Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions

Journal Article · Mon Mar 13 00:00:00 EDT 2023 · PRX Quantum

DOI:https://doi.org/10.1103/prxquantum.4.010326· OSTI ID:2423167

Chen, Yu-An ^[1]; Xu, Yijia ^[1]

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1. National Institute of Standards and Technology (NIST), College Park, MD (United States); Univ. of Maryland, College Park, MD (United States)

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 Chen et al., 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 transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice. We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios r = 1 + 1/2k for positive integers k, utilizing the trivialness of quantum cellular automata in two spatial dimensions. Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1. When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional 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.

• Research Organization: National Quantum Information Science (QIS) Research Centers (United States). Quantum Systems Accelerator (QSA)
• Sponsoring Organization: National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); US Army Research Office (ARO); USDOE
• Grant/Contract Number: AC02-05CH11231
• OSTI ID: 2423167
• Journal Information: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 1 Vol. 4; ISSN 2691-3399
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Physics