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arXiv:2510.00370

Low Depth Color Code Circuits with CXSWAP gate

Satoshi Yoshida, Craig Gidney, Matt McEwen, Adam Zalcman

Published Oct 1, 2025• 0 citations
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Abstract

We present two new types of syndrome extraction circuits for the color code. Our first construction, which after [M. McEwen, D. Bacon, and C. Gidney, Quantum 7, 1172 (2023)] we call the semi-wiggling color code, promises to mitigate leakage errors by periodically interchanging the roles of bulk data and measurement qubits. The second construction reduces circuit depth relative to [C. Gidney and C. Jones, arXiv:2312.08813 (2023)] by employing the CXSWAP gate instead of CNOT. This optimization leads to $\sim 10\%$ improvement in teraquop footprint under the uniform error model with the physical error rate $p=0.1\%$.

DOI
10.48550/arxiv.2510.00370

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We present two new types of syndrome extraction circuits for the color code. Our first construction, which after [M. McEwen, D. Bacon, and C. Gidney, Quantum 7, 1172 (2023)] we call the semi-wiggling color code, promises to mitigate leakage errors by periodically interchanging the roles of bulk data and measurement qubits. The second construction reduces circuit depth relative to [C. Gidney and C. Jones, arXiv:2312.08813 (2023)] by employing the CXSWAP gate instead of CNOT. This optimization leads to ∼10%\sim 10\%∼10% improvement in teraquop footprint under the uniform error model with the physical error rate p=0.1%p=0.1\%p=0.1%.
Quantum error correction (QEC) codes are traditionally defined and searched for without specifying the manner in which its syndrome extraction circuits are executed using elementary gates and measurements. We show how morphing circuits introduced in Refs. [1-3] provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate (ISWAP versus CNOT) and number of physical qubits. We discuss morphing circuits in code optimisation among Abelian two-block group algebra (2BGA) codes, handling boundaries for 2D codes, codes with single-shot properties, and improving performance in stability experiments against measurement and reset errors. We show that alternating syndrome extraction circuits - executed with alternating time-reversed rounds - can be viewed as a two-round morphing circuit whose fault-tolerant properties are computationally much easier to examine than non-alternating syndrome extraction circuits. Our methods find new codes and syndrome extraction circuits of practical interest, including Abelian 2BGA morphing circuits with better code parameters and connectivity than existing circuits. [1] Matt McEwen, Dave Bacon, and Craig Gidney. Relaxing hardware requirements for surface code circuits using time-dynamics. Quantum, 7:1172, 2023. [2] Craig Gidney and Cody Jones. New circuits and an open source decoder for the color code, 2023. [3] Mackenzie H. Shaw and Barbara M. Terhal. Lowering connectivity requirements for bivariate bicycle codes using morphing circuits.
Syndrome extraction in the planar color code is complicated by high weight stabilizers and hook errors that can reduce the circuit-level distance. With a single auxiliary qubit per plaquette, any spatially uniform circuit halves the circuit-level distance. We propose a single-auxiliary syndrome extraction circuit with color-dependent gate schedules that avoids all malign hook errors in the bulk, thereby preserving the full circuit-level distance. The circuit has minimal depth: all stabilizers of the same Pauli type are measured in parallel in six time steps. Furthermore, this schedule can be readily applied to the XYZ color code circuit, yielding an improved temporal distance. We find that at the boundary, no single hook error alone reduces the distance; instead, only certain combinations of hook errors do, which we call fractional hook errors. We demonstrate through Monte Carlo simulations over a range of circuit-level noise models and physical error rates that our circuit outperforms the previous state of the art.
Color-dependent syndrome extraction schedules have been proposed to mitigate hook errors in the triangular color code. Recent work by Kishony and Fowler (arXiv:2603.28852) identifies distance-reducing hook errors by the type of anyon for which they provide a shortcut and chooses the CNOT ordering for stabilizers of each color to avoid these errors. We identify an additional class of distance-reducing hooks, which we call junction hook errors, that is not captured by this classification. The two data qubits of a junction hook error terminate anyon strings of two different colors at the point where the three colored strings of a logical operator meet, which we call the junction. As such, the hook error does not correspond to a single type of anyon, yet it still reduces the distance. For distances d=7,9,11d=7,9,11d=7,9,11, we verify that every non-corner plaquette admits a scheduled hook error that, together with d−2d-2d−2 single data qubit faults, produces an undetectable logical error using d−1d-1d−1 faults. Each of these is a junction hook error. Because of this, every stabilizer, including those in the bulk, still requires additional protection from hook errors (such as flag qubits) to restore the full circuit-level distance with this schedule. A minimum weight logical operator has at most one junction, however, so junction hook errors cannot accumulate on one minimum weight logical operator in the way that single-color malign hook errors can. This helps explain why the schedule still improves the circuit-level distance to d−⌊(d+3)/6⌋d-\lfloor (d+3)/6\rfloord−⌊(d+3)/6⌋, rather than the d/2d/2d/2 distance associated with spatially uniform color code scheduling.
Color codes are promising quantum error-correction (QEC) codes because they have an advantage over surface codes in that all Clifford gates can be implemented transversally. However, the thresholds of color codes under circuit-level noise are relatively low, mainly because measurements of their high-weight stabilizer generators cause an increase in the circuit depth and, thus, substantial errors are introduced. This makes color codes not the best candidate for fault-tolerant quantum computing. Here, we propose a method to suppress the impact of such errors by optimizing weights of decoders using conditional error probabilities conditioned on the measurement outcomes of flag qubits. In numerical simulations, we improve the threshold of the (4.8.8) color code under circuit-level noise from 0.14% to around 0.27%, which is calculated by using an integer programming decoder. Furthermore, in the (6.6.6) color code, we achieve a circuit-level threshold of around 0.36%, which is almost the same value as the highest value in the previous studies employing the same noise model. In both cases, the effective code distance is also improved compared to a conventional method that uses a single ancilla qubit for each stabilizer measurement. Thereby, the achieved logical error rates at low physical error rates are almost one order of magnitude lower than those of the conventional method with the same code distance. Even when compared to the single-ancilla method with a higher code distance, considering the increased number of qubits used in our method, we achieve lower logical error rates in most cases. This method can also be applied to other weight-based decoders, making the color codes more promising as candidates for the experimental implementation of QEC. Furthermore, one can utilize this approach to improve a threshold of wider classes of QEC codes, such as high-rate quantum low-density parity-check codes. Published by the American Physical Society 2024
We present a fast and effective framework for analysing and designing syndrome extraction circuits (SECs). Our approach is based on left--right circuits, a general SEC design that maintains low depth and avoids gate interleaving constraints by staggering XXX and ZZZ stabiliser checks. Initially proposed for specific classes of codes, we generalise this construction to arbitrary CSS codes and optimise the circuit structure to achieve low qubit idling time, large effective distance, and reduced minimum-weight failure mechanisms. A key component of our framework is the formal notion of residual errors and their associated distance metrics, which form lightweight proxies for capturing error propagation and quantifying the potential harm of circuit-level errors. Applying our automated framework to diverse classes of codes, we observe consistent improvements in logical performance of up to an order of magnitude compared to existing single-ancilla SEC designs. Moreover, by directly applying these proxies, we prove that no non-interleaving SEC can achieve circuit distance 121212 for the gross code, and we identify an explicit circuit that we conjecture achieves distance 111111, exceeding previously known constructions.
We present two new color code circuits: one inspired by superdense coding and the other based on a middle-out strategy where the color code state appears halfway between measurements. We also present ``Chromobius'', an open source implementation of the möbius color code decoder. Using Chromobius, we show our new circuits reduce the performance gap between color codes and surface codes. Under uniform depolarizing noise with a noise strength of 0.1%0.1\%0.1%, the middle-out color code circuit achieves a teraquop footprint of 1250 qubits (vs 650 for surface codes decoded by correlated matching). Finally, we highlight that Chromobius decodes toric color codes better when given *less* information, suggesting there's substantial room for improvement in color code decoders.