Understanding the growth step of the magic state cultivation protocol

Question

I'm having trouble following the description of the growth phase in the magic state cultivation protocol from Gidney2024. I would really appreciate a more detailed explanation of what's happening in Figure 9 of Gidney2024. In particular,

• Do the data qubits being initialized as bell-pairs have to be along the blue/green faces of the color code?
• Page 5 of the paper says: "The Bell pairs are chosen so that they determine the stabilizers whose colors match the boundary that is being moved [JBH16].". What does this mean?
• Figure 9's caption says that the missing red stabilizers are initialized by the next measurement cycle. Why? And what will the initialization be?
• In section 2.6, the paper says: "That said, there are still two kinds of fast feedback needed by cultivation. First, when growing the color code patch, stabilizers prepared non-deterministically need to be forced into a specific eigenstate before the next layer of T gates.". (1) By "next layer of gates", is the paper referring to the T gates applied during the check-uncheck operation being performed while growing the fault distance; and (2) Which stabilizers are being prepared non-deterministically? And what will the feedback be for these stabilizers?

Answer 1

Do the data qubits being initialized as bell-pairs have to be along the blue/green faces of the color code?

Yes, because the blue-green boundary is the one that is moving outward.

Page 5 of the paper says: "The Bell pairs are chosen so that they determine the stabilizers whose colors match the boundary that is being moved [JBH16].". What does this mean?

Two of the sides of the triangle aren't moving, they are just extending. The third side is being moved outward as part of growing the code. Along that side there are blue and green stabilizers. And the stabilizers determined by the Bell pairs in the new area are the blue and green ones. The statement is saying that it is not a coincidence, and in fact important, that the boundary being extended and the stabilizers being prepared are both blue+green.

Figure 9's caption says that the missing red stabilizers are initialized by the next measurement cycle. Why? And what will the initialization be?

The red stabilizers must be initialized because the color code isn't complete until all its stabilizers are known.

For cultivation to work the eigenstates need to be specific values, so that the later controlled transversal H_XY gates don't entangle the cat state with the stabilizers. The measurement randomizes those eigenstates, and then fast feedback would be used to force them to the correct signs.

is the paper referring to the T gates applied during the check-uncheck operation being performed while growing the fault distance; and (2) Which stabilizers are being prepared non-deterministically? And what will the feedback be for these stabilizers?

Yes.

The red stabilizers are the ones prepared non-deterministically.

The feedback will be the set of Pauli operations that flip the stabilizers to have the correct sign, which can be solved for by Gaussian elimination and then saved in a lookup table.