Semi-Wiggling Color Code Overview

Updated 2 October 2025

The semi-wiggling color code is a method that alternates roles of data and measurement qubits to improve leakage error mitigation in two-dimensional quantum circuits.
It implements CXSWAP gate substitutions to reduce circuit depth, achieving approximately a 10% footprint improvement compared to standard midout designs.
Designed for leakage-prone hardware like superconducting qubits, this approach enables more robust fault-tolerant operations in next-generation quantum systems.

The semi-wiggling color code refers to a family of syndrome extraction circuits and hardware-tailored operational protocols for two-dimensional color codes, in which the periodic interchange of data and measurement qubits is used to enhance leakage error mitigation, and circuit depth is minimized via specific gate substitutions (notably the CXSWAP gate). While the term “semi-wiggling” has appeared in several contexts, its most recent and precise technical usage is the midout-type color code circuit presented in (Yoshida et al., 1 Oct 2025), where only a subset of the bulk (not boundaries) undergoes systematic role alternation per cycle. This methodology offers key hardware- and performance-related advantages, particularly for superconducting architectures or other hardware subject to leakage errors.

1. Circuit Design: Semi-Wiggling Role Interchange

The semi-wiggling color code circuit modifies the standard midout syndrome extraction by periodically exchanging the role of data and measurement qubits. In a conventional midout circuit, each stabilizer (typically weight six, for the honeycomb color code) is “shrunk” in a sequence of CNOT gates, culminating in a measurement layer, with measurement ancillas and data qubits maintaining fixed roles. The semi-wiggling variant modifies the directionality of the final CNOT in the sequence: rather than shrinking a 2-body to a 1-body stabilizer on the same qubit, the reversed CNOT shifts the stabilizer’s endpoint into an adjacent column, thus assigning the measurement role to a new physical qubit in the subsequent round.

This operationally “wiggles” the measurement responsibility through the code bulk, while boundaries retain their conventional pattern for proper code connectivity—a motivation for the “semi” qualifier.

Aspect	Standard Midout	Semi-Wiggling Midout
Data/Meas. Role	Fixed per qubit	Periodically alternated
Stabilizer Movement	Static “shrinking”	Shifted via reversed CNOT
Leakage Mitigation	Limited (fixed qubits)	Global (all become measured)
Qubit Utilization (Bulk)	Static	Alternating

2. Leakage Error Mitigation

Leakage errors—where a physical qubit leaves the computational subspace—cannot be corrected by standard stabilizer QEC and, if unchecked, quickly compromise the reliability of error-corrected circuits. The principal engineering motivation for semi-wiggling is to ensure that every qubit is periodically subject to measurement and reset operations, thereby scrubbing leaked qubits from the code. In effect, all bulk qubits become measurement qubits every few syndrome extraction cycles. By ensuring that leakage cannot persist on unmeasured (“data”) positions indefinitely, this protocol makes the system more robust to leakage, with the benefit increasing with the “wiggling” frequency.

Practically, the mitigation protocol is realized in hardware by including reset gates whenever a qubit’s role is that of a measurement ancilla. Boundary qubits, which require special stabilizer relations for code integrity, do not undergo such role alternation, hence the “semi” aspect of the construction (Yoshida et al., 1 Oct 2025).

3. Circuit Depth Reduction via CXSWAP Gates

In tandem with semi-wiggling, (Yoshida et al., 1 Oct 2025) introduces a further circuit depth reduction by substituting pairs of adjacent CNOTs with the CXSWAP gate—a hardware-native gate for several platforms, equivalent (up to single-qubit Cliffords) to ISWAP. Given CXSWAP₍₁,₂₎ = CNOT₍₁,₂₎CNOT₍₂,₁₎, two CNOT layers are contracted into one CXSWAP; similarly, CNOTs can be decomposed as CNOT₍₁,₂₎ = CXSWAP₍₂,₁₎SWAP₍₁,₂₎ (and analogously with swapped labels).

Applying these identities, the midout circuit is reconstructed to reduce the total gate depth (steps per syndrome extraction from 8 to 7), while maintaining only nearest-neighbor interactions (after boundary SWAP routing), which is vital for 2D hardware layouts. Reduction of total gate depth directly translates to a reduction in circuit error accumulation per extraction cycle—a core metric governing fault-tolerance thresholds.

Circuit Version	Circuit Depth (Steps)	Teraquop Footprint at p=0.1%	Improvement
Standard Midout	8	Baseline	—
CXSWAP Midout	7	~10% lower vs. baseline	10%

4. Quantitative Performance and Simulation Results

Under a uniform physical error model with rate $p=0.1\%$ , the semi-wiggling midout circuit and its CXSWAP-optimized descendant yield logical error probabilities scaling as $P_L \sim p^{\lfloor (d+1)/2\rfloor}$ , where $d$ is the code distance. With physical qubits scaling as $O(d^2)$ , teraquop (TQ) metrics—measuring the logical error rate per $10^{12}$ operations—are compared via a fit:

$\log P_L = a \sqrt{n} + b,$

where $n$ is the code qubit count, and $a, b$ are fit parameters determined from simulation. The CXSWAP-based circuit achieves a reduction in TQ footprint of approximately 10% at $p=0.1\%$ compared to the standard midout construction (Yoshida et al., 1 Oct 2025), as shown in Figs. 4-5 therein.

Although the “semi-wiggling” construction does not necessarily lead to an improvement in TQ metrics over the standard midout circuit in the uniform error model, it is expected to significantly outperform under leakage-prone physical implementations—a feature not captured by such simplified simulations.

5. Comparison to Other Circuit and Decoding Strategies

The semi-wiggling color code is related to, but distinct from, other recent circuit innovations. The “middle-out” and “superdense” color code circuits (Gidney et al., 2023) pursue different aims: the middle-out circuit achieves higher density (qubits per logical operation), while the superdense circuit fuses stabilizer measurements via Bell-multiplexed ancillas, reducing CNOT layers. Neither directly addresses leakage mitigation via role alternation of data and measurement qubits.

In decoding, approaches such as the Möbius decoder (Sahay et al., 2021) and VibeLSD (Koutsioumpas et al., 21 Aug 2025) focus exclusively on optimal matching and ensemble message-passing decoding, respectively, to close the logical error rate gap to the surface code. These are complementary to the semi-wiggling circuit design, which operates at the circuit and hardware interface level and is agnostic to the specifics of classical decoding so long as syndrome extraction remains faithful to code commutativity and stabilizer structure.

6. Practical and Experimental Implications

The construction is believed to be especially advantageous for platforms (e.g., superconducting transmons, nitrogen-vacancy centers) where leakage outside the computational basis (|0⟩/|1⟩) is a serious error channel, and specialized reset protocols can be selectively applied only to measurement qubits. By ensuring all qubits are periodically reset, semi-wiggling can keep the effective leakage error rate within the code below the single-round threshold, enabling fault tolerance in regimes otherwise inaccessible to standard circuits.

When combined with depth-reducing gate substitutions (CXSWAP) and hardware-native layouts (planar connectivity via 2D nearest-neighbor constraints), the semi-wiggling color code approach offers a route to higher syndrome extraction rates and more robust operation under real device constraints. These improvements are expected to be most significant in noisy intermediate-scale quantum (NISQ) regimes and first-generation FTQC experiments.

7. Summary Table

Feature	Semi-Wiggling Circuit (Yoshida et al., 1 Oct 2025)	Conventional Midout Circuit	Middle-Out Circuit (Gidney et al., 2023)
Data/Meas. Role	Periodically alternated (bulk)	Fixed	Fixed
Leakage Mitigation	Yes (via periodic resets)	No	No
Depth Reduction	Possible (with CXSWAP substitution)	No	No
Boundary Handling	Static, matching original code	Static	Static
TQ Footprint (p=0.1%)	Similar to midout; 10% lower with CXSWAP	Baseline	Higher for equivalent distance
Targeted Error Model	Leakage errors and uniform depolarizing	Uniform depolarizing	Uniform depolarizing

References

Semi-wiggling and CXSWAP-based circuits: (Yoshida et al., 1 Oct 2025)
Middle-out and superdense circuits: (Gidney et al., 2023)
Möbius decoder: (Sahay et al., 2021)
VibeLSD and decoding performance: (Koutsioumpas et al., 21 Aug 2025)

In sum, the semi-wiggling color code is a circuit construction where periodic interchange of bulk data and measurement qubit roles provides intrinsic leakage mitigation, while CXSWAP-based depth reduction further minimizes logical error rates under circuit-level noise models. The approach is especially relevant for next-generation quantum processors where leakage and shallow depth are critical constraints, and it integrates naturally with modern decoding and syndrome extraction pipelines.