A Map of What Quantum Computers Can Do

Plain-language explainer for doi:10.5281/zenodo.20667176 (#408)

The central idea in one sentence

The Fano plane — 7 points, 7 lines — is the correct geometric framework for understanding qubits, and it reveals a three-level hierarchy of quantum computational power that the standard textbook picture misses entirely.

What the Fano plane has to do with qubits

The 3-qubit Pauli group (the group of all tensor products of Pauli matrices $I$, $X$, $Y$, $Z$) has 63 non-identity elements. These fall into 7 groups of 9, one for each point of the Fano plane. The 7 Fano lines correspond to the 7 stabiliser generators of the GHZ state — and by the result of Paper 363, these generators produce the Steane quantum error-correcting code.

The Fano plane is not just a mnemonic. It is the actual geometry that governs which multi-qubit Pauli operators commute, which anti-commute, and which are “magic” (not in the stabiliser group).

The three-level hierarchy

The Fano plane reveals three distinct levels of quantum resource:

Level Name What you need Protected by
0 Stabiliser Clifford gates only Nothing (classically simulable)
1 Standard magic T gate ($e^{i\pi/4}$) Wigner negativity
2 Associamancy SPIN opcode ($i\sqrt{7}/2$) Topological (Weyl group $D_6$)

Level 0 is the “free” resource. Level 1 is what makes standard quantum computers powerful. Level 2 — associamancy — is new to this paper: it requires hardware with $G_2$ symmetry (the SevenQ register or a native $G_2$-symmetric Hamiltonian) and provides topological protection unavailable at Levels 0 and 1.

The jump from Level 1 to Level 2 is the Schur boundary: the set of quantum states whose hidden symmetry group contains a genuinely complex irreducible representation (Frobenius-Schur indicator $\nu_2 = 0$).

The Origami ISA in one paragraph

The Origami ISA is the five-opcode instruction set (SPLIT, SPLAT, TWIST, FLIP, FLOP) that covers Levels 0 and 1. The 731 ISA extends it with SPIN and BIND to reach Level 2. The SevenQ — a 7-qubit register with the 7 qubits corresponding to the 7 Fano points — is the minimal hardware for Level 2.

This paper is the entry point designed to introduce all of this to a quantum computing researcher who knows stabiliser states and T gates but has never heard of the Origami ISA.

For the full technical treatment, see doi:10.5281/zenodo.20667176