Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes

TL;DR

The paper introduces non-Abelian qLDPC codes realized as Clifford-stabilizer theories on 2D Poincaré CW complexes, governed by a higher-form twisted gauge theory and a spacetime path-integral formalism based on cup products. It shows that constant-rate 2D hypergraph-product codes can host native non-Clifford gates via addressable gauging measurements, enabling a magic-state fountain that yields about $\Theta(\sqrt{n})$ disjoint CZ magic states in $O(d)$ rounds on $n$ qubits, while maintaining code distance $d = \Omega(\sqrt{n})$. A detailed construction relates thickened and skeleton HGP codes, demonstrates how 0-form subcomplex symmetries give rise to addressable transversal CZ gates, and derives non-Abelian fusion and Borromean ring braiding within the topological framework. The approach unifies topological, combinatorial-TQFT, and LDPC paradigms, suggesting scalable paths to universal fault-tolerant quantum computation with 2D qLDPC architectures and pointing to future extensions to higher dimensions and more general non-Abelian codes.

Abstract

A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical non-Clifford gates, one needs to resort to 3D product-code constructions. In this work, we extend Kitaev's framework of non-Abelian topological codes on manifolds to non-Abelian qLDPC codes (realized as Clifford-stabilizer codes) and the corresponding combinatorial topological quantum field theories (TQFT) defined on Poincaré CW complexes and certain types of general chain complexes. We also construct the spacetime path integrals as topological invariants on these complexes. Remarkably, we show that native non-Clifford logical gates can be realized using constant-rate 2D hypergraph-product codes and their Clifford-stabilizer variants. This is achieved by a spacetime path integral effectively implementing the addressable gauging measurement of a new type of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford gates and become higher-form symmetries when lifted to higher-dimensional CW complexes or manifolds. Building on this structure, we apply the gauging protocol to the magic state fountain scheme for parallel preparation of $O(\sqrt{n})$ disjoint CZ magic states with code distance of $O(\sqrt{n})$, using a total number of $n$ qubits.

Figures (14)

• Figure 1: A diagram summarizing the relations between various concepts appeared in this paper.
• Figure 2: (a) Illustration of a higher-form ($k$-form) symmetry in a $\mathsf{d}$-dimensional manifold or Poincaré CW complex. The symmetry operator is supported on a codimension-$k$ submanifold/subcomplex corresponding to a $(\mathsf{d}-k)$-cycle $\eta^*_{\mathsf{d}-k}$. (b) Illustration of 0-form subcomplex symmetries on a 2D square complex $X$ which has a 'book-like' structure: a purple edge (book hinge) are adjacent to multiple squares (book pages). There can exist a large number of codimension-0 (top dimensional) subcomplexes corresponding to 2-cycles such as $\bar{\eta}^*_2$ and $\bar{\eta}'^*_2$. Each of these 2-cycles can support a symmetry operator, leading to addressable transversal Clifford gate such as $\widetilde{\text{CZ}}_{\bar{\eta}^*_2}$.
• Figure 3: (a) Anatomy of a $k$-handle in $r$ dimensions. (b) A 0-handle in 2D, $D^{0}\times D^{2}$, whose boundary is $S^{1}$. (c) A 1-handle in 2D, $D^{1}\times D^{1}$, with attaching region $D^{1}$ (purple). (d) Attaching the 1-handle to the 0-handle along $D^{1}$ produces a $(0,1)$-handlebody homeomorphic to the annulus $S^{1}\times D^{1}$.
• Figure 4: Plumber’s view of classical codes. (a) A parity check $\mathsf{C}_j$ represented as a vertex (square). (b) The check $\mathsf{C}_j$ is mapped to a “2-handle,” $\mathsf{C}_j = S^{2} \times I \times D^{5}$. (c) A bit $\mathsf{B}_i$ associated with a hyper-edge consisting of multiple edges incident on a single circle. (d) The bit $\mathsf{B}_i$ is mapped to a “3-handle,” obtained from a 3-sphere $S^{3}$ with several 3-disks $D^{3}$ removed and then thickened by $D^{5}$; its multiple legs attach to neighboring 2-handles. (e) A bipartite Tanner graph with check nodes (squares) and bit nodes (circles), equivalently viewed as a hypergraph with checks on vertices and bits on hyper-edges (red). (f) Thickening the Tanner graph yields a handlebody in which 3-handles are attached to adjacent 2-handles according to the boundary map of the Tanner graph, by gluing along the attaching regions $S^{2} \times D^{5}$ of the 3-handles.
• Figure 5: Anatomy of the dressed "3-handle." (a) Realistic three-dimensional representation of the "3-handle". The 3-sphere is viewed as the interior of a 3-ball whose boundary is identified to a single point. Three thickened 3-balls $D^{3}\times D^{5}$ (gray) are removed, leaving thickened 2-sphere boundaries $S^{2}\times D^{5}$. (b) On the dressed core, the "3-handle" contains two 1-handles connecting the three thickened $S^{2}$ boundaries; together with the three removed $D^{3}$ regions, these form a single 3-ball $D^{3}$ (highlighted by pink dashed lines). The complement in the 3-sphere is another 3-ball, $D^{3}=S^{3}\setminus D^{3}$, which becomes the 3-handle upon thickening by $D^{5}$. (c) Deformation retraction of the two 1-handles and the 3-handle to two 1-cells and one 3-cell, respectively, with the dressed "3-handle" retracting to its core. The resulting cell connects to three neighboring "2-cells" $S^{2}$ and three 0-cells $p$ on adjacent "2-handles". The drawing may be further deformed into the abstract representation shown on the right, matching the style of Fig. \ref{['fig:dictionary']}(d).

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof