Enhanced Fault-tolerance in Photonic Quantum Computing: Comparing the Honeycomb Floquet Code and the Surface Code in Tailored Architecture

TL;DR

This work investigates fault-tolerant quantum computation for photon-mediated hardware by directly comparing the surface code and the honeycomb Floquet code implemented on two spin-optical quantum computing (SPOQC) architectures: CZ-SPOQC and $M_{Z Z}$-SPOQC. Using a unified noise model that includes photon loss, photon distinguishability, and spin decoherence, the study shows a photon loss threshold of $6.3\%$ for the honeycomb Floquet code on the tailored architecture, nearly double the surface code's $3.3\%$ threshold on the previous architecture, while reducing resource requirements. The authors also analyze fault-tolerant regions in loss-decoherence-distinguishability space, finding the honeycomb code offers substantially larger fault-tolerant volumes—demonstrating the value of co-designing quantum error-correcting codes with hardware-native operations. Overall, the results advocate Floquet codes as a promising route for scalable photonic FTQC and highlight the practical benefits of aligning syndrome measurements and error-correcting strategies with the underlying hardware primitives.

Abstract

Fault-tolerant quantum computing is crucial for realizing large-scale quantum computation, and the interplay between hardware architecture and quantum error-correcting codes is a key consideration. We present a comparative study of two quantum error-correcting codes - the surface code and the honeycomb Floquet code - implemented on the spin-optical quantum computing architecture, either with controlled-Z operations or with direct parity measurements. This allows for a direct comparison of the codes using consistent noise models. Notably, we achieve a loss threshold of 6.3% with the honeycomb Floquet code implemented on our tailored architecture, almost twice as high as the loss threshold obtained with the surface code on the previous architecture, all the while requiring less physical qubits. This finding is particularly significant given that photon loss is the primary source of errors in photon-mediated quantum computing. Moreover, we benchmark the general performances of the two codes in a multi-error setting by computing the volume of the fault-tolerant region, and show that the fault-tolerant region of the honeycomb code is over twice as large as that of the surface code.

Figures (11)

• Figure 1: Linear-optical interferometer implementing the unitary $U(\varphi)$ used in the RUS subroutine. Optical mode brought together implement balanced beam-splitters and the light-blue rectangle is a phase-shifter adding a phase $\varphi$. The nature of the operation depends on $\varphi$: if $\varphi=\pi/2$, the target operation is a $\mathsf{CZ}$ gate; if $\varphi=0$, it's an $\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$ measurement of the spins.
• Figure 2: Schematic representation of the RUS subroutine in the presence of physical noise. Step (1) corresponds to the emission of photons, step (2) to the unitary transformation on the photonic modes $U(\varphi)$, and step (3) to photonic detection. (a)$\mathsf{CZ}$-SPOQC architecture. The cycle stops after a $\mathtt{Loss_2}$ detection event, in which case the phases of both spin qubits are lost, or after a Success detection event, in which case the spin qubits get entangled through a $\mathsf{CZ}$ gate with lowered fidelity due to distinguishability. (b)$\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$-SPOQC architecture. The cycle stops after a Success detection event leading to a noisy $\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$ measurement on the spin qubits. For both architectures, decoherence acts throughout the cycle, inducing a continuous dephasing depending on the decoherence time $T_2$. For both architecture, we fix a maximum amount of cycle repetitions $T_{\max}$. If the RUS subroutine did not stop before $T_{\max}$ cycles, the subroutine is aborted. In this case no entanglement is generated.
• Figure 3: Honeycomb Floquet code defined by a sequence of $\mathsf{M}_{\mathsf{X}\mathsf{X}}$, $\mathsf{M}_{\mathsf{Y}\mathsf{Y}}$ and $\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$ measurements on red, green and blue edges of a tri-colorable hexagonal lattice, with architecture layout for the $\textbf{(a)}$$\mathsf{CZ}$-SPOQC and $\textbf{(b)}$$\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$-SPOQC architectures. The white arrows represent quantum emitters and the wavy red arrows represent linear-optical RUS operations between emitters.
• Figure 4: Implementation of an $\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$ measurement for the $\textbf{(a)} \ \mathsf{CZ}$-SPOQC and $\textbf{(b)}$$\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$-SPOQC (right) architectures, where $m$ denotes the measurement result.
• Figure 5: Logical error rate $\varepsilon_L$ as a function of photon loss $\varepsilon$ for different distances in the $\mathsf{CZ}$-SPOQC architecture (pink gradient) and $\mathsf{M}_{\mathsf{Z}\mathsf{Z}}$-SPOQC architecture (blue gradient). Thresholds values are reported on the top of the plots and marked by the black dashed lines. These curves are obtained by averaging out the mismatch between the measure of the logical observable $\mathsf{X}$ obtained through a data readout at the end of the QEC cycles, and the predicted observable by the MWPM decoder higgott2023sparse.