Single-Mode Grid-Code Operations Approaching 1% Logical Errors, Part 1: Theory

Grid codes encode quantum information in bosonic states of an oscillator. By taking advantage of the redundancy in a single physical system, they offer a hardware-efficient route for fault-tolerant quantum computing [1]. The quality of logical grid state preparation and measurement (SPAM) is critical to achieve logical fidelities useful for quantum computing applications. Currently, the experimentally demonstrated schemes for SPAM in the superconducting architecture reach an infidelity of about 10%. State preparation of grid states based on sequences of echoed conditional displacements (ECD) using an auxiliary qubit [2] is limited by the trade-off between incoherent errors on the auxiliary qubit and coherent grid-state errors arising from the limited depth of the preparation circuit. Logical measurements based on a sequence of two ECDs [3] are limited by their lack of robustness against correctable photon loss errors affecting the oscillator's state. Here, we discuss improved SPAM protocols and benchmarking showing increased fidelities, using multi-round schemes, auxiliary mid-circuit measurements and postselection.

[1] M. Lemonde et al., arXiv : 2409.05813 (2024)

[2] A. Eickbusch et al. Nat. Phys. 18, 1464–1469 (2022)

[3] B. Royer et al., Phys. Rev. Lett. 125, 260509 (2020)