Why does the Solovay-Kitaev theorem use the operator norm?

Question

In this review explaining the Solovay-Kitaev theorem, it is stated that the theorem uses the operator norm to define closeness between unitaries. This is then used to determine if a particular set of gates is universal. That is, we require the condition

$$d(U,S) := \sup _{\|\psi\|=1}\|(U-S) \psi\|<\epsilon$$

to decide if the unitary $S$ we constructed from our gate set is a good approximation of the desired unitary $U$.

My question is - why is the operator norm the right choice here? Shouldn't one consider a notion like the diamond distance where we consider the action of $U$ and $S$ on entangled states and see if those outputs are close?

Answer 1

For a linear map $A$ on the Hilbert space (e.g., $A=U-S$), the two norms are equivalent -- they merely differ by a square.

This is a simple exercise, using that the norm which is maximized over all density operators $\rho$ in the diamond norm is convex in $\rho$ (due to the triangle inequality), and thus acquired its maximum on pure states -- for which the trace norm (which is maximized in the diamond norm) and the Hilbert space $2$-norm (which is maximized in the operator norm) are identical (up to a square).