An Analysis of Suzuki-Trotter Decompositions for Quantum Thermodynamics (2024)

Undergraduate: Rachel Emrick

Faculty Advisor: Jingfang Huang
Department: Mathematics, Physics

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Calculating the time evolution and thermodynamics of a quantum system requires exponentiation linear operators, which are represented as matrices. Therefore, it is crucial to understand how to compute matrix exponentials, particularly of large matrices. The matrices used in common quantum mechanics applications require us to employ operator splitting to write them as the sum of two non-commuting pieces. Typically, the exponential of each piece can be calculated, but the exponential of their sum cannot. To that end, this work explores the use of Suzuki-Trotter decompositions to tackle the problem of matrix exponentiation._x000D_
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In particular, we examine the accuracy of Suzuki-Trotter decompositions when used on one diagonal matrix T and one non-diagonal matrix V, such that H = T + V. This is the structure of the Hamiltonian of a three-dimensional two-body quantum problem, which can be split into a diagonal potential energy matrix and a constant potential energy matrix. We consider the effects that different structures in T and V have on accuracy for various decompositions when changing the number of time steps, coupling strength, or matrix size._x000D_
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Our general recommendation for the kinds of matrices we studied is the fourth-order decomposition Q3. Making the problem “harder” by changing its parameters sometimes lessens the advantage that higher-order methods have over lower-order methods, but almost never makes the higher-order methods worse.