Fault-tolerant modular quantum computing with surface codes using single-shot emission-based hardware

TL;DR

The paper tackles scalable fault-tolerant quantum computation by leveraging fully distributed surface codes with single-shot emission-based entanglement. It introduces four-module, single-shot GHZ generation schemes and two distillation families (memory-based and optical) to realize high-fidelity stabilizer checks across a modular network, supported by realistic color-center hardware noise models. The key result is that direct optical protocols, especially DC‑GHZ, achieve fault-tolerance thresholds up to about $p_{ ext{th}} \,\approx\, 0.25\%$ (PNR) or $0.20\%$ (non‑PNR), outperforming prior Bell‑pair fusion approaches, and enabling scalable modular quantum computing with modest hardware upgrades. This work provides a concrete, hardware-grounded pathway to fault-tolerant distributed quantum computing, including detailed threshold analyses, cut-off optimizations, and clear directions for improving indistinguishability, detection efficiency, and coherence times.

Abstract

Fault-tolerant modular quantum computing requires stabilizer measurements across the modules in a quantum network. For this, entangled states of high quality and rate must be distributed. Currently, two main types of entanglement distribution protocols exist, namely emission-based and scattering-based, each with its own advantages and drawbacks. On the one hand, scattering-based protocols with cavities or waveguides are fast but demand stringent hardware such as high-efficiency integrated circulators or strong waveguide coupling. On the other hand, emission-based platforms are experimentally feasible but so far rely on Bell-pair fusion with extensive use of slow two-qubit memory gates, limiting thresholds to $\approx 0.16\%$. Here, we consider a fully distributed surface code using emission-based entanglement schemes that generate GHZ states in a single shot, i.e., without the need for Bell-pair fusions. We show that our optical setup produces Bell pairs, W states, and GHZ states, enabling both memory-based and optical protocols for distilling high-fidelity GHZ states with significantly improved success rates. Furthermore, we introduce protocols that completely eliminate the need for memory-based two-qubit gates, achieving thresholds of $\approx 0.19\%$ with modest hardware enhancements, increasing to above $\approx 0.24\%$ with photon-number-resolving detectors. These results show the feasibility of emission-based architectures for scalable fault-tolerant operation.

Figures (14)

• Figure 1: Modular toric surface code. (a) A direct translation of the monolithic surface code of distance $d=4$ into a modular architecture, where each module (teal circles) hosts a data qubit and a communication qubit (dark blue and yellow circles in (b), respectively). Each stabilizer spans four modules, which share GHZ states generated using the proposed single-shot emission-based protocols. As each module has only one communication qubit (yellow), simultaneous measurement of neighboring stabilizers is not possible. Thus, the stabilizers are measured in four sequential rounds $R^{X,Z}_{1,2}$, as shown. Logical operators are also shown for both encoded logical qubits. (b) Distributed stabilizer measurement circuits using a GHZ state in the form $\ket{\Phi^+_4}=\tfrac{1}{\sqrt{2}}(\ket{0000}+\ket{1111})$. The stars (in yellow) connected via links denote entanglement generation on those communication qubits, either a fresh GHZ state or a distilled GHZ state using distillation protocols. The joint parity is extracted by local operations within each module, followed by classical post-processing. The measurement outcomes for the stabilizer syndrome are calculated as products of individual outcomes. (c) Energy level diagram of the communication (emitter) qubit, showing the ground states $E_0$ and $E_1$ (corresponding to dark and bright states) and the excited states $E_{0e}$ and $E_{1e}$, along with their associated linewidths. For the noise model (see App. \ref{['app:Noise_model']}), this is the emitter model we have followed.
• Figure 2: Hardware setup for the emission-based scheme that generates entanglement links—Bell pairs, W states, and GHZ states of weight-4. Emitter (communication) qubits $c_1,c_2,c_3,c_4$ are excited with a laser pulse to emit photons. These photons propagate through a network of beam splitters (stage-1 and stage-2), which mix the modes of all emitted photons. We observe detection patterns as photon counts at detectors $\text{D}_1,\text{D}_2,\text{D}_3,\text{D}_4$ or at detectors $\text{D}_1',\text{D}_2',\text{D}_3',\text{D}_4'$ depending on which states are created. Post-selecting for specific patterns generates the target states. Creation operators for photonic number states $\hat{p}_i^{\dagger},\hat{q}_i^{\dagger}$ and $\hat{r}_i^{\dagger}$ ($i=1,2,3,4$) are indicated at various input and output ports of the beam splitters.
• Figure 3: Quantum circuits for memory-assisted distillation protocols. Communication qubits (yellow wires) act as photon-emitting qubits, which create entangled states, while auxiliary memory qubits (teal wires) assist with the storage of entangled states for the memory-assisted distillation. Elementary entanglement links are indicated by starred connections, labeled with the raw state produced (GHZ, Bell, or W). Post-selection is performed on measurement outcomes $m_1,m_2,m_3,m_4$ with the indicated valid patterns, and the protocols are executed in the RUS manner as shown in the sub-figures. (a) Bell $\to$ GHZ distillation: a base GHZ link is distilled with auxiliary Bell pairs. (b) $\bm{W} \to$ GHZ scheme: a base GHZ link is distilled with auxiliary $W$ state. (c) $\bm{W} \to \bm{W}$ distillation: two raw $W$ states are locally rotated and distillation projects the base state into a GHZ state. (d) GHZ $\to$ GHZ distillation: two raw GHZ states are created, and the new one is used to distill the prior one.
• Figure 4: GHZ state infidelity ($1-F_\mathrm{GHZ}$) for varying bright-state parameter ($\alpha$) for ES-2 hardware parameter set, coherence times $T=10^6$, and physical error-rate of $p=10^{-3}$.
• Figure 5: Success rates for GHZ state ($P_\mathrm{succ}$) for varying bright-state parameter ($\alpha$) for ES-2 hardware parameter set, coherence times $T=10^6$, and physical error-rate of $p=10^{-3}$.