sebeaumont/hilbert

Hilbert spaces via a language processor for Dirac notation

Dirac Notation for Quantum Theory and Machine Learning

The intent of this project is to provide a language processor that takes Dirac (Bra-Ket) notation expression and can manipulate, either symbolically or via matrix representation, such expressions. There are two main objects in this work; firstly parsing and building a syntax tree of the expressions, which are normally thought of as vectors and operators in a Hilbert space and secondly all the mathematical objects, definitions and results that are entailed.

We also wish to explore at least the SU(2) group of symmetries under the operation of multiplication of unitary matrices and the eigenspace (spectra) of observables which e.g. may be used to represent the evolution of qubits in the unitary quantum theory. Tensor products of state vectors should be supported to represent composite systems up to some practical computational limit.

It may be fruitful to extend this to Hilbert spaces of arbitrary dimension so that more general theories that might be relevant to machine learning and other applications may be explored. Indeed one thesis under exploration is that there is much to share between these two fields of enquiry.

The implementation language is Idris where all required mathematical objects, (vectors, matrices, groups and their algebras) will be defined along with the grammar, language processor, AST and semantics. Which, rather than ad-hoc definitions will leverage the expressiveness of dependent types and mathematical logic in order to rigorously define and state proofs of all relevant results.

This at the same time provides some guarantees of correctness but perhaps more importantly places the entire endeavor under a rigorous framework which allows it to be appreciated and inspected providing a good pedagogical opportunity for both the subject matter and the techniques of type based mathematical logic.

Visualizations of the mathematical objects should also be explored to facilitate understanding and exploration.

---

Simon Beaumont ([email protected]) October 2018.