Resource-Efficient Modular Quantum Computation

• The paper introduces a Bell measurement-based protocol that cuts inter-module entanglement usage by nearly 40%, enabling more resource-efficient quantum operations.
• The protocol confines noise to module interfaces during lattice surgery, maintaining logical error protection and mitigating the formation of hook errors.
• Simulations show that the approach achieves fault tolerance at lower entanglement rates and scales efficiently for larger code distances in modular architectures.

Resource-efficient modular quantum computation denotes approaches that enable large-scale, high-fidelity quantum information processing by decomposing hardware and logical circuits into interacting but individually manageable modules. This paradigm addresses the central challenge of scaling quantum systems—physical, architectural, and algorithmic constraints on qubit numbers, control resources, interconnects, and operational fidelity—by explicitly considering the entanglement distribution, control overhead, and error rates incurred as quantum operations are partitioned across physically or logically separate processor modules. Recent protocols emphasize minimizing entanglement and control resources per module, confining noise to interfaces, exploiting modular error-correction strategies, and leveraging efficient compilation and scheduling techniques for inter-module operations to achieve fault tolerance and algorithmic universality at scale.

1. Lattice Surgery with Bell Measurements: Protocol Architecture

A fault-tolerant modular quantum computer based on the surface code requires non-local logical operations—such as lattice surgery mergers and splits—across logical qubits stored on different processor modules. The protocol described in the referenced work (Haug et al., 15 Oct 2025) introduces an approach where all non-local operations required for lattice surgery are reduced to Bell measurements performed at the interface between modules, replacing previous schemes that required an extensive set of direct inter-module gates or Bell pairs.

The protocol is constructed as follows:

• For each interface between surface-code patches, the syndrome extraction circuit is represented (via ZX calculus) such that, in the bulk of each module, ancilla qubits interact locally with their data qubit neighbors as usual.
• At the interface, each module prepares a local ancilla, and ancilla pairs (one per module) are projected onto Bell states via a two-qubit Bell measurement. This is formally a simultaneous measurement of $X_1X_2$ and $Z_1Z_2$ on the interface ancillas.
• The outcomes of these Bell measurements both generate the necessary entanglement to merge code patches and supply the stabilizer eigenvalues required for error correction.

The Bell measurement is implemented by a short circuit: ancilla preparation, local CX/CZ gates to the relevant data qubits, followed by a two-qubit measurement in the Bell basis, rather than distributing pairs of entangled qubits per individual data qubit across the interface.

2. Entanglement Cost and Resource Advantages

The primary resource advantage of this protocol is in the dramatic reduction of inter-module entanglement consumption:

• In benchmark schemes, a round of syndrome extraction across a distance-$d$ interface requires $2d-1$ Bell pairs per round; the new protocol requires only $d$ such pairs.
• This reduction, quantified via circuit-level simulation, results in ≈40% savings in entanglement usage at fixed logical error rate for a wide range of surface code distances.
• For specific instances, while the previous approach required, for example, 57 or 61 Bell pairs per round for distances 29 or 31 respectively, the Bell-measurement protocol achieves comparable logical error performance using only 35 or 39 pairs for codes of distance 35 or 39.
• The mathematical resource scaling follows: interface entanglement consumption per syndrome round is $d$ instead of $2d-1$.

This efficiency is crucial in realistic modular architectures where the generation of high-fidelity Bell pairs is rate- and fidelity-limited, and local entanglement (within a module) can be produced at much higher rates and quality.

3. Noise Confinement and Error Correction

The protocol leverages the structure of the surface code and properties of Bell measurement circuits to confine link noise to the interface:

• Noise introduced during the interface Bell measurement affects only the ancilla qubits involved and does not propagate into the data qubits of each module.
• Circuit implementation employs alternating gate sequences for syndrome measurement rounds at the interface—specifically, sequences A and B as defined in the work—mitigating formation of distance-reducing “hook errors” in the syndrome graph that could otherwise degrade logical error protection.
• This strategy prevents chains of hook errors from aligning with logical operators, thereby preserving the effective code distance. Numerical simulations show that under the alternating protocol, the effective distance at the interface $d_{\mathrm{eff}}$ attains $d$ (with observed ratios $d_{\mathrm{eff}}/d \approx 0.9$), as opposed to a reduction to approximately $(d+1)/2$ under non-alternating schedules.

Logical error rates, analyzed via depolarizing error models with local gate error rates $p_{\mathrm{loc}}$ and link-specific error rates $p_{\mathrm{link}}$, confirm that for constant entanglement rates, the Bell-measurement protocol achieves lower logical error rates than previous alternatives—particularly for link error rates in regimes relevant to realistic photonic or remote-connection platforms.

4. Thresholds, Performance, and Scalability

The fault-tolerance threshold and overall performance under realistic noise and entanglement rates are key for practical implementation:

• Under uniform depolarizing noise, both the new and benchmark protocols yield similar error thresholds (0.53% for the benchmark vs. 0.52% for the Bell-measurement protocol).
• In regimes where inter-module noise ($p_{\mathrm{link}}$) dominates, the Bell-measurement approach maintains a meaningfully higher threshold (e.g. 17.4% with direct gate implementation), effectively decoupling link-specific errors from the bulk code operations within modules.
• For a fixed target logical error probability (e.g., $10^{-12}$), simulations show that the protocol achieves this target at lower entanglement rates and with fewer Bell pairs per round than any previously reported protocol, which is a significant advance for systems limited by photonic entanglement distribution speed or fidelity.
• The scaling of resource requirements enables practical implementation for larger code distances and larger module sizes.

5. General Applicability Beyond the Surface Code

The protocol is extensible to a wider array of modular quantum computing schemes:

• The principle of substituting direct cross-module gates with Bell measurements (fusion operations) can be applied to any scenario where quantum circuits are divided across modules and inter-module connectivity or entanglement is a limited, noisy, or costly resource.
• The ZX calculus and circuit partitioning strategy underlying this protocol provide a general template for minimizing the required number of module-crossing links via spider splitting, making the interface as lightweight as possible.
• Alternating syndrome extraction schedules may be beneficial for any error-correction code or LDPC-based modular design where interface hook errors threaten logical distance.
• These design methodologies are immediately applicable for modular platforms such as superconducting QPUs in separate cryostats, photonic-link-bridged ion traps, and atom-based multi-module quantum systems—where inter-module Bell measurements are often more practical than high-fidelity, direct two-qubit gates.

6. Significance for Fault-Tolerant Modular Architectures

This protocol supports modular architectures where:

• Each module is a local, small or medium-scale quantum processor with high-fidelity local operations and error correction.
• Modules are interconnected by links that support probabilistic or heralded entanglement, with relatively higher noise or lower bandwidth than intra-module connections.
• Efficient “surface-code merging” or logical–state transfer between modules is critical to algorithmic universality, QEC cycles, or distributed computation.
• The protocol’s entanglement savings mean that approaching the surface code threshold, or matching the best-known logical error suppression rates, becomes feasible even with restricted inter-module hardware.
• Noise is confined to well-characterized interfaces, and the overall system remains scalable with improved error budgets and reduced architectural demands.

The described protocol demonstrates that, by integrating measurement-based logic with interface-specific noise isolation, and optimizing the sequence and scheduling of inter-module operations, resource-efficient and robust modular quantum computation is viable even under stringent hardware constraints. This approach is a foundation for future research into distributed and scalable quantum computation beyond traditional bulk-monolithic architectures.