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The famous 3 - qubit Greenberger, Horne and Zeilinger state: $|GHZ_{3} \rangle = \frac{1}{\sqrt{2}}[|000\rangle + |111\rangle]$.

A stabiliser for $|GHZ_{3} \rangle$ is the 3 - tensor product X Pauli operators: $S_{1} = \sigma_{X} \otimes \sigma_{X} \otimes \sigma_{X}$.

Then, $S_{1} |GHZ_{3}\rangle = |GHZ_{3}\rangle$.

The question is how does one determine all possible stabiliser for $|GHZ_{3} \rangle$?

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    $\begingroup$ Do you mean operators whos +1 eigenstate is GHZ state? If thats the case then there are infinite such operators. But if you want only tensor product of paulis, then you can just iterate over all 4^3, doing matrix vector multiplication and checking if thats correct. $\endgroup$
    – FDGod
    Sep 21, 2023 at 9:40
  • $\begingroup$ @FDGod Only tensor products of Pauli operators. I think I know what you're saying but just for clarity, are you able to provide a reasonably verbose answer? I promise to accept it. $\endgroup$
    – Physkid
    Sep 21, 2023 at 9:43
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    $\begingroup$ I’ll try to write a detailed answer — with “tricks/intution” to figure out if a state is invariant upto a global phase for which pauli operators — sometime tomorrow if no one answers in the meantime. Its 3 AM and I should sleep now lol. $\endgroup$
    – FDGod
    Sep 21, 2023 at 10:09
  • $\begingroup$ @FDGod Thank you for the kindness. Sleep is important. $\endgroup$
    – Physkid
    Sep 21, 2023 at 10:17

2 Answers 2

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You've noticed the global flip stabilizer $X_1X_2X_3$. You can also notice that the $Z$ parity of any two bits always agrees: There is no amplitude on terms like $|001\rangle$ where the second and third bits have different parities. So you also have parity stabilizers $Z_1Z_2$ and $Z_2Z_3$. This is a three-qubit state, and we've listed three independent stabilizers, so we're done: every other stabilizer is the product of a subset of these stabilizers. For example, $Z_1Z_3=(Z_1Z_2)(Z_2Z_3)$.

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You can use software tools. For example, stim.Tableau.from_state_vector:

import stim

ghz_vector = [0] * 8
ghz_vector[0b_000] = 0.5**0.5
ghz_vector[0b_111] = 0.5**0.5

t = stim.Tableau.from_state_vector(ghz_vector, endian='little')

print("Stabilizer generators are:")
for k in range(len(t)):
    print(t.z_output(k))
Stabilizer generators are:
+XXX
+ZZ_
+Z_Z

Under the hood what stim is doing is building a circuit to uncompute the state:

  1. Find a non-zero amplitude and use X gates to move it to the start of the vector. (GHZ state already has a non-zero amplitude at the start.)

  2. While there is another non-zero amplitude in the vector:

    • Use CNOT gates to move the second non-zero amplitude to the second entry of the vector (CNOTs instead of NOTs so that the first entry doesn't move).
    • Use S gates on the least significant qubit until the first and second entries have the same phase.
    • Use a Hadamard gate on the least significant qubit to destructively interfere the second entry to 0. Assuming the input was a stabilizer state, this will cut the number of non-zero amplitudes exactly in half.
  3. The accumulated circuit $C$ is now a Clifford circuit that uncomputes the states. It maps the stabilizers of the state to single-qubit stabilizers $Z_q$. Running it backwards maps each $Z_q$ back to a stabilizer. The stabilizers are $C^{-1} Z_q C$ for each qubit $q$.

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