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Analogue Hamiltonian simulation is a promising near-term application ofquantum computing and has recently been put on a theoretical footing. InHamiltonian simulation, a physical Hamiltonian is engineered to have identicalphysics to another - often very different - Hamiltonian. This is qualitativelysimilar to the notion of duality in physics, whereby two superficiallydifferent theories are mathematically equivalent in some precise sense.However, existing characterisations of Hamiltonian simulations are notsufficiently general to extend to all dualities in physics. In particular, theycannot encompass the important cases of strong/weak andhigh-temperature/low-temperature dualities. In this work, we give threephysically motivated axiomatisations of duality, formulated respectively interms of observables, partition functions and entropies. We prove that theseaxiomatisations are equivalent, and characterise the mathematical form that anyduality satisfying these axioms must take. A building block in one of ourresults is a strengthening of earlier results on entropy-preserving maps tomaps that are entropy-preserving up to an additive constant, which we provedecompose as a direct sum of unitary and anti-unitary components, which may beof independent mathematical interest.

A mathematical framework for quantum Hamiltonian simulation and duality