CXSWAP Gate: Quantum Circuit Optimization

• The CXSWAP gate is a two-qubit unitary that generalizes the SWAP operation by integrating CNOT and single-qubit Clifford operations, enabling optimized quantum circuit designs.
• It reduces circuit depth in color code syndrome extraction by consolidating consecutive CNOT layers, thereby lowering logical error rates and shrinking qubit footprints by roughly 10% under a 0.1% error model.
• The gate enhances error correction through preserved Pauli operator mappings and effective leakage mitigation, making it suitable for superconducting and spin qubit architectures.

The CXSWAP gate is a quantum logic gate with structure and applications that extend and generalize the SWAP operation to richer circuit paradigms. It offers reductions in circuit depth, optimizations in quantum error correction layouts, and compatibility with hardware primitives in superconducting and spin qubit architectures. The CXSWAP is defined as a composite gate built from a CNOT and a SWAP, or alternatively as a Clifford conjugation of an ISWAP. Its propagation properties and its implementation enable improvements in quantum error correction, notably within color code circuits and leakage mitigation strategies.

1. Formal Definition and Structure

The CXSWAP gate is defined as a two-qubit unitary that combines the action of a CNOT gate and a SWAP gate. The relationship to the ISWAP gate is precisely mapped using single-qubit Clifford conjugation:

$$\text{CXSWAP} = (S^\dagger \otimes H S^\dagger) \cdot \text{ISWAP} \cdot (H \otimes I)$$

where $S^\dagger$ is the conjugate phase gate, $H$ is the Hadamard gate, and $I$ is the identity. The CXSWAP maps Pauli X and Z operators to Pauli operators of the same type, a property crucial for circuit and stabilizer analysis.

An additional identity connects the CXSWAP and CNOTs directly:

$$\text{CXSWAP}_{1,2} = \text{CNOT}_{1,2} \cdot \text{CNOT}_{2,1}$$

Further, circuit propagation relations provide:

$$\text{CNOT}_{1,2} = \text{CXSWAP}_{2,1} \cdot \text{SWAP}_{1,2}, \quad \text{CNOT}_{1,2} = \text{SWAP}_{1,2} \cdot \text{CXSWAP}_{1,2}$$

These algebraic relations underlie the optimization strategies for circuit construction in quantum error correction.

2. Circuit Depth Reduction in Color Codes

The use of the CXSWAP gate enables the combination of consecutive CNOT circuit layers into single CXSWAP layers. Specifically, in syndrome extraction circuits for the color code, the introduction of CXSWAP in place of adjacent pairs of CNOTs contracts the circuit depth:

• In the midout color code circuit, the depth is reduced from 8 layers to 7.
• In the superdense circuit, the depth is reduced from 10 to 9 layers.

The reduction in circuit depth is directly linked to a decrease in logical error rates, since each gate and layer provides an additional opportunity for error propagation. Empirical simulations (under a uniform error model with $p=0.1\%$) show a 10% improvement in the so-called "teraquop footprint"—the number of qubits required for achieving a logical error rate of $10^{-12}$—when CXSWAP is employed (Yoshida et al., 1 Oct 2025).

Circuit Variant	Depth (CNOT)	Depth (CXSWAP)	Footprint Improvement
Midout	8	7	~10%
Superdense	10	9	~10%

This feature becomes particularly important in scaling fault-tolerant quantum computation.

3. Error Propagation and Leakage Mitigation

A key advantage of the CXSWAP gate over the ISWAP and even CNOT is the mapping of Pauli X and Z operators under conjugation, which preserves the stabilizer structure during syndrome extraction. This simplifies error analysis and propagation:

• The CXSWAP preserves the form of X and Z operators, facilitating straightforward stabilizer circuit design.
• In practical architectures (notably superconducting qubits), the CXSWAP can be calibrated for improved fidelity relative to CNOT or CZ gates.

For error mitigation:

• Shorter circuit depth minimizes total error accumulation and exposure to leakage errors, since qubits spend less time unmeasured.
• Incorporation of CXSWAP aids in leakage management by enabling frequent qubit resets, which are essential in hardware that suffers from leakage outside the logical subspace.

Experiments indicate that the CXSWAP gate yields a more favorable error footprint than CNOT in these contexts (Yoshida et al., 1 Oct 2025).

4. The Semi-Wiggling Color Code Construction

The semi-wiggling color code is introduced as a strategy for leakage error suppression. In this construction:

• The roles of data and measurement qubits are periodically interchanged ("wiggled") in the bulk lattice.
• By flipping the direction of a final CNOT or CXSWAP step, a two-body stabilizer is shifted across columns, thus regularly resetting data qubits.

This enables reset-based leakage mitigation over complete circuit cycles, which is not possible in conventional color code layouts where data qubits remain unmeasured for several rounds.

Though it does not reduce depth compared to the base midout circuit, the semi-wiggling code improves the practical efficiency of leakage suppression.

Approach	Depth Change	Leakage Mitigation
Midout + CXSWAP	Reduced	Improved
Semi-wiggling color code	Unchanged	Enhanced

A plausible implication is that reset-based hardware error mitigation strategies become more effective when combined with the semi-wiggling scheme.

5. Hardware Implementation and Clifford Relations

The CXSWAP gate is inherently Clifford, constructed via single-qubit Clifford operations and native two-qubit gates such as ISWAP, CZ, and CNOT. In superconducting architectures, where ISWAP- and iSWAP-type interactions can be implemented directly, the Clifford relations allow flexible compilation of complex gates:

• The CXSWAP can be synthesized with fewer gates compared to a cascade of two CNOTs, with the hardware primitive directly matching the Clifford unitary.
• Propagation of SWAPs and CNOTs through the circuit can be optimized using the algebraic identities given above, enabling reduced footprint and improved coherence.

The propagation properties further streamline the design of stabilizer circuits such as those for color code error correction.

6. Comparison with Conventional Gates

Compared to the standard CNOT-based circuits, CXSWAP-based circuits show:

• Reduced number of two-qubit gates and lower circuit depth.
• Enhanced leakage error management due to more frequent qubit resets.
• Potential for higher fidelity in error-prone hardware by matching calibration errors and leakage channels.

While simulations assume similar underlying error rates for CNOT and CXSWAP, in practice, CXSWAP may enable more accurate implementation, further contributing to reduction in logical error rates and resource count for large-scale fault-tolerant quantum computation.

7. Implications for Fault-Tolerant Quantum Computation

The use of the CXSWAP gate constitutes a significant optimization for color code circuits and possibly other stabilizer codes. The reduction in teraquop footprint (~10% as reported for $p=0.1\%$) demonstrates its impact on scalable quantum computing resources. The additional leakage mitigation capability afforded by the semi-wiggling color code structure and frequent resets, combined with favorable propagation properties, positions the CXSWAP as an essential primitive for the next generation of error correction and fault-tolerant circuit compilation.

This suggests broader applicability in quantum architectures where leakage errors, circuit depth, and stabilization of syndrome extraction are primary limiting factors.