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This paper that proves the completeness of the ZX-Calculus introduces different gates:

enter image description here

and

enter image description here

However, they seem very cryptic to me (except maybe the rule E). What is the intuition (what they mean, and how they where obtained) behind these rules? I guess some of them (maybe the last one) has something to do with trigonometry, but it's not really trivial when looking at them.

And is there a way to "decompose" them to remember/understand them easily? I can indeed see some structures that appear in several places (like a red node with one or two attached nodes with same angles), but it's still a bit hard for me to make sense of this…

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Since this paper, there have been several different axiomatisations, arguably simpler than this one. For instance in: https://arxiv.org/pdf/2007.13739.pdf and https://arxiv.org/pdf/1812.09114.pdf. In the last one, in particular, all these rules (except (E)) are replaced by:

enter image description here

where angles on one side of the equation are bound in a non-trivial way to those on the other side. This rule can be understood as a representation of Euler angles (https://en.wikipedia.org/wiki/Euler_angles).

If you are interested specifically in the axioms you mention, here are some remarks:

The supplementarity rule (SUP) was introduced in https://arxiv.org/pdf/1103.2811.pdf and proven necessary in https://arxiv.org/pdf/1506.03055.pdf. This is an equation that proves useful in the study of the $|W\rangle$ state, expressed as a ZX-diagram.

The rule (E) was introduced in https://arxiv.org/pdf/1702.01945.pdf. In a way, it comes from the fact that $e^{i\frac\pi4}$ can also be written as $\frac{1+i}{\sqrt2}$.

The meaning of rules (C) and (A) are discussed in the paper at Section 7 (p.31).

Up to yanking of the wires and simplification using the spider rules, (C) can be seen as commutation of two unitaries, one that is controlled by a qubit, and the other that is "anti-controlled" by the same qubit.

Decomposing rule (A) the appropriate way, it can be seen as an expression of the following equation : $\begin{pmatrix}1&0&0&0\\0&1&1&0\end{pmatrix}\circ\left(\begin{pmatrix}1\\a\end{pmatrix}\otimes\begin{pmatrix}1\\b\end{pmatrix}\right) = \begin{pmatrix}1\\a+b\end{pmatrix}$. You can find more detail in this Section 7 I mentionned.

Finally there's (BW). To understand this equation, we first need to realise that the diagram:

enter image description here

represents the shearing operator $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. It so happens that we can decompose any 2D rotation as a sequence of 3 shear operations along the two canonical axes (see e.g. this link). Up to the usual ZX-rules for simplification, and up to rearranging of the shear operations, (BW) represents the special case of a rotation by $-\pi/2$:

$\begin{pmatrix}0&1\\-1&0\end{pmatrix} = \begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1&0\\-1&1\end{pmatrix}\begin{pmatrix}1&1\\0&1\end{pmatrix}$

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  • $\begingroup$ Thank you very much, also for pointing the simpler axiomatisatio (it also answered another question I had on Euler formulas), I'll read these links! I'm also curious to know if there is a systematic way to compute a (if possible simple) "normal form" of a circuit in general ZX for any diagram size. With matrices it is always possible to obtain such form (involving exponentially many operations), so I guess it may be possible to do the same thing in ZX (again with maybe exponentially many nodes). But I guess it is another question ^^ $\endgroup$ Jun 19 at 9:37
  • $\begingroup$ @LéoColisson you're welcome. There indeed exist several normal forms for ZX-diagrams. Feel free to ask in a separate question if you have specific question about these. $\endgroup$ Jun 23 at 15:56

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