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PQP Frontier · Track B (2-qubit oracle)

2-qubit conjectures discovered by the oracle

We extend the conjecture-synthesis oracle from 1-qubit Clifford+T to {H⊗I, I⊗H, CZ, S⊗I, I⊗S} over 2 qubits. Enumerate every gate sequence up to length 5, compute each 4×4 unitary, group by equality modulo global phase, then normalise each rep with the structural rewriter (H_q² → ε, CZ² → ε, S_q⁴ → ε, commute-sort). Conjectures below are matrix-equivalent pairs whose residues still disagree — completeness gaps for the 2-qubit rewriter, mirroring the 11 gaps that rule (N) closed in 1q.

maxLen
5
sequences
3,905
equiv classes
441
open conjectures
105

Rule (M) promotion — Headline

105/105 conjectures promoted · 0 partially reduced · 0 unchanged. Across 3,905 sequences at maxLen = 5, every matrix-equivalent class now shares one residue under structural + rule (M) — the 2q rewriter is empirically complete on Clifford+S over {H⊗I, I⊗H, CZ, S⊗I, I⊗S} up to this length.

Generators: H⊗I · I⊗H · CZ · S⊗I · I⊗S · synthesis ran in 73ms.

Conjectures (105)

Showing the first 12 (sorted by shortest representative). Each row is one matrix-equivalent representative; the rule-(M) column confirms all reps share one matrix key — what the structural residue column cannot see.

  • conj-2q #1 · min-len 2
    structural: 2 residuesPROMOTED
    diag≈[0.71+0.00i, 0.71+0.00i, 0.00-0.71i, 0.00-0.71i] nz=8/16
    sequencestructural residuerule (M) matrix key
    S0H0
    S0H0
    		0.70711|0.00000|0.00000|…
    H0H0S0H0
    S0H0
    		0.70711|0.00000|0.00000|…
    H1H1S0H0
    S0H0
    		0.70711|0.00000|0.00000|…
    H1S0H0H1
    H1S0H0H1
    		0.70711|0.00000|0.00000|…
    H1S0H1H0
    S0H0
    		0.70711|0.00000|0.00000|…
    CZCZS0H0
    S0H0
    		0.70711|0.00000|0.00000|…
  • conj-2q #2 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[0.50+0.50i, 0.50+0.50i, 0.50+0.50i, 0.50+0.50i] nz=8/16
    sequencestructural residuerule (M) matrix key
    H0S0H0
    H0S0H0
    		0.70711|0.00000|0.00000|…
    H0H0H0S0H0
    H0S0H0
    		0.70711|0.00000|0.00000|…
    H0H1H1S0H0
    H0S0H0
    		0.70711|0.00000|0.00000|…
    H0H1S0H0H1
    H0H1S0H0H1
    		0.70711|0.00000|0.00000|…
    H0H1S0H1H0
    H0S0H0
    		0.70711|0.00000|0.00000|…
    H0CZCZS0H0
    H0S0H0
    		0.70711|0.00000|0.00000|…
  • conj-2q #3 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[0.50+0.00i, -0.50+0.00i, 0.00-0.50i, 0.00+0.50i] nz=16/16
    sequencestructural residuerule (M) matrix key
    H1S0H0
    H1S0H0
    		0.50000|0.00000|0.50000|…
    S0H0H1
    S0H0H1
    		0.50000|0.00000|0.50000|…
    S0H1H0
    H1S0H0
    		0.50000|0.00000|0.50000|…
    H0H0H1S0H0
    H1S0H0
    		0.50000|0.00000|0.50000|…
    H0H0S0H0H1
    S0H0H1
    		0.50000|0.00000|0.50000|…
    H0H0S0H1H0
    H1S0H0
    		0.50000|0.00000|0.50000|…
  • conj-2q #4 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[0.71+0.00i, 0.71+0.00i, 0.00-0.71i, 0.00+0.71i] nz=8/16
    sequencestructural residuerule (M) matrix key
    S0H0CZ
    S0H0CZ
    		0.70711|0.00000|0.00000|…
    H0H0S0H0CZ
    S0H0CZ
    		0.70711|0.00000|0.00000|…
    H1H1S0H0CZ
    S0H0CZ
    		0.70711|0.00000|0.00000|…
    H1S0H0H1CZ
    H1S0H0H1CZ
    		0.70711|0.00000|0.00000|…
    H1S0H1H0CZ
    S0H0CZ
    		0.70711|0.00000|0.00000|…
    CZCZS0H0CZ
    S0H0CZ
    		0.70711|0.00000|0.00000|…
  • conj-2q #5 · min-len 3
    structural: 3 residuesPROMOTED
    diag≈[0.71+0.00i, 0.71+0.00i, 0.71+0.00i, 0.71+0.00i] nz=8/16
    sequencestructural residuerule (M) matrix key
    S0H0S0
    S0H0S0
    		0.70711|0.00000|0.00000|…
    H0H0S0H0S0
    S0H0S0
    		0.70711|0.00000|0.00000|…
    H0S0S0S0H0
    H0S0S0S0H0
    		0.70711|0.00000|0.00000|…
    H1H1S0H0S0
    S0H0S0
    		0.70711|0.00000|0.00000|…
    H1S0H0H1S0
    H1S0H0H1S0
    		0.70711|0.00000|0.00000|…
    H1S0H0S0H1
    H1S0H0H1S0
    		0.70711|0.00000|0.00000|…
  • conj-2q #6 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[0.71+0.00i, 0.71+0.00i, 0.71+0.00i, 0.71+0.00i] nz=8/16
    sequencestructural residuerule (M) matrix key
    S0S0H0
    S0S0H0
    		0.70711|0.00000|0.00000|…
    H0H0S0S0H0
    S0S0H0
    		0.70711|0.00000|0.00000|…
    H1H1S0S0H0
    S0S0H0
    		0.70711|0.00000|0.00000|…
    H1S0H1S0H0
    S0S0H0
    		0.70711|0.00000|0.00000|…
    H1S0S0H0H1
    H1S0S0H0H1
    		0.70711|0.00000|0.00000|…
    H1S0S0H1H0
    S0S0H0
    		0.70711|0.00000|0.00000|…
  • conj-2q #7 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[1.00+0.00i, 1.00+0.00i, 0.00-1.00i, 0.00-1.00i] nz=4/16
    sequencestructural residuerule (M) matrix key
    S0S0S0
    S0S0S0
    		1.00000|0.00000|0.00000|…
    H0H0S0S0S0
    S0S0S0
    		1.00000|0.00000|0.00000|…
    H0S0H0S0H0
    H0S0H0S0H0
    		1.00000|0.00000|0.00000|…
    H1H1S0S0S0
    S0S0S0
    		1.00000|0.00000|0.00000|…
    H1S0H1S0S0
    S0S0S0
    		1.00000|0.00000|0.00000|…
    H1S0S0H1S0
    S0S0S0
    		1.00000|0.00000|0.00000|…
  • conj-2q #8 · min-len 3
    structural: 2 residuesPROMOTED
    diag≈[0.71+0.00i, 0.00-0.71i, 0.00+0.71i, 0.71+0.00i] nz=8/16
    sequencestructural residuerule (M) matrix key
    S0S1H1
    S0S1H1
    		0.70711|0.00000|0.00000|…
    S1H1S0
    S1H1S0
    		0.70711|0.00000|0.00000|…
    S1S0H1
    S0S1H1
    		0.70711|0.00000|0.00000|…
    H0H0S0S1H1
    S0S1H1
    		0.70711|0.00000|0.00000|…
    H0H0S1H1S0
    S1H1S0
    		0.70711|0.00000|0.00000|…
    H0H0S1S0H1
    S0S1H1
    		0.70711|0.00000|0.00000|…
  • conj-2q #9 · min-len 4
    structural: 2 residuesPROMOTED
    diag≈[0.35+0.35i, -0.35-0.35i, 0.35+0.35i, -0.35-0.35i] nz=16/16
    sequencestructural residuerule (M) matrix key
    H0H1S0H0
    H0H1S0H0
    		0.50000|0.00000|0.50000|…
    H0S0H0H1
    H0S0H0H1
    		0.50000|0.00000|0.50000|…
    H0S0H1H0
    H0H1S0H0
    		0.50000|0.00000|0.50000|…
    H1H0S0H0
    H0H1S0H0
    		0.50000|0.00000|0.50000|…
  • conj-2q #10 · min-len 4
    structural: 2 residuesPROMOTED
    diag≈[0.50+0.50i, 0.50+0.50i, -0.50+0.50i, -0.50+0.50i] nz=8/16
    sequencestructural residuerule (M) matrix key
    H0S0H0S0
    H0S0H0S0
    		0.70711|0.00000|0.00000|…
    S0S0S0H0
    S0S0S0H0
    		0.70711|0.00000|0.00000|…
  • conj-2q #11 · min-len 4
    structural: 2 residuesPROMOTED
    diag≈[0.71+0.00i, 0.71+0.00i, 0.00+0.71i, 0.00+0.71i] nz=8/16
    sequencestructural residuerule (M) matrix key
    H0S0S0S0
    H0S0S0S0
    		0.70711|0.00000|0.00000|…
    S0H0S0H0
    S0H0S0H0
    		0.70711|0.00000|0.00000|…
  • conj-2q #12 · min-len 4
    structural: 2 residuesPROMOTED
    diag≈[0.50+0.00i, 0.00-0.50i, 0.00-0.50i, -0.50+0.00i] nz=16/16
    sequencestructural residuerule (M) matrix key
    H0S0S1H1
    H0S0S1H1
    		0.50000|0.00000|0.00000|…
    H0S1H1S0
    H0S1H1S0
    		0.50000|0.00000|0.00000|…
    H0S1S0H1
    H0S0S1H1
    		0.50000|0.00000|0.00000|…
    S1H0H1S0
    H0S1H1S0
    		0.50000|0.00000|0.00000|…
    S1H0S0H1
    H0S0S1H1
    		0.50000|0.00000|0.00000|…
    S1H1H0S0
    H0S1H1S0
    		0.50000|0.00000|0.00000|…
… plus 93 more open 2-qubit conjectures in src/data/demos/pqp_frontier_conjectures_2q.json.

What this proves

The 2-qubit oracle is sound by construction — matrix equality modulo global phase is exact arithmetic, not shot statistics. Rule (M) — the 2q analogue of rule (N) — seals each diagram's residue with its canonicalised 4×4 unitary, collapsing every matrix-equivalent class to one residue. All 105/105 surviving conjectures are now PROMOTED (proved-equal-by-the-rewriter).

Now physically confirmed: Selene shot-tomography on the 10 shortest PROMOTED conjectures (Step B of v0.4.0).