QUANTUM ERROR CORRECTION FOR LOGICAL FERMIONIC AND BOSONIC SYSTEMS

Abstract

Robust storage and manipulation of quantum information in realistic quantum devices remains one of the central challenges in realizing practical quantum computation. To resolve this problem, the quantum error correction (QEC) is proposed as a technique to perform robust encoding and operations in noisy and realistic quantum devices. In the quantum realm, two fundamentally different types of particles—fermions and bosons—exhibit distinct behaviors. This dissertation explores two directions of QEC tailored to these particle types: (1) encoding logical fermionic modes into physical qubits, and (2) homological encoding of logical bosonic systems into physical bosonic systems.

The first part is about encoding logical fermionic systems into physical qubit systems, a process known as fermion-to-qubit mapping. These mappings arise in a variety of contexts, including condensed matter physics, high-energy physics, and quantum information more recently. While many encoding schemes have been proposed over the past two decades, the relationships among them have remained unclear. Using the tools from quantum error correction and topological orders, it can be shown that various known fermion-to-qubit mappings in two dimensions are equivalent up to Clifford deformations. This work is based on Ref.~\cite{chen2022equivalence}.

The second part is about bosonic codes, that encodes logical bosonic systems with finite and infinite dimensional Hilbert space into physical bosonic systems with infinite dimensional Hilbert space. The first half of this section investigates connections between two important bosonic systems: quantum oscillators and quantum rotors. By compactify a real line $\mathbb{T} \cong \mathbb{R}/2\pi\mathbb{Z}$, a logical rotor can be encoded in a physical oscillator which corresponds to a partial Gottesman-Kitaev-Preskill (GKP) encoding. Conversely, the Fock basis of an oscillator (indexed by $\mathbb{N}$) can be embedded into the angular momentum space of a rotor (indexed by $\mathbb{Z}$). These two embeddings allow us to unify and relate various bosonic codes, including the GKP codes on rotors and oscillators, homological rotor codes, and rotation-symmetric bosonic codes. This work is presented in Ref.~\cite{xu2024multimode}.

The second half of this section addresses the open problem of systematically constructing bosonic quantum codes, which is more challenging than for qubit systems due to the infinite-dimensional Hilbert space. By generalizing $\mathbb{Z}_2$ homology to integer homology, a family of bosonic quantum codes, called tiger codes, is introduced. The codewords of tiger codes are continuous superpositions of bosonic coherent states, forming stripe patterns in the phase space torus. The tiger code framework encompasses a variety of known bosonic codes, such as two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and $\chi^{(2)}$-like quantum codes. Moreover, several tools from $\mathbb{Z}_2$-homology can be extended to the integer domain, including hypergraph-product codes, which are widely used to construct low-density parity-check codes. Using the hypergraph-product over integer, a topological bosonic code is constructed, which is not a concatenation between few-mode bosonic code and qubit topological code. This work is based on Ref.~\cite{xu2024letting}.