Bloch Sphere and Rotations

Single Qubit Operations

e^{iAx} = cos(x) + isin(x)A

Thus, Rx(θ)=eiθX/2R_x(\theta) = e^{i\theta X/2}Rx​(θ)=eiθX/2﻿, Ry(θ)=eiθY/2R_y(\theta) = e^{i\theta Y/2}Ry​(θ)=eiθY/2﻿ and Rz(θ)=eiθZ/2R_z(\theta) = e^{i\theta Z/2}Rz​(θ)=eiθZ/2﻿.

R_{\hat{n}}(\theta) = e^{i\theta \hat n\cdot\vec\sigma/2}

In general, we have the above equation where σ⃗=Xi^+Yj^+Zk^\vec\sigma = X\hat i + Y\hat j + Z\hat kσ=Xi^+Yj^​+Zk^﻿.

Some Algebra

X^2 = Y^2 = Z^2 = -iXYZ = I

R_{\hat n}(\alpha) = R_z(\phi)R_y(\theta)R_z(\alpha)R_y(-\theta)R_z(-\phi)\\ = R_z(\phi)R_y(\theta)R_z(\alpha)R_y(\theta)^\dagger R_z(\phi)^\dagger

Theorems

1.

Any arbitrary single qubit unitary operator can be written in the form U=eiαRn^(θ)U = e^{i\alpha}R_{\hat n}(\theta)U=eiαRn^​(θ)﻿.

2.

Suppose UUU﻿ is a unitary operation over a single qubit then ∃ α,β,γ,δ\exists\ \alpha, \beta, \gamma, \delta∃ α,β,γ,δ﻿ such that U=eiαRn^(β)Rm^(γ)Rn^(δ)U = e^{i\alpha}R_{\hat n}(\beta)R_{\hat m}(\gamma)R_{\hat n}(\delta)U=eiαRn^​(β)Rm^​(γ)Rn^​(δ)﻿.

3.

There exists unitaries A,B,CA, B, CA,B,C﻿ for any given unitary UUU﻿ such that ABC=IABC = IABC=I﻿ and U=eiαAXBXCU = e^{i\alpha} AXBXCU=eiαAXBXC﻿.