Identifying quantum resources in encoded computations

J. Davis

doi:10.1038/s41534-026-01263-6

Identifying quantum resources in encoded computations

2026-05-29

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DOI：10.1038/s41534-026-01263-6

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Characterizing the origin of quantum computational advantage amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts. Fault-tolerant computation, however, requires error-correcting codes to encode logical information into a larger, physical quantum system. This makes 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 framework to correctly identify quantum resources in general encoded computations. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetry of the code space is contained within the symmetry 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 show that its negativity provides a measure of magic for the logical content of a state and that its marginals are modular measurement distributions associated with conjugate Zak patches.

Keywords:

Quantum resources

encoded computations

Wigner function

quantum advantage

phase-space negativity

Identifying quantum resources in encoded computations

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