A mathematical framework for quantum Hamiltonian simulation and duality

TL;DR

This paper develops a general mathematical framework for quantum Hamiltonian simulation and duality, unifying various duality concepts through axiomatic characterizations involving observables, partition functions, and entropies.

Contribution

It introduces three equivalent axiomatizations of duality in quantum physics and characterizes the form of dualities satisfying these axioms, extending previous entropy-preserving map results.

Findings

Axiomatizations of duality in terms of observables, partition functions, and entropies

Characterization of dualities as mathematical transformations satisfying the axioms

Decomposition of entropy-preserving maps into unitary and anti-unitary components

Abstract

Analogue Hamiltonian simulation is a promising near-term application of quantum computing and has recently been put on a theoretical footing. In Hamiltonian simulation, a physical Hamiltonian is engineered to have identical physics to another - often very different - Hamiltonian. This is qualitatively similar to the notion of duality in physics, whereby two superficially different theories are mathematically equivalent in some precise sense. However, existing characterisations of Hamiltonian simulations are not sufficiently general to extend to all dualities in physics. In particular, they cannot encompass the important cases of strong/weak and high-temperature/low-temperature dualities. In this work, we give three physically motivated axiomatisations of duality, formulated respectively in terms of observables, partition functions and entropies. We prove that these axiomatisations are equivalent, and characterise the mathematical form that any duality satisfying these axioms must take. A building block in one of our results is a strengthening of earlier results on entropy-preserving maps to maps that are entropy-preserving up to an additive constant, which we prove decompose as a direct sum of unitary and anti-unitary components, which may be of independent mathematical interest.