Physical (or "analogue" - from analogous) Hamiltonian simulation involves engineering a Hamiltonian of interest in the laboratory, and studying its properties experimentally (somewhat analogous to building a model of an aeroplane and studying it in a wind tunnel). This is often touted as the most promising near-term application of quantum computing technology, because (it is claimed) it does not require a scalable, fault-tolerant quantum computer. Indeed, Hamiltonian simulation is already being done in the laboratory.
Despite this, the theoretical basis for Hamiltonian simulation is surprisingly sparse. Even a precise definition of what it means to simulate a Hamiltonian was lacking! By drawing on techniques ranging from Hamiltonian complexity to Jordan algebra homomorphisms, we put analogue Hamiltonian simulation on a sound theoretical footing for the first time.
This is far more fruitful than a mere mathematical tidying-up exercise. In recent work with Gemma de las Cuevas [Science 2016], we used this new theoretical insight to prove that there exist universal spin models that are capable of simulating any other classical spin system, and showed that the simple 2D Ising model with fields is such a universal model. In very recent work with Ashley Montanaro and Stephen Piddock [appearing on the arXiv shortly], we extended these results to the more challenging quantum case, first constructing a rigorous theory of quantum Hamiltonian simulation, then proving that simple spin models such as the 2D Heisenberg model are universal quantum Hamiltonians. Along the way, we take a first step towards proving that error correction is not necessary in Hamiltonian simulation. As well as the obvious practical applications, our results have deep implications for the (non-)role of symmetries, spatial dimension and locality in physics.