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# Questions tagged [zx-calculus]

The ZX-calculus is a high-level and intuitive graphical language for pure qubit quantum mechanics (QM), based on category theory. (arXiv: 1602.04744)

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### How to represent a general 3-qubit state as a symmetric ZX-diagram with 14 parameters?

A general pure 1-qubit state can be written as a ZX-diagram like this: Correspondingly, for a general pure 2-qubit state: How can a general pure 3-qubit state be written as a ZX-diagram? Two things ...
• 858
1 vote
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### Deformation rules for ZX "pipe" diagrams

If I consider the 3-dimensional structures from https://arxiv.org/abs/1905.08916, these would form a nice 3-dimensional structure for surface code layouts, except I'm not sure what the equivalence ...
• 2,116
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### Using ZX calculus to postselect the first round of a distance-2 rotated surface code

I'm translating gate model language to ZX calculus diagram using such notation, to describe the first round of measurement of such a distance-2 rotated surface code below. I wrote this diagram where ...
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### $CNOT$ teleportation in ZX-calculus: how to simplify my circuit further?

I am stuck in simplyfing the following cNOT teleportation in ZX-calculus. I don't know how to proceed further. The circuit I start from is taken from this thesis (Fig 2.14, page 22). Which property ...
• 1,806
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### Getting intuition on the state-injection relations for the generalized $\exp(-iP \pi/8)$ $T$-gates (ideally using ZX calculus)

In Litinsky's paper, there are many circuits relations, like the one below. The left handside represents the "rotation" $\exp(-i P \phi)$ with $\phi=\pi/8$ with similar definitions for the ...
• 1,806
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### Global (Ising) Gates and ZX-calculus representation

I could find from this source -- but also from other works on ZX-calculus -- the following extract: This looks to me as a generalisation of a 2-qubit Ising gate to an $n$-qubit global Ising gate. ...
• 2,030
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### Zx graphs in Stim

It is possible to create an ASCII graph in Stimzx for a ZX calculus diagram. I would like to create something like this but I cannot seem to recreate the correct ASCII format. When I do something ...
• 38
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### Rewriting a contradictory ZX loop into an independent pi node

Consider the following ZX graph: If you perform tensor contraction on this graph, you get the zero tensor. Therefore it should be equivalent to this graph: How do you rewrite the graph, axiom by ...
• 40.1k
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### How do I get correct measurement probabilities in ZX calculus?

I'm learning ZX-calculus, but I'm getting confused when trying to obtain some simple results to compute probabilities for different outcomes. Here's a simple example where I'm getting lost. Here, <...
• 221
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### Does the ZX calculus allow for Y-axis rotations?

I'm trying to understand how Y-axis rotations are represented in ZX Calculus. In the paper, wikipedia, everywhere I look, it's as if there is no such thing as Y-axis rotations, only X and Z. I ...
• 133
1 vote
63 views

### ZX-calculus: meaning of no horizontal edge

Consider the following ZX-diagram: As you can notice, there are some nodes, belonging the same qubit, which are not connected by any edge (neither blue or black). What is the meaning of that (during ...
• 2,030
1 vote
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### PyZX optimisation steps for Clifford circuits

Given the following ZX-diagram It should represent some random Clifford circuit (LC means Local Clifford). As far as I got, any Clifford circuit can be transformed into a ZX-diagram like the above, i....
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### Can H-boxes have a copy-like rule in the ZH-calculus with respect to $\pi$ gates?

In the ZX-calculus we have the following rule, which I think it is known as the copy-rule (grey/white colours may be interchanged; with respect to the usual red/green notation the translation is white ...
• 2,357
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### Is it possible to implement the ZX-calculus bialgebra rule without adaptivity or post-selection?

In the ZX-calculus, one of the fundamental rules of the diagrammatic reasoning is known as the bialgebra rule and it is described by the given diagrammatic equation: Question: Can we implement this ...
• 2,357
1 vote
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### ZX Calculus -- proving the most basic of identities

I'm trying to show the following equivalence in the ZX calculus: This is equivalent to showing that $$|0\rangle - i|1\rangle = |+\rangle + i|-\rangle.$$ I want to do this using the rules listed on ...
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### ZX-calculus: pi-copy rule not required for completeness?

In ZX-calculus, the $\pi$-copy rule is quite famous, and is used for instance here: However, this paper never introduces this rule, and says that this set is enough to prove the Clifford completeness ...
• 666
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### ZX-Calculus: understand clifford+T/general ZX rules

This paper that proves the completeness of the ZX-Calculus introduces different gates: and However, they seem very cryptic to me (except maybe the rule E). What is the intuition (what they mean, and ...
• 666
862 views

### Explain the representation of the CNOT gate in ZX-calculus

In ZX-calculus, the CNOT gate is represented by this: Can someone show me why this is true, using just the basic rewriting rules? All books/papers I have seen simply take it without proof, but I can'...
• 383
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### ZX calculus: What do diamond and loop mean?

Recently, I started to study practical application of ZX calculus but I am confused by meaning of "diamond" and "loop". Issue no. 1: There are these rules: B-rule and D-rule But this example seems ...
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### What are some applications of the ZX calculus?

Recently, I came across ZX calculus. It is an interesting method to describe quantum circuits. However, it seems to me, too complicated for day-to-day use in circuit design (something like to program ...
• 14.7k
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### ZX-Calculus: how to prove this simple equation between two very small circuits

Short version: How could I prove in ZX-calculus that these two diagrams are equal (up to a global phase), using axioms from this paper (Fig. 1) for example? Any intuition is welcome! Long version:...
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### ZX-calculus : measurement and output probabilities

I'm discovering ZX-Calculus, and it seems to be much easier to do computations on circuit that would take much more time with the usual formalism. However, I can't find a nice way to represent ...
• 666
479 views

### Diagrammatic Quantum Reasoning: Proving the loop equation using yanking equations

I'm trying to study the book: Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning, and would like some help with Exercise 4.12: The relevant equations are as ...
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### Controlled Hadamard gate in ZX-calculus

What is the representation of the CH gate in ZX-calculus? Is there a general recipe for going from a ZX-calculus representation of a gate to the representation of the controlled version?
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### Graphical Calculus for Quantum Circuits

So far I have read a little bit about zx-calculus & y-calculus. From Reversible Computation: The zx-calculus is a graphical language for describing quantum systems. The zx-calculus is an ...
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