Need One Bell-pair Only (NOBOL) for Low-Overhead Fault-Tolerant Quantum Computing

Fault-tolerant quantum computing relies on encoding logical qubits into blocks of physical qubits to protect against noise. A major bottleneck in scaling these systems—whether in monolithic quantum processors or distributed quantum computing (DQC) architectures—is the high overhead required to perform logical CNOT operations between non-neighboring or remote qubits. Traditional methods, such as lattice surgery or transversal gate operations, often require a large number of Bell pairs or extensive SWAP operations, which scale linearly with the code distance and increase the risk of decoherence.

The authors introduce NOBOL (Need One Bell-pair Only), a novel framework that enables a logical CNOT operation between two distant logical qubits using exactly one shared Bell pair. The key insight is that logical operations in CSS codes do not need to act on the entire code block. Instead, they only need to interact with the physical qubits that support the logical X or Z operators. By focusing exclusively on these subsets, NOBOL minimizes the required entanglement resources.

NOBOL is agnostic to the specific CSS code used, making it compatible with various architectures, including surface codes. The authors provide circuit realizations that are depth-optimal, achieving logarithmic depth (O(log τ), where τ is the weight of the logical operator). This structural flexibility allows the protocol to be mapped efficiently onto diverse hardware constraints, reducing the time qubits must remain coherent during the operation.

Because NOBOL reduces the redundancy of the operation, it introduces new pathways for error propagation. To maintain fault tolerance, the authors propose two primary containment strategies:

• Repetition Codes: Encoding the halves of the Bell pair in small repetition codes to localize bit-flip or phase-flip errors within specific sub-blocks.
• Intermittent Stabilization: Performing adaptive stabilizer measurements throughout the circuit execution to detect and correct errors before they propagate across the entire logical block.

These techniques allow researchers to tune the trade-off between resource overhead (ancilla qubits) and execution latency (circuit depth) based on the specific requirements of their quantum system.