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Factorizing rotation operator into finite products

  1. Feb 21, 2014 #1
    I know that any unitary operator on the spin-1/2 Hilbert space can be identified as a fixed-axis rotation operator up to a global phase, i.e.,
    $$U=e^{i\alpha}R_{\hat{n}}(\theta)=e^{i\alpha}e^{-i\theta\mathbf{\hat{n}}\cdot\mathbf{\sigma}/2\hbar}$$

    And it's also a known result that any fixed-axis rotation operator can be factorized into 3 Euler rotations z-y-z
    $$R_{\hat{n}}(\theta)=R_{\hat{z}}(\cdot)R_{\hat{y}}(\cdot)R_{\hat{z}}(\\cdot)$$

    Now there is a more general claim that as long as [itex]\mathbf{\hat{u}}[/itex] and [itex]\mathbf{\hat{v}}[/itex] are not parallel, arbitrary [itex]R_{\hat{n}}[/itex] can be factorized into a finite series of alternate rotations around [itex]\mathbf{\hat{u}}[/itex] and [itex]\mathbf{\hat{v}}[/itex]
    i.e.,
    $$
    R_{\hat{n}}(\theta)=R_{\hat{u}}(\beta_1)R_{\hat{v}}(\gamma_1)\cdots R_{\hat{u}}(\beta_k)R_{\hat{v}}(\gamma_k)
    $$
    for some fixed k (which only depends on [itex]\mathbf{\hat{u}}[/itex] and [itex]\mathbf{\hat{v}}[/itex]) and [itex]\beta_i[/itex] and [itex]\gamma_i[/itex]

    I attempted to prove this general claim but to no avail. Apparently this is a pretty important claim that's was used in Nielsen and Chuang's Quantum Computation and Quantum Information (in section 4.5 where they try to prove that H, S, pi/8 and CNOT can approximate any single-qubit gate to arbitrary accuracy). I've also searched online for this result, but didn't find any reference to it. Any pointers will be greatly appreciated
     
  2. jcsd
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