Single-shot preparation of hypergraph product codes via dimension jump

TL;DR

The paper addresses the challenge of reliably preparing the codespace for constant-rate hypergraph product (HGP) quantum LDPC codes, which is a bottleneck in fault-tolerant implementations. It introduces a two-stage, single-shot protocol that uses a thickened code, formed via a homological product with a classical code, to perform a dimension-jump from a 2D to a 3D complex and then collapses back to the original HGP codespace, enabling constant-depth initialization with $O(√n)$ spatial overhead. The method relies on metachecks and a spacetime-decoding perspective to tolerate measurement and data errors, with a secondary improvement via a star-code variant that reduces spatial overhead further. Numerical simulations provide evidence of fault-tolerance and threshold-like behavior under local stochastic noise for selected LDPC ensembles, and the work outlines pathways to generalize the approach and integrate it with fault-tolerant architectures. Overall, the protocol offers a practical route to fast, error-tolerant preparation of LDPC HGP code states, laying groundwork for scalable, high-speed quantum initialization in architectures capable of long-range connectivity.

Abstract

Quantum error correction is a fundamental primitive of fault-tolerant quantum computing. But in order for error correction to proceed, one must first prepare the codespace of the underlying error-correcting code. A popular method for encoding quantum low-density parity-check codes is transversal initialization, where one begins in a product state and measures a set of stabilizer generators. In the presence of measurement errors however, this procedure is generically not fault-tolerant, and so one typically needs to repeat the measurements many times, resulting in a deep initialization circuit. We present a protocol that prepares the codespace of constant-rate hypergraph product codes in constant depth with $O(\sqrt{n})$ spatial overhead, and we show that the protocol is robust even in the presence of measurement errors. Our construction is inspired by dimension-jumping in topological codes and leverages two properties that arise from the homological product of codes. We provide some improvements to lower the spatial overhead and discuss applications to fault-tolerant architectures.

Figures (8)

• Figure 1: A high-level schematic of the state-preparation protocol is illustrated. The protocol involves fault-tolerant code-switching between the target HGP code and a "thickened" code with enhanced error-correcting properties. Local operations and classical communication (LOCC) in the form of measurements and feedback will be used to correct errors.
• Figure 2: The hypergraph product of two classical linear codes is depicted in a 2D layout. Gray circles and squares denote classical bits and parity checks respectively. Solid black circles, red squares, and blue squares denote qubits, $X$-checks, and $Z$-checks respectively. The connectivity of the resulting HGP code is inherited from its classical codes, as shown for a subset of edges.
• Figure 3: The thickened code \ref{['eq:3D HGP H_X and H_Z']} for the HGP code in Fig. \ref{['fig:2D HGP layout']} with $z=3$ is illustrated. Three copies of the original HGP code are connected (green) by intermediate qubits and $Z$-checks, corresponding to the $(Q,S)$ and $(S_X,X)$ spaces of \ref{['eq:3D HGP complex']}. Intermediate $Z$-checks connect pairs of identified qubits along adjacent HGP sheets, and intermediate qubits connect pairs of identified HGP $X$-checks. The 1D repetition code is overlayed above for reference. To reduce clutter, we only display a subset of intermediate qubits and $Z$-checks.
• Figure 4: The dimensional collapse in stage 2 is depicted. All qubits other than a boundary HGP sheet are measured in the $X$ basis (denoted by lightning bolts). The intermediate qubits adjacent to the boundary sheet are connected by green edges to the boundary $X$-checks.
• Figure 5: The interpretation of the bulk $X$-check Tanner graph as a spacetime code is depicted. Gray rectangles denote HGP sheets, and black circles denote the intermediate qubits connected to each $X$-check in its neighboring HGP sheets. The arrow of time runs from the opposite HGP boundary to the unmeasured one. The combination of each HGP sheet with its subsequent intermediate qubits can be interpreted as a round of noisy syndrome measurement.

Theorems & Definitions (9)

• Definition 2.1: Confinement Quintavalle_2021
• Definition 2.2: Soundness Campbell_2019
• Lemma 1.1: Stage 1 residual $X$ error
• proof
• Lemma 2.1: Lemma 1 of Quintavalle_2021
• Lemma 2.2: Stage 2 residual $Z$ error
• proof
• Theorem 3.1: Full protocol residual error
• proof