Encodings of Observable Subalgebras

Simulating complex systems remains an ongoing challenge for classical computers, while being recognised as a task where a quantum computer has a natural advantage. In both digital and analogue quantum simulations the system description is first mapped onto qubits or the physical system of the analogue simulator by an encoding. Previously mathematical definitions and characterisations of encodings have focused on preserving the full physics of the system. In this work, we consider encodings that only preserve a subset of the observables of the system. We motivate that such encodings are best described as maps between formally real Jordan algebras describing the subset of observables. Our characterisation of encodings is general, but notably holds for maps between finite-dimensional and semisimple $C^{*}$-algebras. Fermionic encodings are a pertinent example where a mathematical characterisation was absent. Our work applies to encodings of the the full CAR algebra, but also to encodings of just the even parity sector, corresponding to the physically relevant fermionic operators.