Universal adapters between quantum LDPC codes

TL;DR

This work develops a universal adapter framework enabling joint logical Pauli measurements across arbitrary quantum LDPC codes and even between different codes, addressing a core challenge in fault-tolerant computation with LDPC codes.The core technical contributions include the SkipTree $\mathsf{basis}$ transformation, a repetition-code adapter that braids multiple code supports into a single sparse graph, and a toric-code adapter that implements addressable CNOT gates via Dehn twists, all while preserving LDPC structure and code distance.The framework achieves quasilinear space overhead in the general case ($\tilde O(td)$ per joint measurement) and reduces overhead for geometrically local codes, with concrete two-code and intra-code examples demonstrating practical overheads and modular reuse of components for multi-code architectures.

Abstract

We propose the repetition code adapter as a way to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks, thus providing a flexible tool for fault-tolerant computation with quantum LDPC codes. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The construction achieves joint logical Pauli measurement of $t$ weight $O(d)$ operators using $O(d)$ time and $\tilde O(td)$ additional qubits and checks, up to a factor polylogarithmic in $d$. As a special case, for some geometrically-local codes in fixed $D\ge2$ dimensions, only $O(td)$ additional qubits and checks are required instead. By extending the adapter in the case $t=2$, we also construct a toric code adapter that uses $O(d^2)$ additional qubits and checks to perform addressable logical CNOT gates on arbitrary LDPC codes via Dehn twists. To obtain these results, we develop a novel weaker form of graph edge expansion and the $\mathsf{SkipTree}$ algorithm, which ensures a sparse transformation between different weight-2 check bases for the classical repetition code.

Figures (20)

• Figure 1: (a) For an arbitrary LDPC code (drawn as a rectangular patch), gauging logical measurement williamson2024gauging of a logical Pauli operator works by attaching a stabilizer state defined on an appropriate auxiliary graph (gray area) to the qubit support of the logical operator (gray circles) creating a deformed code in which the logical operator becomes a stabilizer. Only some graph vertices, which represent checks, connect 1-to-1 to the logical support. This set of vertices is referred to as a port. In this work, we construct additional tools: (b) The ports of several auxiliary graphs can be connected together with carefully chosen adapters (curved edges) while keeping the deformed code LDPC. This measures the product of logical Pauli operators without measuring any individually. Adapters can connect operators in the same codeblock or separate blocks and regardless of their structure. Mathematically, our adapter construction relies on a sparse basis transformation for the classical repetition code, which we call the $\mathsf{SkipTree}$ algorithm. (c) Similarly, an arbitrary LDPC code can be merged with a toric code (or other codes) to perform logical gates.
• Figure 2: Example Tanner graphs with qubit sets drawn as circles and check sets drawn as squares. (a) A generic stabilizer code, (b) a CSS code, and (c) the distance-$d$ toric code. In the toric code, each set of qubits and checks contains $d$ objects.
• Figure 3: The deformed code created during auxiliary graph surgery of logical operator $\overline{Z}$ supported on qubits $\mathcal{L}$. Edge qubits $\mathcal{E}$ are introduced and vertex checks $\mathcal{V}=\{A_v\}_v$ and cycle checks $\mathcal{U}=\{B_c\}_c$ are measured. Here $G$ is the incidence matrix of the graph $\mathcal{G}$ defining these checks, $N$ is a cycle basis satisfying $NG=0$, and $M$ encodes the support gained by some of the original stabilizers $\mathcal{S}$, specifically, those that have $X$-type support on $\mathcal{L}$, see Eq. \ref{['eq:deformed_checks']}. The matrix $F$ has elements $F_{qv}$ for all $q\in\mathcal{L}$ and $v\in\mathcal{V}$ and $F_{qv}=1$ if and only if $f(q)=v$. The rest of the qubits and checks of the original code are drawn on the left but their Tanner graph connectivity does not change during the code deformation.
• Figure 4: An example of the steps going into constructing a graph $\mathcal{G}$ satisfying all the desiderata of Theorem \ref{['thm:graph_desiderata']} starting from a graph $\mathcal{G}_0$ satisfying just desiderata 0, 1, and 2. (a) The graph $\mathcal{G}_0$. (b) The thickened graph $\mathcal{G}^{(L)}_0$, here with $L=3$. Note the edges (drawn lighter gray) connecting the corresponding vertices in adjacent layers. (c) A cycle basis for $\mathcal{G}^{(L)}_0$ includes every length four cycle constructed from edge $e=(i,j)$ in layer $l$, the copy of edge $e$ in adjacent layer $l+1$, and the lighter gray edges connecting the copies of $i$ and $j$. The cycle basis of $\mathcal{G}^{(L)}_0$ also includes (highlighted) one cycle for each cycle in a basis of $\mathcal{G}_0$, and each such cycle can be put into any one of the layers independently. A highlighted cycle is equivalent to its copies in other layers by adding to it the length four cycles between layers. (d) Long cycles in the basis can now be cellulated by adding edges (dashed) and including the resulting triangles in the cycle basis instead.
• Figure 5: An example spanning tree with nodes labeled according to the $\mathsf{SkipTree}$ algorithm, Algorithm \ref{['alg:skiptree']}. The root node is labeled $0$ on the far left, and first-type nodes are represented with solid circles, while last-type nodes are dashed. Path $i$ is the unique shortest path in the spanning tree from node labeled $i$ to the node labeled $i+1(\mathrm{mod}\text{\space}15)$. The proof of Theorem \ref{['thm:skiptree']} involves arguing that all such paths are length three or less. Examples of all cases encountered in that proof are included in this figure -- path 12 for case (1), path 3 for case (2), path 0 for case (3), path 6 for case (4), path 14 for case (a), path 13 for case (b), path 11 for case (c), path 10 for case (d), and path 7 for case (e).

Theorems & Definitions (53)

• Definition 1: Cheeger constant
• Definition 2: Relative Expansion
• Lemma 3
• proof
• Definition 4: Thickening
• Theorem 5: Graph Desiderata
• proof
• Lemma 6: Decongestion Lemma
• Theorem 7
• proof