Defining, classifying and optimising fermion–qubit mappings

Abstract

This thesis presents a comprehensive framework for defining, classifying, and optimising fermion–qubit mappings, a crucial step in simulating fundamental physics on quantum computers. This work surveys the existing catalogue of fermion–qubit mappings, exposing gaps and devising new optimisation routines to aid fermionic simulation through a consistent notation that accounts for all of the mappings in the literature. We unify the distinct operator– and state–based notations for fermion–qubit mappings throughout the field in one definition, providing a versatile description that underpins the entire thesis. Our notation explores the intersection of two significant categories of mappings – Pauli–based mappings, which include ternary tree transformations, and the product–preserving mappings, which include classical, affine, and linear encodings of the Fock basis. Deriving the explicit formula for the Majorana operators of affine encodings shows that they are the intersection of the Pauli–based and classical encodings, while also illustrating the vastness of the set of Pauli–based mappings. We then introduce a classification system to partition the Pauli–based mappings into distinct templates, accounting for trivial labelling symmetries of Pauli operators, fermionic modes, and qubits. The classification allows us to prove that the product–preserving ternary tree transformations are equivalent to a subset of the linear encodings of the Fock basis, uniting two significant, yet disparate areas of active research. We then define optimisation problems for fermion–qubit mappings, resulting in new strategies to increase the efficiency of fermionic simulation. Our classification system helps establish the link between fermionic labelling and typical cost models of qubit Hamiltonians. For the Jordan–Wigner transformation, the optimal fermionic labellings for cost functions of Pauli weight correspond to solutions of well–known problems in graph theory. We use the solution to the edgesum problem to find the optimal Jordan–Wigner transformation for square fermionic lattices, revealing a latent reduction of 13.9% in the average Pauli weight of the resulting qubit Hamiltonian. Finally, in pursuit of further efficiency, we extend our overarching definition to include ancilla–qubit mappings, introducing a method for incrementally adding ancilla qubits to further reduce the Pauli weight of qubit Hamiltonians. We demonstrate that just two extra qubits can achieve a 37.9% reduction in average Pauli weight versus the standard ancilla–free approach, or 27.9% over our optimal ancilla–free strategy. In conclusion, this thesis addresses key mathematical and notational gaps in the field and offers a new perspective for designing cost--effective fermion--qubit mappings for fermionic simulations.