Low Depth Color Code Circuits with CXSWAP gate

TL;DR

This work addresses the depth cost of syndrome extraction in color code QEC circuits by introducing semi-wiggling and CXSWAP-based circuit designs. By replacing CNOTs with CXSWAP gates and strategically restructuring the circuit, the CXSWAP midout and CXSWAP superdense variants reduce circuit depth and achieve about a $10\%$ reduction in the teraquop footprint at a physical error rate of $p = 0.1\%$ under a uniform error model. It also presents a leakage-mitigation scheme via semi-wiggle that reassigns data/measurement roles in the bulk without increasing depth, and analyzes the boundary implementation with auxiliary qubits. These results, supported by Stim-based simulations and a Möbius decoder, suggest practical improvements for color-code syndrome extraction on 2DNN hardware, with potential gains from using CXSWAP-like two-qubit gates of higher fidelity.

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 $p=0.1\%$.

Figures (9)

• Figure 1: (a) One cycle of a bulk fragment of the midout circuit consisting of a reset step, 6 CNOT steps, and a measurement step. The X basis detector (red) and Z basis detector (blue) start from a 6-body stabilizer in the reset step, shrink to a one-body stabilizer in the CNOT layers, and are measured in the measurement step. The stabilizer for the color code appears after the 4th step. (b) One cycle of a bulk fragment of the semi-wiggling midout circuit. CNOT direction in the 7th step is flipped relative to the original midout circuit causing a one-column shift of measurement qubits. (c) One cycle of a bulk fragment of the CXSWAP midout circuit. This circuit is obtained by contracting the middle two CNOT steps of (b) to one CXSWAP step in the 4th step, and replacing the other CNOT steps with the CXSWAP steps followed by propagating the SWAP gates to the reset and measurement steps.
• Figure 2: The locations shown in the diagrams on the left and right are alternately used as measurement qubits in each cycle of the semi-wiggling midout circuit.
• Figure 3: (a) The construction of the CXSWAP midout circuit shown in Fig. \ref{['fig:bulk_midout_circuit']} (c) generates diagonal gates on the boundary in the 2nd and 6th steps. (b) The diagonal gates in the 2nd step of (a) are replaced with the horizontal and vertical gates in the 2nd and 3rd steps, respectively. Similarly, the diagonal gates in the 6th steps of (a) are replaced with the vertical and horizontal gates in the 5th and 6th steps, respectively.
• Figure 4: (a) One cycle of a bulk fragment of the original superdense circuit consisting of a reset step, 8 CNOT steps, and a measurement step. The Z detector shrinks to a 2-body stabilizer in the first 4 CNOT steps, and the X detector shrinks to a 2-body stabilizer in the following 3 CNOT steps. The last CNOT step shrinks the 2-body stabilizer to a 1-body stabilizer, which is measured in the last step. (b) A modified superdense circuit that is obtained by swapping the 6th and 8th steps in (a). (c) One cycle of a bulk fragment of the CXSWAP superdense circuit. This circuit is obtained by contracting the middle two CNOT steps of (b) to one CXSWAP step in the 5th step, and replacing the other CNOT steps with the CXSWAP steps followed by propagating the SWAP gates to the reset and measurement steps.
• Figure 5: (a) The construction of the CXSWAP midout circuit shown in Fig. \ref{['fig:bulk_superdense_circuit']} (c) generates diagonal gates on the boundary in the 5th step. (b) The diagonal gates in the 5th step of (a) are replaced with the horizontal and vertical gates in the 4th, 5th, and 6th steps.