Equivalence between fermion-to-qubit mappings in two spatial dimensions

• URL: http://arxiv.org/abs/2201.05153v1
• Date: Thu, 13 Jan 2022 18:59:46 GMT
• Title: Equivalence between fermion-to-qubit mappings in two spatial dimensions
• Authors: Yu-An Chen, Yijia Xu
• Abstract summary: We prove the existence of fermion-to-qubit mappings with qubit-fermion ratios $r=1+ frac12k$ for positive integers $k$. In particular, we discover a new super-compact encoding using 1.25 qubits per fermion on the square lattice.
• Score: 5.173245989087371
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Ref. [1], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary (gLU) transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new super-compact encoding using 1.25 qubits per fermion on the square lattice, which is lower than any method in the literature. We prove the existence of fermion-to-qubit mappings with qubit-fermion ratios $r=1+ \frac{1}{2k}$ for positive integers $k$, where the proof utilizes the trivialness of quantum cellular automata (QCA) in two spatial dimensions. When the ratio approaches 1, the fermion-to-qubit mapping reduces to the 1d Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.

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