Improved energy barrier in higher-dimensional hypergraph product codes

TL;DR

This work analyzes how confinement and related properties yield energy-barrier bounds for quantum LDPC codes, focusing on higher-dimensional hypergraph product (HHGP) constructions. By decomposing logical operators and exploiting tensor-product structure, the authors derive a lower bound on the energy barrier for LDPC HHGP codes that scales with the distances of the underlying classical codes, improving upon bounds obtained from confinement alone. They first establish a general bound for tensor-product codes, then apply it to 3D HHGP codes to show the Z-type barrier is at least the minimum of the constituent distances and, under LDPC assumptions, tied to the underlying distances; a conjectured bound further strengthens this connection. The 4D HHGP case yields analogous improvements for both X and Z logical operators. Overall, the results indicate that macroscopic energy barriers can arise for HHGP codes even when some constituent codes lack such barriers, highlighting a path toward self-correcting regimes via high-dimensional redundancy and distance properties.

Abstract

Single-shot error correction outperforms conventional approaches by requiring only one round of stabilizer measurements for decoding, even in the presence of measurement errors. This capability relates to the confinement property of codes, which provides an energy barrier lower bound. Earlier research established a confinement property for higher-dimensional hypergraph product (HHGP) codes (Quintavalle et al. 2021 PRX Quantum), yielding an energy barrier lower bound for these codes. In this work, by analyzing the structure of logical operators, we show an improved energy barrier lower bound for HHGP codes with low-density parity-check (LDPC) property. Our bound exceeds results derived from confinement alone, and unlike standard hypergraph product codes, these higher dimensional variants can possess macroscopic energy barriers even when the underlying classical codes lack this property. Specifically, our analysis shows that the energy barrier of LDPC HHGP codes is lower bounded by the distance of the underlying classical codes. This bound is tight if the underlying classical codes exhibit system size-dependent distances but constant energy barriers, like 3D and 4D toric codes.

Figures (2)

• Figure 1: A composite system consisting of a 2D quantum repetition code (Region A) and a 1D quantum repetition code arranged in a snake pattern (Region B). The overall code exhibits an energy barrier determined by the 2D quantum repetition code, which scales as the linear dimension of Region A. However, confinement properties are influenced by the 1D quantum repetition component, resulting in the absence of confinement for the composite system.
• Figure 2: The 3D toric code can be structured as a cubic lattice (with periodic boundary conditions). From left to right, the figure illustrates its decomposition into a-slices, b-slices, and c-slices. The 3D Toric code encodes 3 logical qubits, with corresponding 3 logical Z operators. Each Z elementary canonical logical operator is supported on a single slice, with slices of the same color representing logical operators equivalent up to stabilizers. Generally, given any 3D hypergraph product code, one can create such a decomposition, although the structure within each slice will be more complicated and depends on the underlying classical codes.

Theorems & Definitions (23)

• Definition 1
• Definition 2: Soundness, Definition 3 in campbell2019theory
• Lemma 3: Lemma 3 in campbell2019theory
• Lemma 4: Lemma 5 and Lemma 6 in campbell2019theory
• Definition 5: Confinement, Definition 1 in Quintavalle2021single
• Lemma 6: Theorem 1 in Quintavalle2021single
• Definition 7: Expander graph
• Lemma 8
• proof
• Lemma 9