Identifying quantum resources in encoded computations

Abstract: What is the origin of quantum computational advantage? Providing answers to this far-reaching question amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts, with direct applications to the development of quantum devices. The advent of universal quantum computers, however, relies on error-correcting codes to protect fragile logical quantum information by robustly encoding it into symmetric states of a quantum physical system. Such encodings make the task of resource identification more difficult, as what constitutes a resource from the logical and physical points of view can differ significantly. Here we introduce a general framework which allows us to correctly identify quantum resources in encoded computations, based on phase-space techniques. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetries of the code space are contained within the transformations of the physical space, resulting in an object capable of describing the logical content of any physical state, both within and outside the code space. We illustrate our general construction with the Gottesman--Kitaev--Preskill encoding of qudits with odd dimension. The resulting Wigner function, which we call the Zak-Gross Wigner function, is shown to correctly identify quantum resources through its phase-space negativity. For instance, it is positive for encoded stabilizer states and negative for the bosonic vacuum. We further prove several properties, including that its negativity provides a measure of magic for the logical content of a state, and that its marginals are modular measurement distributions associated to conjugate Zak patches.