Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 QuantumDiscreteStateToZXDiagram

Source Notebook

Convert a pure quantum discrete state into a ZX-diagram

Contributed by: Jonathan Gorard

ResourceFunction["QuantumDiscreteStateToZXDiagram"][QuantumDiscreteState[…]]

converts the specified pure QuantumDiscreteState into an equivalent ZXDiagramObject.

Details

If ResourceFunction["QuantumDiscreteStateToZXDiagram"] succeeds in converting the specified QuantumDiscreteState into a ZX-diagram, it will return a ZXDiagramObject expression.
ResourceFunction["QuantumDiscreteStateToZXDiagram"] works by partially tracing out each qubit in the specified QuantumDiscreteState individually. Every qubit of the general form {1,ⅇⅈα} is converted into a Z-spider generator in the resulting ZXDiagramObject, and every qubit of the general form is converted into an X-spider generator in the resulting ZXDiagramObject. Any qubits that are not found to be equivalent to one of these two forms are omitted.
Whenever the specified QuantumDiscreteState cannot be converted into an equivalent MakeZXDiagramObject via the above procedure, ResourceFunction["QuantumDiscreteStateToZXDiagram"] will return Indeterminate.
Since it is representing a pure QuantumDiscreteState, every generator/spider produced within the resulting ZXDiagramObject will have an input arity of 0 and an output arity of 1.
In the resulting ZXDiagramObject, the following properties are supported:
"LabeledGraph"graph form of the ZX-diagram with phases labeled
"UnlabeledGraph"graph form of the ZX-diagram without phases labeled
"OperatorForm"ZX-diagram represented as a tensor product of generators
"ListForm"ZX-diagram represented as a list of generators
"MatrixForm"ZX-diagram represented as an explicit linear map on qubits
"ZSpiders"list of Z-spiders in the ZX-diagram
"XSpiders"list of X-spiders in the ZX-diagram
"HadamardGates"list of Hadamard gates in the ZX-diagram
"Diamonds"list of black diamonds in the ZX-diagram
"Wires"list of wires in the ZX-diagram
"ZSpiderCount"number of Z-spiders in the ZX-diagram
"XSpiderCount"number of X-spiders in the ZX-diagram
"HadamardGateCount"number of Hadamard gates in the ZX-diagram
"DiamondCount"number of black diamonds in the ZX-diagram
"WireCount"number of wires in the ZX-diagram
ResourceFunction["QuantumDiscreteStateToZXDiagram"] names the ZX generators in the resulting ZXDiagramObject in the conventional way, naming Z-spiders as z1,z2,…; X-spiders as x1,x2,…; black diamonds as d1,d2,…; outputs as o1,o2,…; etc.

Examples

Basic Examples (2) 

Convert a single-qubit pure discrete quantum state into a single-spider ZX-diagram:

In[1]:=
state = ResourceFunction["QuantumDiscreteState"][{1, -1}]
Out[1]=
In[2]:=
diagram = ResourceFunction["QuantumDiscreteStateToZXDiagram"][state]
Out[2]=
In[3]:=
diagram["OperatorForm"]
Out[3]=
In[4]:=
diagram["LabeledGraph"]
Out[4]=

Convert a different single-qubit pure discrete quantum state into a single-spider ZX-diagram:

In[5]:=
state2 = ResourceFunction["QuantumDiscreteState"][{1 + I, 1 - I}]
Out[5]=
In[6]:=
diagram2 = ResourceFunction["QuantumDiscreteStateToZXDiagram"][state2]
Out[6]=
In[7]:=
diagram2["OperatorForm"]
Out[7]=
In[8]:=
diagram2["LabeledGraph"]
Out[8]=

Convert a more complicated three-qubit pure discrete quantum state into a three-spider ZX-diagram:

In[9]:=
state = ResourceFunction["QuantumDiscreteState"][{0, (1 + I) Sqrt[2], 0, (1 - I) Sqrt[2], 0, 0, 0, 0}]; diagram = ResourceFunction["QuantumDiscreteStateToZXDiagram"][state]
Out[9]=
In[10]:=
diagram["OperatorForm"]
Out[10]=
In[11]:=
diagram["LabeledGraph"]
Out[11]=

Convert a different four-qubit pure discrete quantum state into a four-spider ZX-diagram:

In[12]:=
state2 = ResourceFunction["QuantumDiscreteState"][{Sqrt[2], 0, Sqrt[2] E^((3 I \[Pi])/5), 0, Sqrt[2] E^((I \[Pi])/4), 0, Sqrt[2] E^((17 I \[Pi])/20), 0, Sqrt[2] E^((I \[Pi])/6), 0, Sqrt[2] E^((23 I \[Pi])/30), 0, Sqrt[2] E^((5 I \[Pi])/12), 0, Sqrt[2] E^(-((59 I \[Pi])/60)), 0}]; diagram2 = ResourceFunction["QuantumDiscreteStateToZXDiagram"][state2]
Out[12]=
In[13]:=
diagram2["OperatorForm"]
Out[13]=
In[14]:=
diagram2["LabeledGraph"]
Out[14]=

Scope (1) 

If the specified QuantumDiscreteState object cannot be expressed as a ZX-diagram, QuantumDiscreteStateToZXDiagram will return Indeterminate:

In[15]:=
state = ResourceFunction["QuantumDiscreteState"][{"RandomPure", 3}]
Out[15]=
In[16]:=
ResourceFunction["QuantumDiscreteStateToZXDiagram"][state]
Out[16]=

Publisher

Jonathan Gorard

Version History

  • 1.0.0 – 26 April 2021

Source Metadata

  • Citation:
    • Bob Coecke and Ross Duncan (2011), "Interacting Quantum Observables: Categorical Algebra and Diagrammatics"
    • Jonathan Gorard, Manojna Namuduri and Xerxes Arsiwalla (2020), "ZX-Calculus and Extended Hypergraph Rewriting Systems I: A Multiway Approach to Categorical Quantum Information Theory"
    • Jonathan Gorard, Manojna Namuduri and Xerxes Arsiwalla (2021), "ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplification"

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