← Quantum Reed–Muller code

Entry facet

Representation ladder

R1 checks through R5 cross-model views, with canonicality and provenance.

  1. R1

    Checks

    Sparse Pauli generators in symplectic form, including stabilizer and gauge rows.

    • symplectic-quotient derived by qeclib.gf2.nullspace + qeclib.distance.exact_distance v1

      The four X rows and ten Z rows form fourteen independent commuting symplectic generators. Their centralizer modulo their span leaves the all-one X representative paired with a weight-three Z representative; exact CSS enumeration finds no lighter nontrivial logical operator.

      References: steane1996quantumrm

  2. R2

    Structure

    CSS pairs, chain complexes, lattices, and other structure behind the checks.

    • punctured-boolean-evaluation canonical

      Label the qubits by the fifteen nonzero four-bit strings. Evaluating each coordinate bit across those labels gives four independent weight-eight X checks. Appending the all-one evaluation selects the logical coset; the binary orthogonal complement then gives the ten Z checks.

      References: steane1996quantumrm

    • tetrahedral-color-code curated from kubica2015universal

      The same incidence data can be drawn as a tetrahedral three-dimensional color-code patch. Cells and faces make the check supports geometric; this is a dictionary for the Reed–Muller evaluation table, not a second stabilizer code with merely coincident parameters.

      References: kubica2015universal

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    No representation at this rung is recorded for this entry.

  4. R4

    Circuit & detectors

    Named syndrome-extraction circuits and detector error models.

    No representation at this rung is recorded for this entry.

  5. R5

    Cross-model views

    Executable or illuminating algebraic, ZX, fermionic, tensor, and statistical dictionaries.

    • triorthogonal derived by qeclib.triortho.transversal_t_candidate + coset-weight enumeration v1

      Put the odd all-one logical row above the four even X-check rows. Every pair and every triple has even overlap. The check-row coset has weight zero modulo eight, while its all-one complement has weight seven modulo eight; this fixes both transversality and the logical T-versus-T-dagger convention.

      References: bravyihaah2012magic, bravyi2005magic

    • gauge-fixed-double curated from bravyi2015doubled

      A common doubled-color-code subsystem exposes the Steane and fifteen-qubit blocks as different gauge choices. Measuring the checks that distinguish those choices turns code switching into a syndrome-conditioned operation rather than an unexplained change of stabilizer tables.

      References: bravyi2015doubled, anderson2014conversion

    • tensor-network-encoder curated from cao2022quantumlego

      Regard the one-to-fifteen encoding isometry as a tensor with one logical leg and fifteen physical legs. Operator pushing recovers the same stabilizer and logical cosets, while contracting multiple copies creates a new code whose global parameters must be recomputed rather than inherited from this block.

      References: cao2022quantumlego

    • phase-sensitive-xp curated from webster2022xp

      The binary symplectic table records Pauli supports but discards the eighth-root phases accumulated by a transversal T-type layer. XP stabilizer notation retains those phases, making it a complementary language for diagonal logical action rather than a different origin for the code.

      References: webster2022xp