Parsimonious Quantum Low-Density Parity-Check Code Surgery

TL;DR

This work introduces a method to construct an ancilla system of qubit size to measure an arbitrary logical Pauli operator of weight in any qLDPC stabilizer code, which immediately reduces the asymptotic overhead across various quantum code surgery schemes.

Abstract

Quantum code surgery offers a flexible, low-overhead framework for executing logical measurements within quantum error-correcting codes. It encompasses several fault-tolerant logical computation schemes, including parallel surgery, universal adapters and fast surgery, and serves as the key primitive in extractor architectures. The efficiency of these schemes crucially depends on constructing low-overhead ancilla systems for measuring arbitrary logical operators in general quantum Low-Density Parity-Check (qLDPC) codes. In this work, we introduce a method to construct an ancilla system of qubit size $O(W \log W)$ to measure an arbitrary logical Pauli operator of weight $W$ in any qLDPC stabilizer code. This new construction immediately reduces the asymptotic overhead across various quantum code surgery schemes.

Figures (9)

• Figure 1: Parsimonious Ancilla System. (a) An example of an induced graph of a logical operator. (b) The ancilla system obtained via the decongestion lemma + thickening with height $O(\log^3 |V|)$ such that cycles are now contractible due to the faces (colored) and the congestion per edge is controlled to remain constant. (c) The ancilla system obtained via the parsimonious cone. Specifically, two binary trees are constructed, based on the local structure of edges (top) and vertices (bottom). Shuffles of $O(\log |V|)$ binary trees induce faces (depicted in color and grey) so that the local views are paired up to reveal the global structure. Specifically, any cycle in (a) is embedded in (c) by traversing between the top and bottom binary trees so that the boundaries of the shuffling faces correspond to a cycle in (a) (e.g., colored and grey). Hence, cycles in (a) are products of local, low-density faces in (c).
• Figure 2: Coning. (a) depicts the measurement graph $\sG$ of an $X$-type logical with vertices and edges depicted by dots and lines, respectively. (b) depicts the cone of $\sG$, which is a cell complex obtained by adding an extra vertex (red), and connecting the vertex to all vertices in the original graph. All vertical faces (shaded red) are also added to the cell complex.
• Figure 3: Interpolation Cone. (a) Left $T'$ and right $T$ binary trees denotes those with labels $\bm{\tau}'=\bm{\sigma\tau}, \bm{\tau}$, respectively, where $\bm{\sigma}$ is a SWAP, while the middle denotes the interpolation graph complex $S$ of $T',T$. The vertices of $T',T$ are labeled by $|s'\rangle,|s\rangle$ with bit strings $s$, respectively. Note that, e.g., $|1'0'\rangle = |\tau'(01)\rangle$. (b) denotes the graph $G$ from which $C$ is constructed as described in Theorem \ref{['thm:parsimonious-cone']}.
• Figure 4: Pruned Interpolation Cone. In comparison to Fig. \ref{['fig:interpolation-cone']}, the green branches are not necessary to embed the (subdivision of) graph $G$ within $C$ and are thus pruned.
• Figure 5: Pruning*. The colored leaves form the set $\sL$. (a) denotes a binary tree with trivial labels, and the bit strings denote the vertices within the tree. (b) denotes a $\sL$-pruned* binary tree with trivial labels, and the bit strings denote the remaining vertices. In particular, the vertex $|0\rangle$ in (a) is absent in (b) so that the edge $\|00\rangle$ in (b) has endpoints $|00\rangle$ and $|\varnothing\rangle$.

Theorems & Definitions (77)

• Definition 2.1: Measurement Graph
• Theorem 2.2: Parsimonious Cone
• Theorem 2.3: Deformed Code
• proof
• Definition 3.1
• Definition 3.2: Basis
• Definition 3.3: Cocomplex
• Example 3.1: CSS Codes
• Definition 3.4: Systolic Distance
• Definition 3.5: Chain Map