A finite circuit specification for every authorized set

Equation (eq:TA) is also an algorithmic circuit specification; it uses no quantum access to $\bar{A}$.

For $m\ge 3$, complete the orthonormal family $\{{u^A}_{i,b}\}$ by applying Gram–Schmidt to the computational basis in lexicographic order, omitting zero residuals. Set $t=m-k-1\ge 0$. Define a unitary $U_A$ that sends

and sends the added basis vectors, in their construction order, to the unused computational basis vectors in lexicographic order.

The decoding program is:

1. apply $U_A$ on the $m$ accessible qubits;

2. read the last $t$ qubits in the computational basis;

3. if their string is $0^t$, return the first qubit and discard the $k$ syndrome qubits;

4. otherwise discard all remaining data and return $I_2/2$.

For $t=0$, the third step is unconditional.

This program implements exactly (eq:DA). The successful projector is $T_A{T}_A^{†}$; coherences between it and its orthogonal complement are removed by the complete measurement.

For a fully finite gate description, one may decompose the specified matrix $U_A$ into two-level unitaries by column elimination. When eliminating entries $a,b$, use, if $r=(|a{|}^2+|b{|}^2{)}^{1/2}>0$,

Its product with $(a,b{)}^{\mathsf{T}}$ is $(r,0{)}^{\mathsf{T}}$. After finitely many eliminations the remaining diagonal phases complete the decomposition. Reversing the factors gives $U_A$. Here arbitrary two-level gates on the declared accessible register are admitted. No gate acts on an omitted register. This is not an efficiency or hardware-native compilation claim.

For triples the simpler specification is

whose entries are explicitly the conjugate transpose of (eq:TA) evaluated using (eq:Vmatrix).