# Which feedback is *necessary* for surface-code based and measurement based quantum computing?

It is clear that a Clifford feedback is necessary for performing a non-Clifford gate with measurement-based quantum computing (for example with Lattice surgery in surface codes).

But, given many logical qubits and non-Clifford gates, can you delay any feedback to the very end of the circuit without modifying anything along the way?

Mathematically, can you always find a Clifford operator $$C_2$$ such that $$C_2P=PC_1$$ for any Pauli projection operator $$P$$ and Clifford operator $$C_1$$?

If so, is it true to say that you do not need any real-time feedback (or gate modification) apart of just before the final measurement of the computational qubits?

I'm not sure I understand why your "mathematically" statement implies that you can delay feedback to the end of the circuit. However, there is a construction that allows you to delay feedback until the end of the circuit, but requires you to hold an additional ancilla qubit for every $$T$$ gate in your circuit. Provided your circuit is only polynomially long, this only requires a polynomial space overhead, but it could be significant.
This is from Fig. 25 of Litinski's Game of Surface Codes paper. It allows you to choose whether a teleported $$\pi/8$$ rotation (generalized T-gate) is replaced by its hermitian conjugate ($$-\pi/8$$ rotation) by measuring an ancilla "control" qubit in either the X or Z basis. It also does not require any Clifford correction, but only a Pauli correction.
This means you can execute your entire circuit without any Clifford corrections. The Pauli corrections will change some $$T$$ gates to $$T^\dagger$$ gates when you commute the Pauli corrections to the end of the circuit, but you can decide whether you implement a $$T$$ or $$T^\dagger$$ by changing the measurement basis of that gate's control qubit. The lack of Clifford corrections plus the ability to replace a $$T$$ gate with $$T^\dagger$$ after the teleportation allows you to delay feedback until the end.
Note, however, that you'll have to perform your end-of-the-line measurements sequentially, not in parallel. Measuring the control qubit of the first $$T$$ gate tells you what Pauli correction occurred, which tells you if the subsequent $$T$$ gates should be replaced with $$T^\dagger$$s. So before you measure the control qubit of the $$n$$th $$T$$-gate, you need to have measured all the control qubits of the $$T$$ gates before that $$T$$ gate that could introduce Pauli corrections that affect the $$n$$th $$T$$ gate, so you know if it should be replaced by $$T^\dagger$$.