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4

To study unitary $t$-designs, we define the moment operator with respect to a probability measure $\nu$ as $$M_t(\nu) := \int_{U(d)} U^{\otimes t} (\cdot) (U^{\otimes t})^\dagger d\nu(U) \simeq \int_{U(d)} (U \otimes \bar U)^{\otimes t} d\nu(U).$$ Often, designs are defined via subsets $D\subset U(d)$ endowed with some "canonical" measure. For ...

0

Integrated with other time-critical functions inside the (same) superconducting circuit the advantage will be latency as an classical simulator would introduce those. Compounded latency could create a situation where a set of quantum clifford gates would have an edge not achievable with an external classical simulator. https://en.wikipedia.org/wiki/...

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Are there examples of quantum algorithms only composed of Clifford operations that show [...] A reduction in the "same spirit" of the $n^{800}→n$ for instance. No. An $n$ qubit Clifford+measure circuit with $m$ operations can be simulated in $O(n^2m)$ time (arXiv:quant-ph/0406196) with small constant factors (arXiv:2103.02202).

6

Quantum advantage using Clifford gates Gottesman-Knill theorem applies to stabilizer circuits only, not to all circuits consisting of Clifford gates. The former satisfy the stronger requirements of having a stabilizer input state and using only stabilizer basis measurements. Note that availability of magic states enables one to apply non-Clifford gates using ...

2

Your intuition is correct if you think of the superpositions as classical probability distributions, which is how they are usually described in popularizations. What's unique to quantum mechanics is not superposition as such, but constructive/destructive interference between different computational paths, which leads to certain results appearing more/less ...

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