A Mathematical Framework for Quantum Hamiltonian Simulation and Duality

Area of Science:

• Quantum Information Science
• Theoretical Physics
• Mathematical Physics

Background:

• Analogue Hamiltonian simulation is a key quantum computing application with experimental success.
• Duality in physics relates seemingly different theories.
• Current Hamiltonian simulation definitions don't cover all physical dualities.

Purpose of the Study:

• Generalize the definition of duality in physics.
• Develop a framework applicable to all dualities, including those transforming strong and weak interactions.
• Characterize dual maps on operators and states.

Main Methods:

• Introduced a generalized definition of duality.
• Characterized dual maps for operators and states.
• Proved duality equivalence via observables, partition functions, and entropies.
• Extended results on entropy-preserving maps to include an additive constant.

Main Results:

• A generalized duality definition is established, encompassing strong-weak interaction transformations.
• Equivalence of duality is proven for observables, partition functions, and entropies.
• A new class of maps preserving entropy up to an additive constant is introduced.
• These maps decompose into unitary and antiunitary components.

Conclusions:

• The generalized duality framework broadens the applicability of Hamiltonian simulations.
• The characterization of dual maps provides new tools for theoretical physics.
• The mathematical properties of entropy-preserving maps offer independent interest.