Energy Barrier in Quantum Error Correcting Codes

Abstract

Quantum error correction is necessary for the building of a useful fault-tolerant quantum computer. However, in practice, error correction requires a large amount of fault-tolerance overheads and fast decoding. One promising approach to overcome this challenge is to use codes with a macroscopic energy barrier, which can be used to build self-correcting quantum memory. So far, rigorous bounds on the energy barrier have typically been derived only for specific codes. Determining a lower bound for the energy barrier of codes is generally challenging due to two reasons: there are too many possible error paths (sequences of local errors) that can implement a fixed logical operator, and the logical operator itself is defined only modulo stabilizers. In this thesis, we prove tight bounds on the energy barrier that are applicable to any quantum code obtained from the hypergraph product of two classical codes. If the underlying classical codes are low-density parity-check codes (LDPC), the energy barrier of the quantum code is shown to be the minimum energy barrier of the underlying classical codes (and their transposes) up to an additive $O(1)$ constant.

Hypergraph product codes, at best, preserve the energy barrier of the underlying classical codes. Constructing a quantum code with a macroscopic energy barrier requires the underlying classical codes to feature an extensive energy barrier. However,

determining the energy barrier for classical codes is challenging, although it is comparatively simpler than for quantum codes because of the absence of stabilizer effects. Utilizing a property of tensor product codes, we demonstrate that higher-dimensional hypergraph products may construct quantum codes with a macroscopic energy barrier, even when the underlying classical codes lack this property. Specifically, the 3D hypergraph product enhances the energy barrier of one type of logical operator.