Improved QLDPC Surgery: Logical Measurements and Bridging Codes

TL;DR

This work tackles the high ancilla overhead in fault-tolerant quantum computing with LDPC codes by introducing gauge-fixed QLDPC surgery, which achieves low-space, distance-preserving logical measurements by exploiting Tanner-graph expansion. It develops a modular decoding framework and bridge-based joint measurements to connect different code families, enabling scalable, universal architectures. The approach is validated on the [[144,12,12]] gross code, where a mono-layer ancilla using 103 extra qubits suffices to realize multiple Pauli measurements and Clifford gates with robust fault-tolerance, supported by circuit-level simulations and decoding benchmarks. Together, these advances promise more practical, large-scale fault-tolerant quantum computing with LDPC codes and diverse code families.

Abstract

In this paper, we introduce the gauge-fixed QLDPC surgery scheme, an improved logical measurement scheme based on the construction of Cohen et al. (Sci. Adv. 8, eabn1717). Our scheme leverages expansion properties of the Tanner graph to substantially reduce the space overhead of QLDPC surgery. In certain cases, we only require $Θ(w)$ ancilla qubits to fault-tolerantly measure a weight $w$ logical operator. We provide rigorous analysis for the code distance and fault distance of our schemes, and present a modular decoding algorithm that achieves maximal fault-distance. We further introduce a bridge system to facilitate fault-tolerant joint measurements of logical operators. Augmented by this bridge construction, our scheme can be used to connect different families of QLDPC codes into one universal architecture. Applying our toolbox, we show how to perform all logical Clifford gates on the [[144,12,12]] bivariate bicycle code. Our scheme adds 103 ancilla qubits into the connectivity graph, and one of the twelve logical qubits is used as an ancilla for gate synthesis. Logical measurements are combined with the automorphism gates studied by Bravyi et al. (Nature 627, 778-782) to implement 288 Pauli product measurements. We demonstrate the practicality of our scheme through circuit-level noise simulations, leveraging our proposed modular decoder that combines BPOSD with matching.

Figures (13)

• Figure 1: Diagram of the CKBB scheme for measuring a logical $X$ operator. The original code's Tanner graph is a subgraph of this graph supported on $\mathcal{V},\mathcal{C}^X,$ and $\mathcal{C}^Z$.
• Figure 2: Diagram of the gauge-fixed $X$ ancilla system $\mathcal{G}_X$ with $L = 5$. The original code $\mathcal{G}$'s Tanner graph is a subgraph of this graph supported on $\mathcal{V},\mathcal{C}^X,$ and $\mathcal{C}^Z$. The first layer $(V_1,C_1)$ is called the interface, and the remaining layers and the gauge checks $U_L$ are called the module.
• Figure 3: Space-time diagram and matching graph of a measurement protocol using an $X$ ancilla system. Detectors are denoted by lines connecting different checks and a lightbulb symbol. The errors identified by certain detectors can be decoded separately using a modular decoding approach - see Section \ref{['sec:modular_decoder']}.
• Figure 4: The ancilla system for measuring the joint logical operator $\bar{X}_1\bar{X}_2$, where $\mathop{\mathrm{supp}}\nolimits{\bar{X}_1}=V_0^{(1)}$ and $\mathop{\mathrm{supp}}\nolimits{\bar{X}_2}=V_0^{(2)}$ are disjoint. The joint system is exactly the two systems for measuring $\bar{X}_1$ and $\bar{X}_2$ with the addition of bridge qubits $B$ and gauge checks $U^{B}$. Checks in $C_0^{(1)}\cap C_0^{(2)}$ are each connected to two ancilla qubits, one from $C_1^{(1)}$ and one from $C_1^{(2)}$. Although not strictly necessary, the drawing assumes that the number of layers in the $\bar{X}_1$ and $\bar{X}_2$ systems are the same and the bridge connects the last layers together. In general, the bridge $B$ can be used to connect together any odd layer $j_1$ from the first system and odd layer $j_2$ of second.
• Figure 5: The ancilla system for measuring $\bar{Y}$ where $\bar{X}$ and $\bar{Z}$ overlap on one qubit $q_0$. It largely consists of the ancilla systems for measuring $\bar{X}$ and $\bar{Z}$ separately, but with changes to the first layer, and the addition of the bridge qubits $B$ and more gauge checks $U^B$. The first layer introduces just one check $q_1$ connected to $q_0$ and has sets of checks $V_1^X$ and $V_1^Z$ which are each one smaller than sets in higher layers, e.g. $V_3^X$ and $V_3^Z$.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• proof
• Definition 3: Boundary Cheeger Constant
• Lemma 4: Expansion Lemma
• proof
• Lemma 5
• proof
• Theorem 6
• proof