How to Sum and Exponentiate Hamiltonians in ZXW Calculus

Abstract

This paper develops practical summation techniques in ZXW calculus to reason about quantum dynamics, such as unitary time evolution. First we give a direct representation of a wide class of sums of linear operators, including arbitrary qubits Hamiltonians, in ZXW calculus. As an application, we demonstrate the linearity of the Schroedinger equation and give a diagrammatic representation of the Hamiltonian in Greene-Diniz et al, which is the first paper that models carbon capture using quantum computing. We then use the Cayley-Hamilton theorem to show in principle how to exponentiate arbitrary qubits Hamiltonians in ZXW calculus. Finally, we develop practical techniques and show how to do Taylor expansion and Trotterization diagrammatically for Hamiltonian simulation. This sets up the framework for using ZXW calculus to the problems in quantum chemistry and condensed matter physics.

In Proceedings QPL 2022, arXiv:2311.08375

References in corpus (6)

• The algebra of entanglement and the geometry of composition
• Diagrammatic Design and Study of Ansätze for Quantum Machine Learning
• Quantum Machine Learning using the ZXW-Calculus
• A non-anyonic qudit ZW-calculus
• Categorical Semantics for Feynman Diagrams
• Categorical Quantum Dynamics

Cited by in corpus (6)

• Quantum Picturalism: Learning Quantum Theory in High School
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Speedy Contraction of ZX Diagrams with Triangles via Stabiliser Decompositions
• Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computing
• Addition and Differentiation of ZX-diagrams
• ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces