Constant-Time Surgery on 2D Hypergraph Product Codes with Near-Constant Space Overhead

TL;DR

These gadgets combine the strengths of different approaches to fault-tolerant logical operations: they partially retain the flexibility of surgery while achieving overheads comparable to transversal gates and demonstrate new possibilities in the design of gadgets for fast logical computation.

Abstract

Generalized code surgery is a versatile and low-overhead technique for performing fault-tolerant computation on quantum low-density parity-check (qLDPC) codes. In many settings, surgery exhibits practical space overheads, while its time overhead remains a bottleneck at $O(d)$ syndrome rounds per operation. In this work, we construct surgery gadgets that perform parallel logical measurements on 2D hypergraph product codes in constant time overhead ($O(1)$) and near-constant space overhead ($\tilde{O}(1)$). The reduced time overhead is a result of amortization, as we show, following the formulation by Cowtan et al. (arXiv:2510.14895), that performing $d$ surgery operations in $O(d)$ time is fault tolerant. Our gadgets combine the strengths of different approaches to fault-tolerant logical operations: they partially retain the flexibility of surgery while achieving overheads comparable to transversal gates. Consequently, they are well-suited for near-term experimental realization and demonstrate new possibilities in the design of gadgets for fast logical computation.

Figures (4)

• Figure 1: Constant-time Surgery. We abstractly represent the spacetime volume of $t$ surgery operations, where time flows from left to right. The thin slices represent switching between a sequence of deformed codes, $\tilde{C}[1]...\tilde{C}[t]$, each performing a different logical measurement. Importantly, we only need to spend a single round in each deformed code. To ensure fault tolerance in the presence of noisy measurements, these $t\geq d$ fast surgery measurements must be preceded and followed by $O(d)$ rounds of syndrome measurements in the base code $C$.
• Figure 2: (a) A classical code, $C_\bullet=C_1\rightarrow C_0$ used in the base hypergraph product code $Q$, $(C\otimes D)_\bullet$. A concrete example, the $[7,4,3]$ Hamming code, is shown with a highlighted logical operator $\bar{Z}$. (b) The augmented classical code, $\mathrm{cone}(g)$, which is the classical code $C_\bullet$ attached to an ancilla complex $G_\bullet: G_1\rightarrow G_0\rightarrow G_{-1}$ through a chain map, $g_i:G_i\rightarrow C_i$. The Tanner graph of the augmented $[7,4,3]$ code is shown. Notably, the product of all $Z$ checks of $G_\bullet$ is equal to the previously highlighted $\bar{Z}$ logical. (c) The final deformed code or coned code, $(\mathrm{cone}(g)\otimes D)_\bullet$, that we switch into to perform fast surgery on $(C\otimes D)_\bullet$. $(\mathrm{cone}(g)\otimes D)_\bullet$ is the tensor product of the augmented code in \ref{['fig:toric-code']}(b) with $D_\bullet$. This deformed code contains meta-checks (in $G_{1}^1$ of the chain complex), which check for measurement errors of $Z$ checks. We show the Tanner graph of a concrete example of a deformed code by taking the hypergraph product of the augmented code example in \ref{['fig:toric-code']}(b) with the dual of the $[3,1,3]$ repetition code. This coned code measures the highlighted logical representatives, $\bar{Z}_1, \bar{Z_2}$ and $\bar{Z}_3$, in parallel.
• Figure 3: (a) A toric code, with qubits as edges, $X$ checks assigned to vertices, and $Z$ checks assigned to faces. $\bar{Z}_1$ (solid blue), $\bar{X}_1$ (dashed orange), $\bar{Z}_2$ (solid green), and $\bar{X}_2$ (dashed red) logical operators are highlighted. (b) $\mathrm{cone}(f[i])$ code for measuring the $\bar{Z}_1$ logical (outlined in light blue), whose minimum-weight representatives are in $(C_1\otimes D_0)$. Notice that $\bar{Z}_2$, highlighted in green, remains in the homology of this filled torus and thus, unmeasured. Gray edges are qubits in the spaces $(C_0\otimes D_1)\oplus (C_1\otimes D_0)$, which are associated with qubits from the base toric code, $(C\otimes D)_\bullet$. Orange edges are qubits in $(G_0\otimes D_0)\oplus (G_{-1}\otimes D_1)$ which are qubits from the ancilla complex $(G\otimes D)_\bullet$. Faces are assigned to $Z$ checks. The faces on the surface of the filled torus are the $Z$ checks of the base toric code. Additional faces have been introduced, including solid blue checks that fill each minimum-weight $\bar{Z}_1$ loop. These checks are in $G_1\otimes D_0$ and multiply to equal a $\bar{Z}_1$ representative (see filled $\bar{Z}_1$ loop on the right). Checks in $G_0\otimes D_1$ are highlighted in gray. Vertices are assigned to $X$ checks. Note that the toric code's $X$ checks are deformed into weight-five checks which include qubits from the ancilla complex (orange edges). (c) Decoding graph of $\partial_{\mathrm{cone}(f[i]),3}=M_Z$, where vertices are meta checks (volumes in (b)) and errors may occur on edges, which are $Z$ check measurements (faces in (b)). In particular, vertical edges are checks in $G_1\otimes D_0$ (blue faces in (b)) and horizontal edges are checks in $G_0\otimes D_1$ (gray faces in (b)). This decoding graph is a square lattice with one periodic boundary condition. The outlined blue edges are $Z$ checks whose product is the blue $\bar{Z}_1$ representative in (b). To flip the logical measurement and invalidate no meta checks, one needs a string error (red) that extends all the way around the decoding graph, and is thererfore a measurement error with weight $d$.
• Figure 4: The complete chain complex of a deformed code $\mathrm{cone}(f[i])=(\mathrm{cone}(g[i])\otimes D)_\bullet$. Spaces are color-coded to match the example in Fig. \ref{['fig:toric-code']}. However, here we explicitly display all the spaces of a general coned code that are not shown in Fig. \ref{['fig:toric-code']}, such as $G[i]_{-1}\otimes D_1$ and $G[i]_{-1}\otimes D_0$.

Theorems & Definitions (40)

• Lemma 1: Main results from williamson2024low-overheadIde_2025
• Definition 1: Boundary Cheeger Constant
• Definition 2: Compacted Code
• Lemma 2: Fast hypergraph surgery, Theorem 5.3 in Ref. cowtan2025fast
• Definition 3: Tensor Product of Chain Complexes
• Definition 4: Hypergraph Product Codes
• Lemma 3: Künneth Formula
• Lemma 4: Equation (51), Theorem 17 of Ref. zeng2020minimal
• Lemma 5
• Proposition 1