Correlated decoders for transversal CNOTs

Logical gate implementation with quantum error correction codes can improve fidelity of quantum computation. Logical information stored in quantum error correction codes are protected by error syndrome obtained from stabilizer measurement that capture the errors without destroying the protected logical states. Among all quantum error correction codes, surface codes are a leading candidate for practical implementation due to its high fault-tolerance threshold and the low-depth stabilizer measurement circuits. Recent hardware development on non-local qubit connectivity supports the transversal implementation of logical controlled-NOT (tCNOT) gates on surface codes. In tCNOT gate implementation, physical CNOT gates are applied to each corresponding pair of physical qubits on the control surface code and the target surface code. These physical CNOT gates propagate errors between qubits, thus creating correlated errors in the two surface codes. Therefore, decoders to address correlated errors are needed for error correction with tCNOT gates. 

Previous decoding proposals for tCNOT gates include the single-update decoding, where syndrome after the tCNOT gates is added from one surface code to another surface code. This method is suboptimal as it doubles the noise in the combined syndrome. An alternative scheme, which we call the combined-hypergraph decoding, utilizes all the syndrome of the circuit and captures the correlated errors by decoding graphs with hyperedges. The existing hypergraph decoders applicable to this scheme are typically slow. In our recent paper in a collaboration with the Puri group at Yale (https://arxiv.org/abs/2408.01393), we propose a low-complexity decoding scheme called the ordered decoding. In ordered decoding, we first decode one surface code whose errors before the tCNOT gate are copied to the other surface code by the tCNOT gate. We then apply the identified data qubit errors before the tCNOT gate to the other surface code and correct the residual errors on the other surface code with corresponding updated syndrome. Any efficient graph-based decoders can be used in this scheme. In our paper, we benchmark the performance of the three decoding schemes for the scalable fault-tolerant quantum computation. In particular, we find that our ordered decoding with a minimum-weight perfect matching(MWPM) decoder is highly effective in correcting correlated errors introduced by tCNOT gates, exhibiting a higher fault-tolerance threshold than the single-update decoding and the combined-hypergraph decoding. Additionally, we study teleportation circuits with tCNOT gate, where one logical qubit is measured immediately after the tCNOT gate. In this case, no hyperedge exists in the decoding graph. Regular MWPM decoder can achieve a threshold similar to that of surface code memory.  

 Beyond tCNOT gate, logical CNOT gate can also be implemented by lattice surgery, where only nearest-neighbor interactions are required. We provide a comparison of performance and preliminary resource estimation for lattice surgery CNOT gate and tCNOT with ordered decoding. Our investigation builds toward an analysis for the hardware implementation of transversal gates in large-scale quantum algorithms.