Low Depth Parity Check Gate Set for Quantum Error Correction

Designing QEC schemes with high error threshold reduces the demands on the physical hardware, and makes practical quantum computing more feasible. In this work, we exploit the insight that a QEC code needs not use universal logic gates, but can be simplified to perform solely the task of error detection and correction. By building gates that are fundamental to QEC rather than universal computation, we can boost the threshold and ease the experimental demands on the physical hardware. We call these gates low-depth parity check gates, since they perform a two-body parity check measurement in a single operation, instead of requiring a sequence of 1-and 2-qubit gates. We insert the low-depth parity check gates within a 'widget' that yields directly the measurement of two-body Pauli Product operators (Mpp2), namely the XX and ZZ operators. We present a rigorous formalism for constructing and verifying the low-depth parity check gate set, designed specifically for QEC, which can also be used for N-qubit parity check circuits. We then proceed to apply this technique to the two-body parity check circuits which are necessary for the implementation of the Honeycomb code, which requires only two-body parity check circuits instead of the four-body parity checks needed in the surface code