The compass model on a square lattice provides a natural template for building subsystem stabilizer codes.The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed.We explore threshold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information.We analyze these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models.In these idealized settings, we observe considerably higher thresholds against asymmetric noise.At the circuit level, these codes inherit the bare-ancilla fault tolerance of the Bacon-Shor code.
A Bacon-Shor code is a subsystem quantum error-correcting code on an $L\ifmmode\times\else\texttimes\fi{}L$ lattice where the $2(L\ensuremath{-}1)$ weight-
2L stabilizers are usually inferred from the measurements of $2L(L\ensuremath{-}1)$ weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.
Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalizes to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼5% in an experimentally relevant regime where dephasing errors occur at a rate 100 times greater than bit-flip errors.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing
Z-type stabilizers with
Y-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a
ZY surface code by replacing the
X-type stabilizers with
Y-type stabilizers. We compare it with the standard
ZX surface code under circuit-level Pauli-
X biased noise, with and without an additional gate-based
XX crosstalk noise. We find that for the
ZX surface code, the
X-memory threshold increases monotonically with bias while the
Z-memory threshold decreases and saturates. For the
ZY surface code, the
Y-memory threshold is nearly constant across all bias values. The
Z-memory thresholds of the
ZX and
ZY codes are consistent within the uncertainty. Adding
XX crosstalk reduces the
Z-memory threshold beyond the fitting uncertainty while leaving the
X memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric code under circuit-level depolarizing noise. This variant preserves the almost-linear time complexity of the original while significantly increasing the performance in the fault-tolerance setting. In this noise model, weighting the union-find decoder increases the threshold from
0.38 to
0.62, compared to an increase from
0.65 to
0.72 when weighting a matching decoder. Further assuming quantum nondemolition measurements, weighted union-find decoding achieves a threshold of
0.76 compared to the
0.90 threshold when matching. We additionally provide comparisons of timing as well as low error rate behavior.
Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.