Architectural Approaches to Fault-Tolerant Distributed Quantum Computing and Their Entanglement Overheads

TL;DR

This work analyzes fault-tolerant distributed quantum computing (FT-DQC) architectures by comparing three principal designs (Type I: GHZ-mediated stabilizers, Type II: distributed planar code patches with seam operations, Type III: inter-block operations via lattice surgery or teleportation) using toric and planar surface codes. It derives how entanglement overhead, measured in Bell-pair generation attempts, scales with code distance $d$ and protocol choice, providing closed-form expressions for $N_{ ext{round}}(d)$ and the GHZ-resource cost under realistic link success $p_{ ext{link}}$, distillation $p_{ ext{distill}}$, and parity-acceptance $p_{ ext{parity}}$ models. The analysis reveals a quadratic $O(d^2)$ overhead for Type I and Type III schemes, while Type II exhibits linear-in-$d$ entanglement costs per syndrome round, highlighting the trade-offs between memory-density, interconnect quality, and fault-tolerance under near-term hardware constraints. These results inform co-design decisions across entanglement generation, code choice, and network protocols to enable scalable FT-DQC with current or near-future technologies.

Abstract

Fault tolerant quantum computation over distributed quantum computing (DQC) platforms requires careful evaluation of resource requirements and noise thresholds. As quantum hardware advances toward modular and networked architectures, various fault tolerant DQC schemes have been proposed, which can be broadly categorized into three architectural types. Type 1 architectures consist of small quantum nodes connected via Greenberger-Horne-Zeilinger (GHZ) states, enabling nonlocal stabilizer measurements. Type 2 architectures distribute a large error correcting code block across multiple modules, with most stabilizer measurements remaining local, except for a small subset at patch boundaries that are performed using nonlocal CNOT gates. Type 3 architectures assign code blocks to distinct modules and can perform fault tolerant operations such as transversal gates, lattice surgery, and teleportation to implement logical operations between code blocks. Using the planar surface code and toric code as representative examples, we analyze how the resource requirements, particularly the number of Bell pairs and the average number of generation attempts, scale with increasing code distance across different architectural designs. This analysis provides valuable insights for identifying architectures well suited to fault tolerant distributed quantum computation under near term hardware and resource constraints.

Figures (8)

• Figure 1: Teleported non-local CNOT using a shared Bell pair. Local CNOTs and complementary-basis measurements implement a remote $\mathrm{CNOT}_{c\to t}$ with only classical feedforward across the link; noisy entanglement link errors propagate asymmetrically ($Z$ errors to the control, $X$ errors to the target).
• Figure 3: Expected entanglement link generation attempts per stabilizer round of each type ($X$ or $Z$) $N_{\mathrm{round}}(d)$, versus code distance $d$ for the $GHZ$-mediated distributed setting. The parameters are $p_{\mathrm{link}}=0.5$ and $p=10^{-2}$ and $p_{distill} = 0.5$.
• Figure 4: Expected entanglement link generation attempts per stabilizer round of each type ($X$ or $Z$) $N_{\mathrm{round}}(d)$, versus depolarizing noise parameter $p$ for the $GHZ$-mediated distributed setting. The parameters are $p_{\mathrm{link}}=0.5$ and $d=100$ and $p_{distill} = 0.5$.
• Figure 5: Schematic showing monolithic and distributed implementations of stabilizer measurements in the toric code with periodic boundary conditions. In the monolithic architecture, each stabilizer generator (either $g^{(Z)}$ or $g^{(X)}$) is measured locally using an ancillary qubit that interacts with four neighboring data qubits. In the distributed implementation, stabilizer measurements are performed by preparing and distributing a $GHZ$ state across the involved nodes, followed by local Pauli measurements and classical communication to complete the nonlocal syndrome extraction. This figure is adapted from Ref. 10.1116/5.0200190.
• Figure 6: Left: Multiple surface code blocks are hosted on separate hardware modules, each connected via a reconfigurable pairwise cross-connect. This switching method enables connections between modules or allows them to be patched together. Right: A quantum operation between two surface code patches residing on different modules is mediated along a one-dimensional boundary or seam. Stabilizer measurements that cross the seam are performed using teleportation-based gates (shown in red). Data (open circles) and syndrome (filled circles) qubits located along this boundary are exposed to higher noise due to their participation in nonlocal entanglement generation. This figure is adapted from Refs. PhysRevA.86.032324Ramette2024-ed.