I am not familiar with the Clifford group - I do know that Clifford unitaries can form a unitary 3-design (from this paper) and can be used to approximate Haar random unitaries, but I don't know how good that approximation is. Context for this question: I am studying a quantum circuit that relies on the invariance of the Haar measure to work properly, and I am trying to figure out how well the circuit can be approximated with Clifford gates.
1 Answer
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In general, a unitary $t$-design is defined as a quantity that gives same expectation values as the Haar measure on the unitary group for homogenous polynomial functions (of $U$ and $U^*$) of degree $t$, roughly speaking 1.
So, by approximation one means that when one wants to calculate the expectation value of such a function over the unitary group, one doesn't need to evaluate the expectation by sampling (acc. to the Haar measure) from this continuous group but can just pick elements uniformly from a finite set of unitary matrices that form a $t$-design. In this sense one could say that a $t$-design is good upto the $t$-th moment of the Haar measure.
So, it boils down to the properties of the metric/function you are using in your problem and if it is a homogenous polynomial of degree 3 or less.
