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Numeric Quantum Block Cellular Automata Examples
This tutorial develops complete numeric quantum block cellular automata, examines their returned evolution objects and compares the specialized numerical methods selected for different two-qubit rules.
Initial State Representations
An exact binary list is interpreted as a computational-basis qubit string, so this input means the four-qubit state :
|1001〉
In[1]:=
bits={1,0,0,1};
[bits,QuantumOperator["CH"],1]["States"]1
 | QuantumCA |
Out[1]=
QuantumState
 |  |
|
Define the equivalent 16-amplitude global state vector before passing it to the automaton:
In[2]:=
psi=N@UnitVector[16,10];
[psi,QuantumOperator["CH"],1]["States"]1
| QuantumCA |
Out[2]=
QuantumState
| |
|
Define a two-qubit global state vector for :
|00〉
In[3]:=
psi=N@UnitVector[4,1];
[psi,QuantumOperator["CH"],1]["States"]1
| QuantumCA |
Out[3]=
QuantumState
|  |
|
Machine-real binary-looking values are amplitudes rather than a qubit string:
In[4]:=
bits={1,0,0,0};psi={1.,0.,0.,0.};
[bits,QuantumOperator["CX"],0]["Qudits"],
[psi,QuantumOperator["CX"],0]["Qudits"]
 | QuantumCA |
| QuantumCA |
Out[4]=
{4,2}
The numeric conversion in is significant: an exact evaluates to and is therefore interpreted as a four-qubit string.
N@UnitVector[4,1]
UnitVector[4,1]
{1,0,0,0}
Define a coherent pure-state superposition; numeric state vectors are normalized internally:
In[5]:=
psi=UnitVector[16,1]+IUnitVector[16,16];
[psi,QuantumOperator["RootSWAP"],1]["States"]1
| QuantumCA |
Out[5]=
QuantumState
|  |
|
Pure objects are accepted as another explicit representation. Mixed objects are not supported because their evolution requires density matrices.
QuantumState
QuantumState
A Complete Evolution Object
Create a six-qubit controlled-Hadamard evolution and display the complete returned association as a structured dataset:
In[6]:=
evo=
[{1,0,0,0,0,0},QuantumOperator["CH"],2];Dataset[evo]
| QuantumCA |
Out[6]=
|
Inspect the final quantum state directly:
In[7]:=
| QuantumCA |
Out[7]=
QuantumState
|  |
|
Inspect the final layer of reduced one-qubit Bloch vectors:
In[8]:=
| QuantumCA |
Out[8]=
{{0,0,-1.},{0.103553,0,0.707107},{0.323223,0,0.53033},{0.176777,0,0.75},{0.125,0,0.875},{0,0,1.}}
Numerical Methods
Automatic backend selection follows the structure of the local unitary. Permutations of computational-basis states use the classical path:
In[9]:=
 | QuantumCA |
Out[9]=
Classical
A monomial phase rule uses sparse amplitude associations:
In[10]:=
| QuantumCA |
Out[10]=
Sparse
A tensor product of one-qubit gates uses the product method:
In[11]:=
h=N[1/Sqrt[2]{{1,1},{1,-1}}];
[{1,0,0,0,0,0},KroneckerProduct[h,IdentityMatrix[2]],3]["Backend"]
| QuantumCA |
Out[11]=
Product
A particle-number-preserving RootSWAP rule uses sparse number sectors:
In[12]:=
| QuantumCA |
Out[12]=
NumberConserving
An entangling Clifford rule uses the stabilizer-classified path:
In[13]:=
h=N[1/Sqrt[2]{{1,1},{1,-1}}];cx=N@{{1,0,0,0},{0,1,0,0},{0,0,0,1},{0,0,1,0}};
[{1,0,0,0,0,0},N[KroneckerProduct[h,IdentityMatrix[2]].cx],3]["Backend"]
| QuantumCA |
Out[13]=
Stabilizer
A controlled-Hadamard rule falls back to the general tensor method:
In[14]:=
| QuantumCA |
Out[14]=
Tensor
Comparing Rule Dynamics
Compare the final reduced Bloch vectors generated by three local rules from the same initial layer:
In[15]:=
Dataset@MapFunctionrule,Withevo=
[{1,0,0,0,0,0},QuantumOperator[rule],3],"Rule"rule,"Backend"evo["Backend"],"Final Bloch Vectors"evo["BlochVectors"]-1,{"CX","RootSWAP","CH"}
 | QuantumCA |
Out[15]=
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Quantum Framework Inputs
Use the equivalent raw controlled-Hadamard matrix:
Plotting Controlled-Hadamard Dynamics
Visualize the colored controlled-Hadamard evolution together with each final reduced Bloch vector:
Focus on the Regular color encoding without the matrix: