Talk:Solovay–Kitaev theorem

What about U(2^q)?

The extended version of Solovay–Kitaev theorem shows that this is possible to navigate SU(d) efficiently, but quantum gates on q qubits belong to U(2^q). It seems from reading the paper, they really do mean SU(d). I assume that it is considered to be known that U(1) is efficiently generated by a few phase shift gates, and that $SU(2)\times U(1)=U(2)$, which in turn is extensible to U(2^q) via random coupling gate (e.g. CNOT) or as shown in the extended SK??? Can someone please explain this? · · · Omnissiahs hierophant (talk) 19:41, 31 March 2021 (UTC)

My guess is either what you said (it's straightforward to extend to U) or that all the gates in quantum circuits are considered to be phase-shifted so that all its gates are in SU. Not sure about this, though. Fawly (talk) 22:27, 8 April 2021 (UTC)

Any gate U that is in SU(2) have det(U)=1. The determinant of two gates in series is det(U_1 * U_2) = det(U_1) * det(U_2). Most gates that are listed have determinants -1. So, flipping it back to 1 (so it's in SU(2)) is just multiplying it with whatever other gate. Maybe one can think of SU(2) as a simplification of all hermitian gates where symmetric doublets have been removed. A phase shifting gate P have det(P) = the phase it shifts with, and have no other effect beyond changing the phase along a specific basis axis. Multiplying a hermitian gate with phase shifting gates (one per basis vector?) results in a gate from the group U(2). Alternatively as it is explained here. · · · Omnissiahs hierophant (talk) 02:38, 11 April 2021 (UTC)

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