2.6 Phase gates galore

We have already met the generic phase gate P_\varphi=\begin{bmatrix}1&0\\0&e^{i\varphi}\end{bmatrix} which acts via \begin{array}{lcr} |0\rangle&\longmapsto&|0\rangle \\|1\rangle&\longmapsto&e^{i\varphi}|1\rangle \end{array} but there are three specific examples of P_\varphi that are important enough to merit their own names (two of which are rather confusing, at first glance).

Phase-flip Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&-|1\rangle\end{array}
\frac{\pi}{4}-phase S = \begin{bmatrix}1&0\\0&i\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&i|1\rangle\end{array}
\frac{\pi}{8}-phase T = \begin{bmatrix}1&0\\0&e^{i\frac{\pi}{4}}\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&e^{i\frac{\pi}{4}}|1\rangle\end{array}

Recall that a phase gate P_\varphi is only defined up to a global phase factor, and so we can write its matrix either as P_\varphi = \begin{bmatrix} 1 & 0 \\0 & e^{i\varphi} \end{bmatrix} or as P_\varphi = \begin{bmatrix} e^{-i\frac{\varphi}{2}} & 0 \\0 & e^{i\frac{\varphi}{2}} \end{bmatrix} The first version is more common in the quantum information science community, but the second one is sometimes more convenient to use, as it has determinant 1, and hence belongs to a group called \mathrm{SU}(2). We will occasionally switch to the \mathrm{SU}(2) version of phase gates, and this is where the \frac{\pi}{4}-phase and \frac{\pi}{8}-phase gates get their names, since their \mathrm{SU}(2) versions have e^{\mp i\pi/4} and e^{\mp i\pi/8} (respectively) on the diagonal, even though they are actually of phase \pi/2 and \pi/4 (respectively).

We will soon explain what this group \mathrm{SU}(2) is and how it relates to another important group called \mathrm{SO}(3), but it turns up in many places throughout quantum physics, as well as mathematics in general. Other places you might see \mathrm{SU}(2) appear are when talking about quaternions (which are somehow the next thing in the sequence \mathbb{R}\hookrightarrow\mathbb{C}\hookrightarrow\mathbb{H}) and two of the four “fundamental interactions”, namely electromagnetism and the weak nuclear force, which get bundled together into something known as electroweak interaction.

We will also eventually talk about how this aforementioned relationship between \mathrm{SU}(2) and \mathrm{SO}(3) lets us describe rotations of things in three-dimensional space. The abstract mathematical concept lying behind this is one with a very lofty-sounding title indeed: representation theory of Lie algebras. This lets us formally talk about things like (non-relativistic) spin. As for this application of \mathrm{SU}(2) in studying the electroweak interaction, this is an example of something known as gauge theory.

The remaining gate, the phase-flip Z, is arguably the most important specific phase gate, since it is one of the Pauli operators, which we will now discuss.

While we’re talking about phase, we should also justify why we keep on saying “let us ignore the global phase factors”. In general, states differing only by a global phase are physically indistinguishable, and so it is physical experimentation that leads us to this mathematical choice of only defining things up to a global phase.

If you are more mathematically minded, then we can justify ignoring the global phase in a few other ways. Taking the axiomatic approach, where values of physical observables correspond to eigenvalues of operators, think about how the eigenvalues of a matrix A relate to those of the matrix \mu A, where \mu is a complex number with |\mu|=1. One “high-level” way of dealing with this, in the language of gauge theory, is to talk of invariance under gauge symmetry (here, in particular, we’re talking about U(1) symmetries).