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Wolfram <> Quantum Computation

Wolfram quantum computation framework​
Documentation pages: https://resources.wolframcloud.com/PacletRepository/resources/Wolfram/QuantumFramework/
github page: https://github.com/sw1sh/QuantumFramework
Install the paclet:
In[]:=
PacletInstall["Wolfram/QuantumFramework",ForceVersionInstall->True]
Out[]=
PacletObject
Name: Wolfram/QuantumFramework
Version: 1.4.1

Load the context:
In[]:=
Needs["Wolfram`QuantumFramework`"]
Or, use this short URL to get the latest development version:
In[]:=
PacletInstall["https://wolfr.am/DevWQCF",ForceVersionInstall->True]​​Needs["Wolfram`QuantumFramework`"]

Quantum objects: states, operators, circuits and more

See the tech note on abstractions for quantum fundamentals.
Define a random pure state for a qubit
In[]:=
state=QuantumState["RandomPure"]
Out[]=
QuantumState
Pure state
Qudits: 1
Type: Vector
Dimension: 2

Corresponding state vector
In[]:=
state["StateVector"]
Out[]=
SparseArray
Specified elements: 2
Dimensions: {2}

In[]:=
Normal[state["StateVector"]]
Out[]=
{0.190019-0.491909,-0.848838-0.0373262}
Corresponding density matrix
In[]:=
MatrixForm@state["DensityMatrix"]
Out[]//MatrixForm=
0.278081+0.
-0.142934+0.424643
-0.142934-0.424643
0.721919+0.
Corresponding Bloch vector
In[]:=
state["BlochPlot"]
Out[]=
Corresponding Cartesian coordinates {x,y,z}
In[]:=
state["BlochCartesianCoordinates"]
Out[]=
{-0.285868,-0.849287,-0.443837}
Corresponding spherical coordinates {r,θ,ϕ}
In[]:=
state["BlochSphericalCoordinates"]
Out[]=
{1,2.03067,-1.89548}

Quick Examples: Bloch Plots and Random Walks

Rotate Bloch vector along x-axis by an angle:
In[]:=
Manipulate[​​(QuantumOperator["RZ"[θz]]@QuantumOperator["RY"[θy]]@QuantumOperator["RX"[θx]]@state)["BlochPlot"],{θx,0,2π},{θy,0,2π},{θz,0,2π},ControlPlacement->Top,TrackedSymbols:>{θx,θy,θz}]
Out[]=
​
θx
θy
θz
Wolfram`QuantumFramework`QuantumOperator[RZ[0.]][Wolfram`QuantumFramework`QuantumOperator[RY[0.]][Wolfram`QuantumFramework`QuantumOperator[RX[0.]][state]]][BlochPlot]
Given the initial state
|0〉
, randomly apply rotations with random angles
In[]:=
states=NestList[QuantumOperator[{RandomChoice[{"RX","RY","RZ"}],RandomReal[{-π/10,π/10}]}][#]&,QuantumState["0"],300];
Follow the random motion on the surface of Bloch sphere
In[]:=
coords=#["BlochCartesianCoordinates"]&/@states;
Put all points together and plot them
In[]:=
walk=Table[Show[states[[t]]["BlochPlot"],ListLinePlot3D[coords[[;;t]]]],{t,Length@states}];
In[]:=
ListAnimate[walk,AnimationRunningFalse]
Out[]=

As for the Practice Problems

◼
    • ◼

    • Exploring #1

    Check out Wolfram U courses like Introduction to Finite Mathematics and Introduction to Probability for lots more on these topics:
    Games like this can also be modeled as a Markov process:
    Eventually, either the player or the house run out of chips!

    Exploring #9

    Example Circuit

    What happens to the state of the qubit after each quantum operation?
    What happens to the measurement probabilities if you did a measurement?

    CHSH Game as a Teaching Example

    For more examples using Bell’s theorem and Bell states read our documentation.
    CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt.
    Charlie is a referee sending random 0 or 1 to Alice and Bob
    Let’s look at the statistics of Charlie
    Plot probabilities (quantum predictions)
    Simulate the measurement and count results

    Classical approach to win CHSH game

    Alice and Bob strategy: report only 0, no matter what Charlie sends
    Show some of the possible results:
    Count only the winning results (this classical strategy hits a limit around 75% chance to win)

    Quantum Strategies for the CHSH game

    Alice and Bob can use a Bell state and leverage entanglement to win the game with higher probability than any classical strategy.
    Calculate probabilities

    What about the effects of errors?

    Add noise to Alice’s wire and Bob’s wire:
    Compute the probabilities of each outcome:
    Condense this into the chance to win the game given the noise:
    Plot results:
    Should Bob’s strategy depend on the amount of noise in Alice’s wire?
    Proceed as before:
    Get a parametric function for the chance to win:
    Use the rest of Wolfram Mathematica’s symbolic capabilities to analyze the result: