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Time-dependent Hamiltonians and geometry of operators generated by them
1Seoul National University, Indian Institute of Technology Kanpur, Department of Physics, Kanpur 208016, India and Department of Physics and Astronomy, Center for Theoretical Physics, Seoul 08826, Republic of Korea.
Physical Review. E
|February 20, 2025
Summary
We calculated quantum circuit complexity for time-dependent quantum systems. An equivalence was found between costs for specific harmonic oscillator models, offering insights into quantum evolution.
Area of Science:
- Quantum mechanics
- Quantum information theory
- Complexity theory
Background:
- Understanding quantum system evolution is crucial.
- Quantifying complexity in quantum mechanics is an active research area.
- Time-dependent Hamiltonians present unique challenges.
Purpose of the Study:
- To derive the complexity geometry for time-dependent quantum Hamiltonians.
- To apply Nielsen's geometric formulation to calculate circuit complexity costs.
- To investigate analytical expressions for these costs in specific quantum systems.
Main Methods:
- Utilized Nielsen's geometric formulation of circuit complexity.
- Calculated bi-invariant cost for time-dependent Hamiltonians.
- Regularized Hamiltonian norms to obtain analytical expressions.
- Analyzed specific time-dependent quantum mechanical systems, including harmonic oscillators.
Main Results:
- Obtained analytical expressions for circuit complexity costs.
- Established an equivalence between the total costs for two different time-dependent harmonic oscillator models.
- Demonstrated the application to smooth quench protocols.
Conclusions:
- The geometric approach provides a framework for understanding complexity in time-dependent quantum systems.
- The identified equivalence simplifies cost calculations for certain systems.
- Comparison with information-theoretic quantities like Shannon entropy offers deeper insights.
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