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arXiv:2507.06355 · The European Physical Journal D

Density matrix analysis of systems with periodic Hamiltonians

Soham Sen, Manjari Dutta, Sunandan Gangopadhyay

Published Sep 1, 2025• 0 citations
arXivHTMLPDF
DOI
10.1140/epjd/s10053-025-01070-2

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Abstract Density matrix exponentiation (DME) is a quantum algorithm that processes multiple copies of a program state σ to realize the Hamiltonian evolution e − i σ t . Wave matrix Lindbladization (WML) similarly processes multiple copies of a program state ψ L in order to realize a Lindbladian evolution. Both algorithms are prototypical sample-based quantum algorithms and can be used for various quantum information processing tasks, including quantum principal component analysis, Hamiltonian simulation, and Lindbladian simulation. In this work, we present detailed sample complexity analyses for DME and sample-based Hamiltonian simulation, as well as for WML and sample-based Lindbladian simulation. In particular, we prove that the sample complexity of DME is no larger than 4 t 2 / ε , where t is the desired evolution time and ɛ is the desired imprecision level, as quantified by the normalized diamond distance. We also establish a fundamental lower bound on the sample complexity of sample-based Hamiltonian simulation, which matches our DME sample complexity bound up to a constant multiplicative factor, thereby proving that DME is optimal for sample-based Hamiltonian simulation. Additionally, we prove that the sample complexity of WML is no larger than 3 t 2 d 2 / ε , where d is the dimension of the space on which the Lindblad operator acts nontrivially, and we prove a lower bound of 10 − 4 t 2 / ε on the sample complexity of sample-based Lindbladian simulation. These results prove that WML is optimal for sample-based Lindbladian simulation whenever the Lindblad operator acts nontrivially on a constant-sized system. Finally, we point out that the DME sample complexity analysis in Kimmel et al(2017 npj Quantum Inf. 3 13) and the WML sample complexity analysis in Patel and Wilde (2023 Open Syst. Inf. Dyn. 30 2350010) appear to be incomplete, highlighting the need for the results presented here.
We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by M.V. Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order to the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model-independent. We apply them to spin 1/2 model coupled to reservoir at thermodynamic equilibrium.