Loss-tolerant distributed lattice surgery using fusion networks

The gist Scaling quantum computers to practically relevant logical qubit counts at low error rates requires more qubits than are typically expected to fit on a single quantum processing unit (QPU) [1–4]. This has motivated interest in distributed quantum computing (DQC) as a model for scalable fault-tolerant quantum computation (FTQC), in which multiple QPUs containing matter-based qubits are connected via a photonic network [5–7].

Hybrid Protocols and Interface Distance

The paper constructs hybrid syndrome extraction protocols using the ZX calculus to perform distributed lattice surgery between distributed logical qubits [66]. The results demonstrate that under negligible local noise, independent fusion outcome erasure thresholds of 50% are achievable in the network for distributed lattice surgery, both with an interface mediated by fused Bell pairs and with a cluster-state interface decomposed into a fusion network of linear-chain resource states [Page 31]. Relative to a straight Bell-pair interface geometry, the hybrid protocols increase the merge-observable interface distance from d + 1 to 2d + 1 and restore the perpendicular-observable interface distance from ⌊(d + 1)/2⌋ to d [Page 31].

Resource State Decomposition and Truncation

The paper describes decomposing cluster states into smaller resource states connected by linear-optical fusion measurements, which is termed fusion-based quantum computing (FBQC) [25]. The construction of a hybrid lattice surgery diagram involves clusterising the merge region to produce a fusion network, where the exact resource state decomposition depends on how the incident legs are partitioned [Page 31]. Truncating the linear resource states to a fixed size leaves both interface distances unchanged while lowering the erasure threshold predictably, to 38% for eight-qubit chains and 32% for five-qubit chains [Page 31].

Error Models and Thresholds

The paper introduces several error models, including a heterogeneous error model that combines circuit-level depolarising noise on matter qubits with resource-state errors, photon loss, and fusion failures [Page 31]. The interface detector error model (DEM) is constructed by allowing erasures only on elementary linear-optical fusion outcomes, where each elementary fusion outcome is erased independently with probability pe [Page 31]. The analysis shows that the asymptotic erasure threshold for Bell, BellZigZag, and 3d-Chain protocols under pure erasure noise is 50% [Page 21].

Subthreshold Performance and Noise Regimes

The subthreshold simulations reveal a crossover between two regimes: at pc = pr = 10−3, local errors largely mask the interface-distance differences, with the 3d-Chain giving the strongest suppression of the merge observable though failing to improve over BellZigZag for the perpendicular observable [Page 27]. At lower local noise, the linear-chain suppression factors rise more rapidly as erasure is reduced, exposing the augmented interface distance [Page 27].

Comparison of Geometries

The study compares different interface geometries:

1. Straight interface: The Bell protocol has an interface distance of d + 1 for the merge observable and ⌊(d + 1)/2⌋ for the perpendicular observable [Page 30].

2. Zig-zag interface: The BellZigZag protocol restores the perpendicular-observable distance from ⌊(d + 1)/2⌋ to d while leaving the merge-observable distance at d + 1 [Page 30].

Conclusion and Future Work

The work provides an initial demonstration of the benefits of hybrid syndrome extraction protocols, highlighting the potential to tailor error correction schemes towards dominant noise sources such as photon loss [Page 31]. The principal local-noise limitation of the hybrid interfaces arises from the deterministic boundary fusions between the circuit and resource-state regions [Page 28]. There is a large design space still to explore and alternative interface structures may lead to improved performance [Page 28].

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The gist Scaling quantum computers to practically relevant logical qubit counts at low error rates requires more qubits than are typically expected to fit on a single quantum processing unit (QPU) [1–4].

How it works

The paper constructs hybrid syndrome extraction protocols using the ZX calculus to perform distributed lattice surgery between distributed logical qubits [Page 31]. The construction involves mapping logical level descriptions down to a hybrid protocol designed for heterogeneous hardware, where measurement and world-line annotations fix a hybrid realisation with fusion networks at the interface and circuit-based syndrome extraction elsewhere [Page 31].

Resource State Generation

To generate all-photonic resource states, emitters can be measured out after emission, leaving only photons in the state [79]. The simplest example of this is the generation of a matter-based Bell pair using distributed emitters [Page 16]. Each remote emitter produces an emitter–photon Bell pair. Successfully fusing the two photons then leaves the two emitters in a Bell state up to corrections from the fusion outcomes [Page 16].

Interface Detector Error Models

The interface detector error model (DEM) is constructed by allowing erasures only on elementary linear-optical fusion outcomes, where each elementary fusion outcome is erased independently with probability pe [Page 31]. The analysis shows that the asymptotic erasure threshold for Bell, BellZigZag, and 3d-Chain protocols under pure erasure noise is 50% [Page 21].

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The gist Scaling quantum computers to practically relevant logical qubit counts at low error rates requires more qubits than are typically expected to fit on a single quantum processing unit (QPU) [1–4].

How it works

The paper constructs hybrid syndrome extraction protocols using the ZX calculus to perform distributed lattice surgery between distributed logical qubits [Page 31]. The construction involves mapping logical level descriptions down to a hybrid protocol designed for heterogeneous hardware, where measurement and world-line annotations fix a hybrid realisation with fusion networks at the interface and circuit-based syndrome extraction elsewhere [Page 31].

Conclusion and Future Work

The work provides an initial demonstration of the benefits of hybrid syndrome extraction protocols, highlighting the potential to tailor error correction schemes towards dominant noise sources such as photon loss [Page 31]. The principal local-noise limitation of the hybrid interfaces arises from the deterministic boundary fusions between the circuit and resource-state regions [Page 28].

1. The AI system can perform distributed lattice surgery for logical operations using matter-based qubits and photonic links, achieving a 50% interface-erasure threshold when local noise is absent in hybrid protocols.

2. The system can utilize fusions networks to construct hybrid syndrome extraction protocols by applying ZX calculus transformations to construct them, replacing sequential measurements with cluster states that locally interact with ancilla qubits.

3. The AI can tailor error correction schemes towards dominant noise sources by analyzing how interface distance enhancements are effective in low local noise regimes, suggesting a mechanism for optimizing performance against photon loss.

4. The system can implement circuit-based and measurement-based syndrome extraction diagrams, allowing it to leverage the strengths of both paradigms in heterogeneous matter–photonic architectures via hybrid checks.

5. The AI can derive photon-loss thresholds from erasure thresholds using the formula derived from Eq. (24), enabling precise prediction of performance under realistic noise models incorporating loss and fusion failure.

6. The system can employ fusion boosting to improve photon-loss thresholds, which is particularly useful when local noise is high, allowing it to probe far-subthreshold regimes.

7. The AI can perform subthreshold suppression analysis by calculating the factor of finite-distance suppression from d to d + 2 using Eq. (28), quantifying how increased interface distance buys better logical error suppression as erasure probability decreases.

Networking matter-based quantum processing units (QPUs) offers a promising route to scaling fault-tolerant quantum computers. This requires distributed logical operations to be performed across photonic links, where noise is characteristically different from and stronger than in local QPUs owing to photon loss and probabilistic linear-optical operations. Measurement- and fusion-based quantum computing are designed to be robust against loss and probabilistic photonic operations, suggesting they could complement circuit-based error correction at the interface between networked QPUs. We use ZX calculus transformations to construct hybrid syndrome extraction protocols and apply them to distributed rotated surface code lattice surgery, demonstrating that several protocols attain a 50% interface-erasure threshold when local noise is absent, including circuit-based lattice surgery using fused Bell pairs and hybrid protocols using linear cluster states. Relative to a straight Bell-pair interface geometry, the hybrid protocols increase the merge-observable interface distance from d+1 to 2d+1 and restore the perpendicular-observable interface distance from (d+1)/2 to d. We calculate how the threshold decreases when using truncated resource states and map correctable regions under resource-state errors, local circuit noise, and fusion erasure. We then convert these erasure thresholds into photon-loss thresholds and probe subthreshold performance using fusion boosting. At circuit and resource-state error rates of 10-3, local errors largely mask the interface-distance advantage. As local noise is reduced to 10-4 and below, the linear-chain protocols improve more rapidly with decreasing erasure, suggesting that interface distance enhancements are effective in low local noise regimes.