A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the $N^{1/3}$ distance barrier

TL;DR

It is suggested that it is feasible to apply native logical non-Clifford gates on qLDPC codes or directly inject high-fidelity magic states as resources ('magic state fountain') without the distillation process.

Abstract

We develop a topological theory for fault-tolerant quantum computation in quantum low-density parity-check (qLDPC) codes. We show that there exist hidden simplicial or CW complex structures encoding the topological data for all qLDPC and CSS codes obtained from product construction by generalizing the Freedman-Hastings code-to-manifold mapping. This is achieved by building manifolds from the Tanner graphs of the skeleton classical or quantum codes, which further form a product manifold and an associated thickened product code defined on its triangulation. One can further deformation retract the manifold back to a CW complex which supports a non-topological code with minimal overhead suitable for near-term implementation. Both types of codes admit cohomology operations including cup product which can induce non-Clifford gates. When applying this mapping to a 3D hypergraph product code obtained from the product of 3 copies of good classical expander codes, we obtain non-Clifford logical CCZ gates via constant depth circuits on a code with constant stabilizer weight $w=O(1)$, constant rate $K=Θ(N)$, and polynomial distance $D=Ω(N^{1/3})$. When applied to logical CCZ on 3D homological product codes consisting of the product of a pair of good quantum and classical LDPC codes, we can further improve the distance to $D=Ω(\sqrt{N})$ exceeding the $N^{1/3}$ distance barrier implied by the Bravyi-König bound for conventional topological codes with the aid of non-Euclidean geometries. Our work suggests that it is feasible to apply native logical non-Clifford gates on qLDPC codes or directly inject high-fidelity magic states as resources ('magic state fountain') without the distillation process. For the homological product construction, the fountain can inject $Θ(\sqrt{N})$ magic states in parallel in a single round.

Figures (20)

• Figure 1: Several 'higher structures' across the area of physics, computer science and mathematics are connected through this work.
• Figure 2: Cup product definition on simplicial complexes, where the arrows point from vertices with lower order to vertices with higher order. (a) Cup product of 1-cochains $a \cup a'$ on 2D simplicial complex. One takes product of cocycle values on the red and blue edges respectively in each 2-simplex (triangle). (b) Cup product of 1-cochains $a \cup a' \cup a"$ on a 3D simplicial complex. One takes the product of cocycle values on the red, blue, and green edges respectively in each 3-simplex (tetrahedron).
• Figure 3: (a) Illustration of the cup product and intersection in a 2D triangulation $\mathcal{L}$. The 1-cocycle $\alpha^1$ (blue) and $\beta^1$ (green) corresponds to the support of the logical-$X$ operators $\overline{X}^{(1)}_{\alpha^1}$ and $\overline{X}^{(2)}_{\beta^1}$. The cup product $\alpha^1 \cup \beta^1$ evaluates non-trivially only on the highlighted (grey) triangle, hence the sum $\int_{\mathcal{L}}\alpha^1 \cup \beta^1=1$. This corresponds to a non-trivial intersection of the Poincaré dual 1-cycles $\alpha^*_1$ and $\beta^*_1$ (dashed lines) on the dual triangulation $\mathcal{L}^*$. (b) The triple cup product sum $\int_{\mathcal{L}} \alpha^1 \cup \beta^1 \cup \gamma^1=1$ on a 3D triangulation $\mathcal{L}$, which corresponds a non-trivial triple intersection of the Poincaré dual 2-cycles (membranes) $\alpha^*_2$, $\beta^*_2$ and $\gamma^*_2$. This can also be understood as the triple intersections of the logical-X membranes $\overline{X}^{(1)}_{\alpha^1}$, $\overline{X}^{(2)}_{\beta^1}$, and $\overline{X}^{(3)}_{\gamma^1}$.
• Figure 4: (a) A toy example of a repetition code $\bar{\mathcal{C}}$ with two bits and two checks. One of the checks is redundant. The codeword of both $\bar{\mathcal{C}}$ and $\bar{\mathcal{C}}^T$ are highlighted, labeled as $\bar{\mathsf{a}}_1$ and $\bar{\mathsf{b}}^0$ respectively. (b) The check is mapped to a "1-cell" $S^1$ and the bit is mapped to a "2-cell" $D^1 \times S^1$. (c) When attaching the "2-cells" to the "1-cell" one obtains a torus. The codeword of $\bar{\mathcal{C}}$ is mapped to $\mathsf{a}_2$ while the codeword of $\bar{\mathcal{C}}^T$ is mapped to $\mathsf{b}^1 \sim \mathsf{b}^*_1$. There exist a spurious 1-cocycle $\mathsf{d}^1 \sim \mathsf{d}^*_1$. (d, e) An illustration of a general repetition code defined on a circle mapped to a torus. The spurious 1-cocycle $\mathsf{d}^1 \sim \mathsf{d}^*_1$ has only $O(1)$ size and will hence decrease the distance of the transposed code $\mathcal{C}^T$. (f) Map check to a "2-cell" $S^2$, and bit to a "3-cell" $D^1 \times S^2$. (g) Attaching the cell together give rise to $S^2 \times S^1$. The codeword of $\bar{\mathcal{C}}$ and $\bar{\mathcal{C}}^T$ are mapped to 3-cycle $\mathsf{a}_3$ and 2-cocycles $\mathsf{b}^2$ respectively, which have separated dimensions with the spurious 1-cocycle $\mathsf{d}^1$. (h) Mapping for the general repetition code. The spurious 1-cocycle $\mathsf{d}^1$ does not affect the distance of the transposed code $\mathcal{C}^T$.
• Figure 5: (a) Anatomy of the $k$-handle in $r$ dimension. (b) A 0-handle in 2D $0 \times D^2$ with no attaching region. The boundary is $S^1$. (c) The 1-handle $D^1 \times D^1$ in 2D with attaching region $D^1$ (purple). (d) Attach the 1-handle to the boundary of the 0-handle along the attaching region $D^1$ (purple). This gives rise to a (0,1)-handlebody homeomorphic to an annulus $S^1 \times D^1$.

Theorems & Definitions (38)

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