Modular Architectures and Entanglement Schemes for Error-Corrected Distributed Quantum Computation

TL;DR

This work evaluates fault-tolerant, error-corrected distributed quantum computation using solid-state modular hardware. It compares emission-based and scattering-based GHZ-generation schemes within two toric-code-based modular architectures (WT4 and WT3) and links hardware parameters to logical error rates through hardware-tailored noise models and superoperator QEC simulations. Scattering-based entanglement schemes deliver higher thresholds (up to ~0.35–0.40% in FP), while emission-based methods struggle under near-term parameters, guiding design choices for scalable, distributed quantum processors. The study also provides a concrete methodology for threshold estimation and highlights the importance of GHZ fidelity (>99%) and GHZ-success probability (>10^-4) as gatekeepers for practicality, with implications for future BB-code alternatives and gated logical operations.

Abstract

Connecting multiple smaller qubit modules by generating high-fidelity entangled states is a promising path for scaling quantum computing hardware. The performance of such a modular quantum computer is highly dependent on the quality and rate of entanglement generation. However, the optimal architectures and entanglement generation schemes are not yet established. Focusing on modular quantum computers with solid-state quantum hardware, we investigate a distributed surface code's error-correcting threshold and logical failure rate. We consider both emission-based and scattering-based entanglement generation schemes for the measurement of non-local stabilizers. Through quantum optical modeling, we link the performance of the quantum error correction code to the parameters of the underlying physical hardware and identify the necessary parameter regime for fault-tolerant modular quantum computation. In addition, we compare modular architectures with one or two data qubits per module. We find that the performance of the code depends significantly on the choice of entanglement generation scheme, while the two modular architectures have similar error-correcting thresholds. For some schemes, thresholds nearing the thresholds of non-distributed implementations ($\sim0.4 \%$) appear feasible with future parameters.

Figures (30)

• Figure 1: Weight-4 toric code architecture with one data qubit per module. (a) The structure of a module. It consists of a single communication qubit that allows for optical connection to the other modules and some memory qubits acting as either data qubits of the code or auxiliary qubits for the stabilizer measurements. (b) Multipartite GHZ states used for measuring the stabilizers that spread among modules. (c) One round of $Z$ and $X$ type stabilizer measurements, each divided into two subroutines. The four colors of the squares in the toric-code lattice correspond to the four subroutines as shown in the sequence. The toric-code lattice encodes two qubits with logical operators ($L_{1,2}^{X, Z}$). The corresponding blue and orange lines indicate the qubit involved in implementing the logical operators. (d) Quantum circuits for implementing the distributed $Z$ and $X$ stabilizer measurement. A GHZ state is generated upon demand (star signs) and used to measure the joint parity $ZZZZ$ or $XXXX$ on the stabilizer data qubits via the application of local controlled-$Z/X$ gates. The outcome of the stabilizer measurement is the joint parity outcome from the four measurements, i.e., $m_{X/Z}=m_1\cdot m_2\cdot m_3 \cdot m_4$. (e) Time layers of the stabilizer measurement. The stabilizer measurements are repeated $d$ times, the same as the distance of the square toric code. These $d$ layers constitute one full QEC cycle, and all syndrome data is then sent to a decoder. Thereafter, suitable corrections are applied to the qubits.
• Figure 2: Weight-3 toric code architecture of code distance $d=4$ with two data qubits per module. (a) One round of stabilizer measurements, consisting of measurements for $X$ and $Z$ type stabilizers, each containing four subroutines. The logical qubits of the code are the same as for the WT4 architecture, as shown in Fig. \ref{['fig:weight_4_architecture']}(c). (b) Quantum circuits for the stabilizer measurement. The notation is the same as in Fig. \ref{['fig:weight_4_architecture']}
• Figure 3: Emission-based (EM) scheme. (a) An optical setup with two emitters as communication qubits can be initialized in a CQ-photon entangled state. Then, the emitted photons ($\text{c}_\text{ph}$) are sent to a middle station where an optical Bell measurement is performed using a balanced beam splitter and single photon detection. The detection of an early and a late photon at the detectors successfully heralds a Bell pair between the two CQs ($c_i$). (b) The energy levels of the photon-CQ system. The CQ has two sets of independent transitions: $E_{0}\leftrightarrow E_{{\rm 0e}}$ and $E_{1}\leftrightarrow E_{{\rm 1e}}$ with transition frequencies $\Delta_0$ and $\Delta_1$, respectively. Here, $E_1$ refers to the energy level of the bright-state $|1\rangle$ that emits a photon upon excitation.
• Figure 4: Principles of the reflection (RFL) scheme. (a) System setup of the RFL scheme. It contains a series of atoms, each embedded within a single-side cavity. The fibers and circulators guide the photon to interact with each atom-cavity system successively. The delay loop, beam splitter, and photon detectors are used to detect the photonic qubit in the X basis. (b) Quantum circuit that the reflecting protocol implements. It contains cnot gates and a photon measurement in its X basis. (c) Physical implementation for the cnot gate. It contains four steps, including scattering the photon of the two time bins and implementing Hadamard gates on the atom. (d) The energy levels of the photon-atom-cavity system. The spin has two sets of independent transitions coupled to the input photon: $E_{0}\leftrightarrow E_{{\rm 0e}}$ and $E_{1}\leftrightarrow E_{{\rm 1e}}$. $\omega_{\rm c}$: cavity resonance frequency; $\omega_{\rm ph}$: input photon frequency.
• Figure 5: Principle of the carving (CAR) scheme. (a) System setup of the CAR protocol. It consists of spins embedded inside a two-sided cavity (or waveguide), a light source, beam-splitters, and photon detectors. (b) The operating sequence of the CAR protocol. All the spins are initialized in $|+\rangle$ states. The operating sequence consists of scatterings of single photons through the optical circuit and not gates implemented on spins between every two sequential scattering rounds.