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arXiv:cond-mat/0001304

A Method to Calculate a Delta Function of the Hamiltonian by the Suzuki-Trotter Decomposition

T. Munehisa, Yasuko Munehisa

Published Jan 21, 2000• 2 citations
arXivHTMLPDF

Abstract

We propose a new method to calculate expectation values of a delta function of the Hamiltonian, < Ψ\mid δ(\hat{H} - E)\mid Ψ>. Since the delta function can be replaced with a Gaussian function, we evaluate < Ψ\mid \sqrt{\fracβπ} e^{-β(\hat{H} - E)^2} \mid Ψ> with large βadopting the Suzuki-Trotter decomposition. Errors of the approximate calculations with the finite Trotter number N_t are estimated to be O(1/N_t^K) for the Kth-order decomposition. The distinct advantage of this method is that the convergence is guaranteed even when the state \mid Ψ> contains the eigenstates whose energies spread over the wide range. In this paper we give a full description of our method within the quantum mechanical physics and present the numerical results for the harmonic oscillator problems in one- and three-dimensional space.

DOI
10.48550/arxiv.cond-mat/0001304

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We propose a new method to calculate expectation values of a delta function of the Hamiltonian, < Ψ\mid δ(\hat{H} - E)\mid Ψ>. Since the delta function can be replaced with a Gaussian function, we evaluate < Ψ\mid \sqrt{\fracβπ} e^{-β(\hat{H} - E)^2} \mid Ψ> with large βadopting the Suzuki-Trotter decomposition. Errors of the approximate calculations with the finite Trotter number N_t are estimated to be O(1/N_t^K) for the Kth-order decomposition. The distinct advantage of this method is that the convergence is guaranteed even when the state \mid Ψ> contains the eigenstates whose energies spread over the wide range. In this paper we give a full description of our method within the quantum mechanical physics and present the numerical results for the harmonic oscillator problems in one- and three-dimensional space.
Accurately simulating long-time dynamics of many-body systems is a challenge in both classical and quantum computing due to the accumulation of Trotter errors. While low-order Trotter-Suzuki decompositions are straightforward to implement, their rapidly growing error limits access to long-time observables. We present a framework for constructing efficient high-order Trotter-Suzuki schemes by identifying their structure and directly optimizing their parameters over a high-dimensional space. This method enables the discovery of new schemes with significantly improved efficiency compared to traditional constructions, such as those by Suzuki and Yoshida. Based on the theoretical efficiency and practical performance, we recommend two novel highly efficient schemes at 4th4^{\textrm{th}}4th and 6th6^{\textrm{th}}6th order. We also demonstrate the effectiveness of these decompositions on the Heisenberg model and the quantum harmonic oscillator, and find that for a fixed final time they perform better across the computational cost. Even when using large time steps, they surpass established low-order schemes like the Leapfrog. Finally, we investigate the in-practice performance of different Trotter schemes and find the decompositions with more uniform coefficients tend to feature improved error accumulation over long times. We have included this observation into our choice of recommended schemes.
We explore the errors in the free energy and in operator expectation values when the quantum operator ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$=${\ensuremath{\prod}}_{l=1}^{L}$${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ is approximated by ${\ensuremath{\prod}}_{l=1}^{L}$f, where \ensuremath{\Delta}\ensuremath{\tau}=\ensuremath{\beta}/L and f is an approximant to ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$. We determine analytically the dependence of the resulting errors on \ensuremath{\Delta}\ensuremath{\tau} for \ensuremath{\Delta}\ensuremath{\tau} small, on \ensuremath{\beta} for \ensuremath{\beta} large, and on the size of the system in the limit of a large system. We focus on Trotter approximations, as well as on the expansion of ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ in powers of (\ensuremath{\Delta}\ensuremath{\tau})H. Our results are particularly relevant to Monte Carlo studies of quantum systems, and can be used effectively to eliminate the error due to Trotter-like approximations and to provide a guide for choosing a particular type of approximation.
We solve the solution of the Time-Dependent Schrodinger Equation (TDSE) using the higher-order Trotter-Suzuki Decomposition. We use The Gaussian function as a stationary state in a linear potential system. The TDSE solution using Baker - Campbell - Hausdorff was used to validate the results and to measure the accuracy of the Trotter - Suzuki decomposition. So that the difference between the Baker - Campbell - Hausdorff result and the Suzuki - Trotter decomposition is considered an error. The error of the TDSE solution by the Trotter - Suzuki second-order decomposition was lower than the first order. Meanwhile, the error of the TDSE solution by second-order hybrid will be lower than the second-order Trotter - Suzuki decomposition when the value of dx=0.1 and 0.05 with dt < 0.0001. The error comparison of these three methods is only valid when time t<1.
Trotterization is the most common and convenient approximation method for Hamiltonian simulations on digital quantum computers, but estimating its error accurately is computationally difficult for large quantum systems. Here, we develop a method for measuring the Trotter error without ancillary qubits on quantum circuits by combining the m th - and n th -order ( m < n ) Trotterizations rather than consulting with mathematical error bounds. Using this method, we make Trotterization precision guaranteed, developing an algorithm named Trotter ( m , n ) , in which the Trotter error at each time step is within an error tolerance ε preset for our purpose. Trotter ( m , n ) is applicable to both time-independent and -dependent Hamiltonians, and it adaptively chooses almost the largest step size δ t , which keeps quantum circuits shallowest, within the error tolerance. Benchmarking it in a quantum spin chain, we find the adaptively chosen δ t to be about 10 times larger than that inferred from known upper bounds of Trotter errors. Published by the American Physical Society 2024
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Owing to their simplicity and low overhead, Suzuki-Trotter formulas remain the de facto Hamiltonian simulation methods on current quantum computing platforms. Systematic Trotter errors, however, will quickly become limiting when scaling to larger problems and aiming for higher accuracy. We present a mechanism that removes the systematic error of any kkk-th order Suzuki-Trotter simulation, at the cost of a constant sampling overhead. The key insight is that the Trotter error is itself a coherent dynamics to be reversed, rather than a deviation to be bounded. By identifying the structure of this error in closed form, we carry out that reversal through quasi-probabilistic decompositions. The resulting algorithm, called Probabilistic Trotter Error Reversal (PTER), is unbiased and still improves the gate-count scaling of Suzuki-Trotter formulas while retaining their simplicity. Numerical simulations of a Heisenberg spin chain support the predicted resource advantage already at modest system sizes.