Parity Codes in Space and Time

We present a single framework to describe parity codes in exclusively spatial dimensions as quantum error correction codes, as well as parity codes with a temporal dimension, as quantum circuits. This framework builds on the stabilizer formalism to introduce label tracking of parity information for illustrating and constructing quantum circuits and quantum error correction codes. Additionally, this framework proposes the necessary transformations and conditions to translate between the spatial and temporal versions of a given parity code. This technique is then applied to find the temporal variant of the spatially extended LHZ-code [1] using CNOT gates on a linear nearest-neighbor (LNN) qubit architecture. The resulting quantum circuit provides an efficient implementation of quantum algorithms on LNN architectures, that require the interactions found in all-to-all connected transverse-field Ising models. Two widely used quantum algorithms that require such interactions are the Quantum Approximate Optimization Algorithm (QAOA) [2] and the quantum Fourier transform (QFT). For the QAOA, this is the first proposal with a linear circuit depth while not requiring more than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n^{2}$</tex> CNOT gates per iteration. Our approach for QFT, requiring only LNN connectivity, surpasses previous QFT implementations even on devices with high qubit connectivity in terms of circuit depth while not increasing the gate count in leading order.