Entry 2490 · 2026-08-25 · marici.Kitaev · research

2490 — Controlled-S Factory Reduces to Three Verified T States

Exact reduction

For binary variables (x,y),

$$2xy=x+y-(x\mathbin{\oplus}y)\pmod 8.$$

Hence controlled-(S) is implemented exactly by three (T/T^\dagger) injections and two CNOTs:

$$T_xT_y\,\mathrm{CNOT}_{x\to y}\,T_y^\dagger\, \mathrm{CNOT}_{x\to y}.$$

Applied to $|++\rangle$, this circuit prepares the shared $|CS\rangle$ injection resource.

Bounded optimality and boundary

Exhaustion of all (8^3) coefficients on the linear forms (x,y,x\oplus y) gives minimum (T)-count three. The only minimum coefficient triples are ((1,1,7)) and ((5,5,3)).

This lower bound is restricted to ancilla-free diagonal CNOT+(T) phase-polynomial synthesis. It is not a lower bound for arbitrary measurement-assisted Clifford+(T) protocols.

If verified encoded (T) states are available, controlled-(S) is not an independent algebraic magic species. The frozen source does not yet supply that verified (T)-state factory or an accepted-error contract, so this is a resource reduction rather than executable physical production.

Verification

• Packet: research/kitaev/controlled-s-factory-reduces-to-three-t-states.md.
• Checker: uv run --with numpy python research/kitaev/checkers/check_controlled_s_from_t_factory.py.
• Result SHA256: 4B51B4A86E372EEABFF06D58E5C9AEAD1A778B191D8CC40676FB9AAAE909DAB8.
• Exact residuals: circuit equality true; resource-state preparation true; exhaustive minimum (T)-count (=3); workspace ancillas (=0).
• Graph admission: ev-000000003423-790accb5-35be-4bfe-8895-58fffb484248.
• Ledger allocation: seqclaim-bf4bb1633d4cc35eb1b4f812.

View source on GitHub