The strongly correlated systems we use to realise quantum error-correcting codes may give rise to high-weight, problematic errors. Encouragingly, we can expect local quantum error-correcting codes with no string-like logical operators - such as the cubic code - to be robust to highly correlated, one-dimensional errors that span their lattice. The challenge remains to design decoding algorithms that utilise the high distance of these codes. Here, we begin the development of such algorithms by proposing an efficient decoder for the `Fibonacci code'; a two-dimensional classical code that mimics the fractal nature of the cubic code. Our iterative decoder finds a correction through repeated use of minimum-weight perfect matching by exploiting symmetries of the code. We perform numerical experiments that show our decoder is robust to one-dimensional, correlated errors. First, using a bit-flip noise model at low error rates, we find that our decoder demonstrates a logical failure rate that scales super exponentially in the linear size of the lattice. In contrast, a decoder that could not tolerate spanning errors would not achieve this rapid decay in failure rate with increasing system size. We also find a finite threshold using a spanning noise model that introduces string-like errors that stretch along full rows and columns of the lattice. These results provide direct evidence that our decoder is robust to one-dimensional, correlated errors that span the lattice.
The strongly correlated systems we use to realise quantum error-correcting codes may give rise to high-weight, problematic errors. Encouragingly, we can expect local quantum error-correcting codes with no string-like logical operators - such as the cubic code - to be robust to highly correlated, one-dimensional errors that span their lattice. The challenge remains to design decoding algorithms that utilise the high distance of these codes. Here, we begin the development of such algorithms by proposing an efficient decoder for the `Fibonacci code'; a two-dimensional classical code that mimics the fractal nature of the cubic code. Our iterative decoder finds a correction through repeated use of minimum-weight perfect matching by exploiting symmetries of the code. We perform numerical experiments that show our decoder is robust to one-dimensional, correlated errors. First, using a bit-flip noise model at low error rates, we find that our decoder demonstrates a logical failure rate that scales super exponentially in the linear size of the lattice. In contrast, a decoder that could not tolerate spanning errors would not achieve this rapid decay in failure rate with increasing system size. We also find a finite threshold using a spanning noise model that introduces string-like errors that stretch along full rows and columns of the lattice. These results provide direct evidence that our decoder is robust to one-dimensional, correlated errors that span the lattice.
Efficient and realistic error decoding is crucial for fault-tolerant quantum computation (FTQC) on near-term devices. While decoding is a classical post-processing task, its effectiveness depends on accurately modeling quantum noise, which is hardware-dependent. In particular, correlated bit-flip (X) and phase-flip (Z) errors often arise under circuit-level noise. We introduce the Iterative Reweighting Minimum-Weight Perfect Matching (IRMWPM) decoder, which systematically incorporates such correlations to enhance quantum error correction. Our method leverages fault-detection patterns to guide reweighting: correlated X and Z detection events are identified, and their conditional probabilities update weights on the primal and dual lattices. This iterative procedure improves handling of realistic error propagation in a hardware-agnostic yet noise-aware manner. We prove that IRMWPM converges in finite time while preserving the distance guarantee of MWPM. Numerical results under circuit-level noise show substantial improvements. For distances ≥17 and physical error rates ≤0.001, IRMWPM reduces logical error rates by over 20x with only a few iterations. It also raises the accuracy threshold from 1% to 1.16%, making it practical for near-term real-time decoding. Extrapolated estimates suggest that to reach logical error rate 10−16, IRMWPM requires distance d=31, while standard MWPM needs d=50, implying a major reduction in qubit overhead.
Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+ϵ , which admits a fault-tolerant non-Clifford CCZ gate \cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension 2+ϵ . For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold ( ∼1.7% ) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by 2.95% ) for the MWPM decoder for a particular FSC with Hausdorff dimension DH≈2.966 . The latter can be mapped to a lower bound of the critical point of a confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.
Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure ( pfail∼pd/2 ) for both noise models, where p is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes ( 8.2% for bit-flip noise and 0.46% for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.
The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced by the depolarising noise model. The decoder is obtained by mapping the syndrome of the surface code onto that of the color code, thereby allowing us to adopt more sophisticated color-code decoding algorithms. Analytical arguments and exhaustive testing show that the resulting decoder can find a least-weight correction for all weight d/2 depolarising errors for even code distance d . This improves the logical error rate by an exponential factor O(2d/2) compared with decoders that treat bit-flip and dephasing errors separately. We demonstrate this improvement with analytical arguments and supporting numerical simulations at low error rates. Of independent interest, we also demonstrate an exponential improvement in logical error rate for our decoder used to correct independent and identically distributed bit-flip errors affecting the color code compared with more conventional color-code decoding algorithms.
Quantum computers are highly sensitive to noise due to interactions with the environment, which severely limits the utility of near-term devices. Quantum error correction addresses this challenge by encoding logical information over multiple qubits in order to preserve information in the presence of noise. This facilitates the realization of the theoretical advantages of quantum computing such as exponential speed-up compared to classical computers for certain algorithms. Physical realizations of qubits may have different noise profiles, including biased noise, where phase-flip errors may be more likely than bit-flip errors. We propose a quantum error-correcting code, the XYZ² code, which is tailored to biased noise. The code has high error thresholds against biased noise, below which logical errors are exponentially suppressed with the linear dimension of the code. The important classical task of deriving corrections given partial information about errors is called decoding. We investigate several decoding strategies tailored to specific challenges, ranging from a novel maximum-likelihood sampling decoder, to a model-free, data-driven, neural-network-based decoder, to an efficient multi-step decoder for concatenated codes. We also study the link between quantum error-correcting codes and their statistical-mechanical counterparts by mapping to generalized random-bond Ising models and subsequently deriving the exact results for finite-size corrections under biased noise for moderate code sizes. Quantum error correction is still in its developing stages, and this work provides a useful contribution to the field by bridging a gap between theory and practice. By adapting well-studied topological codes to error models that are closer to reality, this work paves the way for the realization of a fault-tolerant quantum architecture.
Different choices of quantum error-correcting codes can reduce the demands on the physical hardware needed to build a quantum computer. To achieve the full potential of a code, we must develop practical decoding algorithms that can correct errors that have occurred with high likelihood. Matching decoders are very good at correcting local errors while also demonstrating fast run times that can keep pace with physical quantum devices. We implement variations of a matching decoder for a three-dimensional, non-CSS, low-density parity check code known as the Chamon code, which has a non-trivial structure that does not lend itself readily to this type of decoding. The non-trivial structure of the syndrome of this code means that we can supplement the decoder with additional steps to improve the threshold error rate, below which the logical failure rate decreases with increasing code distance. We find that a generalized matching decoder that is augmented by a belief-propagation step prior to matching gives a threshold of 10.5% for depolarizing noise.