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Magic states are used to obtain universal fault tolerant quantum gates.

The process involves distilling high quality magic states from many lower quality magic states using a state distillation circuit. And then using the high quality magic state to perform a non-Clifford logical gate through gate teleportation using a gate distillation circuit.

The fault tolerant state distillation/ gate distillation circuits I am familiar with all seem to use transversal CNOT.

CSS codes are the only stabilizer codes with transversal CNOT? shows only CSS codes have transversal CNOT (examples of CSS codes, and codes equivalent to CSS codes, include all Surface codes, even XZZX surface codes, Steane code, $ [[15,1,3]] $ RM code, Shor code etc...) .

Does this mean that magic states only work for CSS codes? In the sense that to get a fault tolerant gate from magic states it is necessary that the code has transversal CNOT?

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Note that the two qubit gate doesn't need to be specifically transversal. It needs to be fault tolerant; transversal is overkill. You can use any method of fault tolerance, like code deformation or code concatenation.

Supposing you can do all single qubit stabilizer gates, then two qubit stabilizer gates can be divided into five equivalence classes:

  • I-like (equal to doing nothing, up to single qubit stuff on a two qubit system)
  • CX-like (equal to a CX, up to single qubit stuff on a two qubit system)
  • SWAP-like (equal to a SWAP, up to single qubit stuff on a two qubit system)
  • CXSWAP-like (equal to a CX followed by a SWAP, up to single qubit stuff on a two qubit system)
  • MZZ-like (equal to a parity measurement, up to single qubit stuff on a two qubit system)

If you have access to single qubit stabilizer gates, you can consume magic states with any fault tolerant CX-like, CXSWAP-like, or MZZ-like interaction (an "entangling interaction"). SWAP-like and I-like interactions are insufficient because they are not entangling; they cannot map single qubit Pauli product operators into multi qubit Pauli product operators.

Any of the entangling interactions can be used to implement all the other entangling interactions, assuming you can apply multiple instances of the interaction and have access to single qubit effects and clean workspace qubits. The most counter-intuitive of these is building CX out of MZZ; that was one of the key insights in the original lattice surgery paper.

If you don't have access to all single qubit stabilizer effects, you may not be able to transform the entangling interaction you have into an entangling interaction that correctly consumes the magic state. For example, suppose you need an MXX but only have MZZ and S and MZ. Normally you'd use $H$ to turn MZZ into MXX, but here you don't have a way to do $H$. This will have knockon effects like making it hard to get to CX, because the usual construction of CZ out of MZZ has MXX as an intermediate step. That said, it's often the case that you can find some way to make it work; like a stabilizer magic state that both enables your missing single qubit rotation and can be duplicated using your available operations.

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  • $\begingroup$ What do you mean by "stabilizer gates"? Does that just mean Clifford gates? I'm also not so clear on the gate MZZ, what $ 4 \times 4 $ matrix is that? $\endgroup$ Commented Jun 27, 2024 at 19:37
  • $\begingroup$ Stabilizer gates include dissipative effects like measurement, whereas Clifford implies unitary. $\endgroup$ Commented Jun 27, 2024 at 21:20
  • $\begingroup$ ok so a stabilizer gate is unitary if and only if it is a Clifford gate, is that right? Is there a good reference where I can read about "stabilizer gates" I have never seen this term before. Usually I am pretty familiar with the literature, did you coin this term yourself? $\endgroup$ Commented Jun 27, 2024 at 21:47
  • $\begingroup$ Basically yes; it's a term that I use that's not so common. A "stabilizer gate" is a gate that can be efficiently simulated by a stabilizer simulator; a gate that fits natively into the stabilizer formalism. This includes Clifford gates but also many dissipative gates like Pauli basis measurements and resets. $\endgroup$ Commented Jun 27, 2024 at 23:57
  • $\begingroup$ These are probably simple questions but "The most counter-intuitive of these is building CX out of MZZ; that was one of the key insights in the original lattice surgery paper." that all seems interesting, what paper are you referring to? Also what do you mean by "PARITY-like interaction" $\endgroup$ Commented Jul 22, 2024 at 19:58

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