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Fault-tolerant cluster states form the basis for scalable measurement-based quantum computation. Recently, new stabilizer codes for scalable circuit-based quantum computation have been introduced that have very high thresholds under biased noise where the qubit predominantly suffers from one type of error, e.g. dephasing. However, extending these advances in stabilizer codes to generate high-threshold cluster states for biased noise has been a challenge, as the standard method for foliating stabilizer codes to generate fault-tolerant cluster states does not preserve the noise bias. In this work, we overcome this barrier by introducing a generalization of the cluster state that allows us to foliate stabilizer codes in a bias-preserving way. As an example of our approach, we construct a foliated version of the XZZX code which we call the XZZX cluster state. We demonstrate that under a circuit-level-noise model, our XZZX cluster state has a threshold more than double the usual cluster state when dephasing errors are more likely than errors that cause bit flips by a factor of order ~100 or more.

 

Measurement-based quantum computing (MBQC) is an alternative model of quantum computation that is equivalent to the standard gate-based model and is the preferred approach for several optical quantum computing architectures. In MBQC, a quantum computation is executed by preparing an entangled cluster state and then selectively measuring qubits. MBQC can be made fault-tolerant by creating an MBQC computation that executes the standard surface code, an approach known as "foliation." Recent results on gate-based quantum computing have demonstrated that in the presence of biased noise, a modified version of the surface code known as the XZZX code has much higher thresholds than the standard surface code. However, naively foliating the XZZX code does not result in a high-threshold fault-tolerant MBQC, because the foliation procedure does not preserve the noise bias of the physical qubits. To create a high-threshold fault-tolerant MBQC, we introduce a modified cluster state that preserves the bias, and use our modified cluster state to construct an MBQC computation that executes the XZZX code. Using full circuit-level noise simulations, we show that the threshold of our modified MBQC is higher than either the standard fault-tolerant MBQC or the naïve foliated XZZX code in the presence of biased noise, demonstrating the advantage of our approach.