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Physical Review A

Applications of the generalized Trotter formula

Hans A. De Raedt, Bart De Raedt

Published Dec 1, 1983• 240 citations
Publisher

Abstract

We study the properties of Suzuki's systematic approximations to the exponential operator $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ by calculating the thermodynamic functions of three simple quantum models. We demonstrate that the path-integral representation of the partition function obtained from these approximations can be simplified and made more accurate by constructing Hermitian versions of Suzuki's expressions.

DOI
10.1103/physreva.28.3575

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We study the properties of Suzuki's systematic approximations to the exponential operator exp(\ensuremath−\ensuremathβH)\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)exp(\ensuremath−\ensuremathβH) by calculating the thermodynamic functions of three simple quantum models. We demonstrate that the path-integral representation of the partition function obtained from these approximations can be simplified and made more accurate by constructing Hermitian versions of Suzuki's expressions.
The Lie-Trotter formula has been a fundamental tool in quantum mechanics, quantum computing, and quantum simulations. The error estimations for the Lie-Trotter product formula play a crucial role in achieving scalability and computational efficiency. In this note, we present two error estimates of Lie-Trotter product formulas, utilizing Jordan product within Banach algebras. Additionally, we introduce two generalized Lie-Trotter formula and provide two explicit estimation formulas. Consequently, the renowned Suzuki symmetrized approximation for the exponentiated sums follows directly from our main Theorem.
An improved transfer matrix method is proposed for calculating the partition function of quantum spin systems, which is expressed by the M-th approximant of the Suzuki-Trotter transformation with Trotter number M. The point is to incorporate the symmetry property of the Trotter subsystem in calculation. The size of our transfer matrix is reduced very much. Thermodynamics is described by the maximum eigenvalue of the transfer matrix. Our method is applied to the XY model with spin 1/2. We consider the cases of pair and tri spin decompositions with Trotter number up to 4 (M≤4). We can do our calculation analytically in those cases. In the pair spin decomposition we find results which have been obtained by the Monte Carlo simulation and by a transfer matrix method by computer. We find nice results in the tri spin decomposition. Our result in the case M=4 is as good as the result in M=8 by the previous Monte Carlo simulation on comparing them with the exact result of Katsura.
Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of 4He^{4}\mathrm{He}4He. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.
We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.