A Sierpinski Triangle Fermion-to-Qubit Transform

TL;DR

This paper introduces a new fermion-to-qubit encoding based on the Sierpinski tree data structure, which preserves operator locality and encodes fermionic states as computational basis states, enhancing quantum simulation efficiency.

Contribution

It establishes a novel correspondence between classical data structures and fermion-to-qubit transforms, introducing the Sierpinski tree encoding for quantum simulations.

Findings

Matches the operator locality of ternary tree encoding

Encodes fermionic states as computational basis states

Provides a new classical data structure analogy for fermion-to-qubit transforms

Abstract

In order to simulate a system of fermions on a quantum computer, it is necessary to represent the fermionic states and operators on qubits. This can be accomplished in multiple ways, including the well-known Jordan-Wigner transform, as well as the parity, Bravyi-Kitaev, and ternary tree encodings. Notably, the Bravyi-Kitaev encoding can be described in terms of a classical data structure, the Fenwick tree. Here we establish a correspondence between a class of classical data structures similar to the Fenwick tree, and a class of one-to-one fermion-to-qubit transforms. We present a novel fermion-to-qubit encoding based on the recently discovered "Sierpinski tree" data structure, which matches the operator locality of the ternary tree encoding, and has the additional benefit of encoding the fermionic states as computational basis states. This is analogous to the formulation of the Bravyi-Kitaev encoding in terms of the Fenwick tree.