Preparing |T> is measuring (X+Y) by synthesizing Controlled_(X+Y) and then measuring the control bit. To take a step back, can you easily prepare the state |Controlled_(C_XYZ)> defined to be Controlled_(C_XYZ)|+>|Phi>, where |Phi> is a Bell State? If so you can Clifford convert that state to |CCZ>

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Thanks!

Off the top of your head, any idea if quantum communication complexity helps (e.g. reduced expected communication or increased success probability) for this specific problem?

Nice! I'm putting in n=4, k=2, d=2 and leaving everything else blank - no results. Also is H the stabilizer and T the destabilizer?