Time-dependent Hamiltonians and Geometry of Operators Generated by Them

• URL: http://arxiv.org/abs/2405.14410v1
• Date: Thu, 23 May 2024 10:32:29 GMT
• Title: Time-dependent Hamiltonians and Geometry of Operators Generated by Them
• Authors: Kunal Pal, Kuntal Pal,
• Abstract summary: We obtain the complexity geometry associated with the Hamiltonian of a quantum mechanical system. We show that an equivalence exists between the total costs of obtaining an operator through time evolution.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We obtain the complexity geometry associated with the Hamiltonian of a quantum mechanical system, specifically in cases where the Hamiltonian is explicitly time-dependent. Using Nielsen's geometric formulation of circuit complexity, we calculate the bi-invariant cost associated with these time-dependent Hamiltonians by suitably regularising their norms and obtain analytical expressions of the costs for several well-known time-dependent quantum mechanical systems. Specifically, we show that an equivalence exists between the total costs of obtaining an operator through time evolution generated by a unit mass harmonic oscillator whose frequency depends on time, and a harmonic oscillator whose both mass and frequency are functions of time. These results are illustrated with several examples, including a specific smooth quench protocol where the comparison of time variation of the cost with other information theoretic quantities, such as the Shannon entropy, is discussed.

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