Re:Quantum Computing for the Curious
by Strilanc on Tuesday March 19, 2019 @01:28AM
(#19428301)
> why represent states as 2d complex vectors instead of quarterions? Aren't they the same thing?
Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.
That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.
Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.
That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.