Discovering optimal fermion-qubit mappings through algorithmic enumeration

Mitchell Chiew and Sergii Strelchuk

DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK

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Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. A characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians.

$All$ fermion-qubit mappings must use a numbering scheme for the fermionic modes in order for translation to qubit operations. We make a distinction between the unordered labelling of fermions and the ordered labelling of the qubits. This separation shines light on a new way to design fermion-qubit mappings by making use of the enumeration scheme for the fermionic modes. The purpose of this paper is to demonstrate that this concept permits notions of fermion-qubit mappings that are $optimal$ with regard to any cost function one might choose. Our main example is the minimisation of the average number of Pauli matrices in the Jordan-Wigner transformations of Hamiltonians for fermions interacting in square lattice arrangements. In choosing the best ordering of fermionic modes for the Jordan-Wigner transformation, and unlike other popular modifications, our prescription does not cost additional resources such as ancilla qubits.

We demonstrate how Mitchison and Durbin's enumeration pattern minimises the average Pauli weight of Jordan-Wigner transformations of systems interacting in square lattices. This leads to qubit Hamiltonians consisting of terms with average Pauli weights 13.9% shorter than previously known. By adding only two ancilla qubits we introduce a new class of fermion-qubit mappings, and reduce the average Pauli weight of Hamiltonian terms by 37.9% compared to previous methods. For $n$-mode fermionic systems in cellular arrangements, we find enumeration patterns which result in $n^{1/4}$ improvement in average Pauli weight over naïve schemes.

Understanding the behaviour of fermionic systems is one of the major challenges in physics, chemistry, and material science. Fermions arise in a number of different problem areas from studying complex molecules to theories describing the interactions between the building blocks of our universe – quarks and gluons.

The emerging quantum computers open new avenues for simulating fermionic systems achieving scales that previously were intractable to their classical counterparts. Currently, the task of simulating fermionic systems on a quantum computer requires large overheads due to the inherent non-local nature of interactions. Numerous efforts to reduce the simulation complexity on a quantum device established a tradeoff: they reduce the complexity of simulation at a cost of expending valuable quantum resources such as qubits that scale proportionally to the system size.

We introduce a novel way to reduce the complexity of the simulation by exploiting a new degree of freedom – the way to enumerate fermions. The savings come free-of-charge and only require one to generate a fermion labelling scheme. We provide an optimal scheme for the most common two-dimensional layout – the rectangular lattice. Our method allows for much stronger, polynomial reductions of the overhead for natural classes of practical systems.

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