Clifford+Toffoli is a universal gate set. What's a cheap way to approximately (or exactly) get a T state using this gate set? I know that once I have a T state I can catalyze more very easily, but how do I get the first one?

Here's my initial attempt. I tried some random stuff to make anything that overlaps with a T state, then copied it three times to produce three T state, expanded them into phase gradient states, then used kickback from adding into those states to do the T gates of a 15-to-1 factory. This produced a state with an infidelity below $10^{-4.5}$. But obviously this is suboptimal in its size and success rate:

Clifford+Toffoli+M is a universal gate set. What's a cheap way to approximately (or exactly) get a T state using this gate set? I know that once I have a T state I can catalyze more very easily, but how do I get the first one?

Here's my initial attempt. I tried some random stuff to make anything that overlaps with a T state, then copied it three times to produce three T state, expanded them into phase gradient states, then used kickback from adding into those states to do the T gates of a 15-to-1 factory. This produced a state with an infidelity below $10^{-4.5}$. But obviously this is suboptimal in its size and success rate:

Clifford+Toffoli is a universal gate set. What's a cheap way to approximately (or exactly) get a T state using this gate set? I know that once I have a T state I can catalyze more very easily, but how do I get the first one?

Here's my initial attempt. I tried some random stuff to make anything that overlaps with a T state, then copied it three times to produce three phase gradient states, then used kickback from adding into those states to do the T gates of a 15-to-1 factory. This produced a state with an infidelity below $10^{-4.5}$. But obviously this is suboptimal in its size and success rate:

Clifford+Toffoli is a universal gate set. What's a cheap way to approximately (or exactly) get a T state using this gate set? I know that once I have a T state I can catalyze more very easily, but how do I get the first one?

Here's my initial attempt. I tried some random stuff to make anything that overlaps with a T state, then copied it three times to produce three T state, expanded them into phase gradient states, then used kickback from adding into those states to do the T gates of a 15-to-1 factory. This produced a state with an infidelity below $10^{-4.5}$. But obviously this is suboptimal in its size and success rate: