Examples

Here we will show some examples of using ZXCalculus.jl.

Clifford simplification

This example can be found in the appendix of Clifford simplification. Firstly, we build up the circuit by using push_gate!.

using ZXCalculus

function generate_example_1()
    zxc = ZXCircuit(4)
    push_gate!(zxc, Val(:Z), 1, 3//2)
    push_gate!(zxc, Val(:H), 1)
    push_gate!(zxc, Val(:Z), 1, 1//2)
    push_gate!(zxc, Val(:H), 4)
    push_gate!(zxc, Val(:CZ), 4, 1)
    push_gate!(zxc, Val(:CNOT), 1, 4)
    push_gate!(zxc, Val(:H), 1)
    push_gate!(zxc, Val(:H), 4)
    push_gate!(zxc, Val(:Z), 1, 1//4)
    push_gate!(zxc, Val(:Z), 4, 3//2)
    push_gate!(zxc, Val(:X), 4, 1//1)
    push_gate!(zxc, Val(:H), 1)
    push_gate!(zxc, Val(:Z), 4, 1//2)
    push_gate!(zxc, Val(:X), 4, 1//1)
    push_gate!(zxc, Val(:Z), 2, 1//2)
    push_gate!(zxc, Val(:CNOT), 3, 2)
    push_gate!(zxc, Val(:H), 2)
    push_gate!(zxc, Val(:CNOT), 3, 2)
    push_gate!(zxc, Val(:Z), 2, 1//4)
    push_gate!(zxc, Val(:Z), 3, 1//2)
    push_gate!(zxc, Val(:H), 2)
    push_gate!(zxc, Val(:H), 3)
    push_gate!(zxc, Val(:Z), 3, 1//2)
    push_gate!(zxc, Val(:CNOT), 3, 2)

    return zxc
end
ex1 = generate_example_1()

We can draw this ZX-diagram by using

using YaoPlots
plot(ex1)

To simplify ex1, one can simply use

simplified_ex1 = clifford_simplification(ex1)

or explicitly apply the simplification rules:

simplify!(LocalCompRule(), ex1)
simplify!(Pivot1Rule(), ex1)
replace!(PivotBoundaryRule(), ex1)
simplified_ex1 = circuit_extraction(ex1)

And we draw the simplified circuit.

plot(simplified_ex1)

Phase teleportation

This example is an arithmetic circuit from phase teleportation. We first build up the circuit.

using ZXCalculus, YaoPlots
function generate_example2()
    zxc = ZXCircuit(5)
    push_gate!(zxc, Val(:X), 5, 1//1)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 5, 4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 1)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:CNOT), 5, 4)
    push_gate!(zxc, Val(:Z), 4, 1//4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 1)
    push_gate!(zxc, Val(:CNOT), 4, 1)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:Z), 1, 1//4)
    push_gate!(zxc, Val(:Z), 4, 7//4)
    push_gate!(zxc, Val(:CNOT), 4, 1)
    push_gate!(zxc, Val(:CNOT), 5, 4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 3)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:CNOT), 5, 4)
    push_gate!(zxc, Val(:Z), 4, 1//4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 3)
    push_gate!(zxc, Val(:CNOT), 4, 3)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:Z), 3, 1//4)
    push_gate!(zxc, Val(:Z), 4, 7//4)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 4, 3)
    push_gate!(zxc, Val(:CNOT), 5, 4)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 5, 3)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 2)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:CNOT), 5, 3)
    push_gate!(zxc, Val(:Z), 3, 1//4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 2)
    push_gate!(zxc, Val(:CNOT), 3, 2)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 2, 1//4)
    push_gate!(zxc, Val(:Z), 3, 7//4)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 3, 2)
    push_gate!(zxc, Val(:CNOT), 5, 3)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 5, 2)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 1)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:CNOT), 5, 2)
    push_gate!(zxc, Val(:Z), 2, 1//4)
    push_gate!(zxc, Val(:Z), 5, 7//4)
    push_gate!(zxc, Val(:CNOT), 5, 1)
    push_gate!(zxc, Val(:CNOT), 2, 1)
    push_gate!(zxc, Val(:Z), 5, 1//4)
    push_gate!(zxc, Val(:Z), 1, 1//4)
    push_gate!(zxc, Val(:Z), 2, 7//4)
    push_gate!(zxc, Val(:H), 5)
    push_gate!(zxc, Val(:Z), 5)
    push_gate!(zxc, Val(:CNOT), 2, 1)
    push_gate!(zxc, Val(:CNOT), 5, 2)
    push_gate!(zxc, Val(:CNOT), 5, 1)
    return zxc
end
ex2 = generate_example2()
plot(ex2)

We can use phase_teleportation for reducing the number of T gates of a circuit without changing its general structure.

reduced_ex2 = phase_teleportation(ex2)
plot(reduced_ex2)

By using tcount,

tcount(ex2)
tcount(reduced_ex2)

we can see that the number of T gates has decreased from 28 to 8.

Other usages

In the previous sections, we introduced how to use ZXCalculus.jl for quantum circuits using ZXCircuit. Sometimes, one may wish to work with the lower-level graph representation directly.

Working with ZXGraph directly

For advanced users who need direct access to the graph structure, you can work with ZXGraph. However, for most use cases, ZXCircuit is the recommended interface.

using ZXCalculus, YaoPlots, Graphs

# Create a ZXCircuit first
zxc = ZXCircuit(3)
push_gate!(zxc, Val(:H), 1)
push_gate!(zxc, Val(:CNOT), 1, 2)

# Access the underlying ZXGraph if needed for low-level operations
zxg = zxc.zxd  # The internal ZXGraph

# Apply graph-based rules
simplify!(LocalCompRule(), zxc)

Deprecated: ZXDiagram

Note: ZXDiagram is deprecated. For circuit-based operations, use ZXCircuit instead. If you have existing code using ZXDiagram, you can convert it:

zxc = ZXCircuit(old_zxd)  # Convert deprecated ZXDiagram to ZXCircuit