, and a volume of fluid (VoF) model is used to capture the free surface, combining the k-ε equations. A finite volume method (FVM) is utilized to discretize both the RANS equations and VoF transport equation. The pressure implicit split operator (PISO) method is set as the velocity-pressure coupling equation. The overset mesh technique is utilized to simulate ship motions in waves. A DTMB5415 ship is selected as a case study to predict its pitch and heave responses in regular and irregular waves at different wave length and wave steepness. The ship is free to move in the pitch and heave directions. The CFD (Computational Fluid Dynamics) results are found to be in good agreement with the strip theory and experimental data. It can be found that the CFD method presented in this study can provide a theoretical basis and technical support for green design and manufacture of ships.