## Friday, September 17, 2010

### M Theory Lesson 358

A nice paper by Michel Planat shows how the Weyl group $W(E_8)$ of $E_{8}$ may be generated using gates for three qubits, namely the Toffoli gate and the two gates $I \otimes S$ and $S \otimes I$, where $S$ is a $4 \times 4$ matrix with entries $\pm 1$. The matrix $S$ may be related to a simple braiding matrix $R$ by the relation $RS = F_2 \otimes I$, for $F_2$ the usual Hadamard gate, or Fourier transform. Similarly, $W(E_7)$ is given by three simple three qubit gates, including the Toffoli gate. As Planat remarks:
Indeed, the unitary realization of $W(E_8)$ with quantum gates (of the GHZ type) is much different from the Weyl group one gets from the Lie algebra of $E_8$.
He also defines two gates to realise $W(E_6)$.

1. Some of this in Michel Planat looks familiar. I am trying to connect this with W(E_8) as diag[H_4,H_4] plus permutations on 8 letters in an explicit manner. The 120 dim so(16) and the 128 Clifford CL(7) define the two Hadamard matrices. This paper seems to reduce this in a funny way, but I have not yet seen how that is done.

LC

2. Good to see you thinking about these things, Lawrence. I find that Planat has an unuusual style, but he is one of few specialists in arithmetic quantum information.

3. Of course not everything is mathematical. I am trying to craft a particular argument about things. It is one of those things similar to what might be called "Newton's apple," or Einstein's thought experiments about riding along a light beam and falling in elevators. In fact it goes back further, and the following bit by Galileo has always struck me by its elegance and clarity:

Simp.

There can be no doubt but that one and the same body moving in a single medium has a fixed velocity which is determined by nature and which cannot be increased except by the addition of momentum [/impeto/] or diminished except by some resistance which retards it.

Salv.

If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

Simp.

You are unquestionably right.

Salv.

But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see[108] how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.

/From: Dialogues Concerning Two New Sciences/ [1638]

4. Well, I am glad to see more real physicists on this corner of the blogosphere these days. There have been too many so called mathematical physicists for the last few decades.