11

The best possible textbook reference at the moment is Coecke and Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. It is written by one of the two inventors of the ZX calculus (Bob Coecke), and one of the people who has contributed the most to the development of ...


9

In answer to your first question, yes there is a way to build controlled-$H$ in ZX. It more or less mirrors the construction in quantum circuits. First, one can construct controlled-$Z$-phase gates using the technique described in the "Quantum Computing" chapter of mine and Coecke's book (Picturing Quantum Processes, p. 687): n.b. the white dots are $Z$-...


7

The ZX calculus was not designed to be a programming language, or a language in which to specify things at a high level. It is a language for reasoning about things on a relatively low level, albeit without digging into the actual physics. What it was designed for is hinted at in your observations, and by the name of the ZX calculus itself. It can ...


6

The easiest way to do this would be to use the general controlled unitary construction given in this post: Given a decomposition for a unitary $U$, how do you decompose the corresponding controlled unitary gate $C(U)$? translate the circuit to ZX and hopefully simplify the circuit using ZX rewrite rules to get a nicer expression.


5

I am going to assume that $|0\rangle$ is represented by , and $|1\rangle$ by . Then, represents the bit-flip, or NOT gate. What it does is: $|x\rangle\mapsto |x\oplus1\rangle$. This can be verified easily in the graphical calculus, using the "spider" rule: and Now, by definition, CNOT controls NOT: $|x_1,x_2\rangle\mapsto|x_1,x_1\oplus x_2\rangle$. In ...


5

First, there is a neat way of performing the controlled Hadamard, with only two occurrences of the $T$ gate: Regarding your second question, I can give an answer that will get a tad technical, based on the paper arXiv:1805.05296. We are going to use the following construction that we generalise to $n$ wires: The reason we denote this operator like this ...


4

You already put Selinger's survey, so here are a couple more links. Baez and Stay: Baez and Stay is a survey article. It covers monoidal, braided, symmetric and dagger categories. For the example related to quantum computation focus on either Hilb or cobordism. The appropriate string diagrams for these are included along with the sections for those types of ...


4

If you use the symbol of triangle and lambda box in ZX, see e.g. my paper with KangFeng Ng arXiv:1706.09877, then there are two simple ways of representing the CH gate in ZX-calculus, one of them even need not any decomposition of the H gate (diagrams should be read from top to bottom): If you don't want any H showing in CH, then you have another diagram ...


2

If you agree to treat diagrams up to a constant factor, then you can ignore loops and diamonds. As you correctly guessed, it's a normalization constant. For a multi-qubit system, you represent an identity operator with several wires. If you trace them, you get dimension equal to $2^n$, and in the diagram you represent this dimension as $n$ disjoint loops ...


2

If all the angles in your diagram are multiples of $\pi/2$, you should use the rules from "A Simplified Stabilizer ZX-calculus" instead of the original rules in the paper you linked. But I'll work with them since it's what you asked. Because you don't care about non-zero scalar factors (such as global phase), you can simplify the rules by dropping any ...


2

Here is a screenshot of a possible proof: You can ignore the scalars if you want. The idea is to: disconnect the red node using the spider rule turn it into a green node with a Hadamard node decompose the hadamard gate use the copy rule to get rid of the red node The above proof uses the rule from the second paper. If you want to use the other rule, the ...


2

In the ZX calculus, the closest thing to a graph that measures an observable is a graph that post-selects that observable to be in its $+1$ eigenbasis. If you are attempting to understand a surface code lattice surgery computation in terms of a ZX graph this is kind of annoying. You need to be able to figure out which postselections are just shorthands for ...


2

Here is the solution. The trick is to use "the only connectivity matters" rule. The swap rule of 4.9 helps us reorder the inputs, which then makes it topologically equivalent to the next diagram (match the first and second wires of the states).


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