#### Hadamard gate: π/2 FT

Hadamard gate: π/2 FT

QuantumMultiwayCompile[<|"Operator"->{{1,1},{1,-1}},"Basis"->{{1,0},{0,1}}|>,{{1,0},{1,0},{0,1}},1,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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The edges correspond to multiplications in the matrix multiply.

Fourier[{1,-1,2,2}]

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{2.+0.,-0.5-1.5,1.+0.,-0.5+1.5}

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Sqrt[2]Fourier[{2,1}]

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{3.,1.}

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#### For MW case, introduce {0,-1} to represent -|1>

For MW case, introduce {0,-1} to represent -|1>

#### Each state is a point in branchial space; for every state, need an additional point in branchial space

Each state is a point in branchial space; for every state, need an additional point in branchial space

Each state is like a vector; there is a reverse vector needed.

#### Arbitrary Fourier

Arbitrary Fourier

FTGraph[{a_,b_},t_:1]:=QuantumMultiwayCompile[<|"Operator"->{{1,1},{1,-1}},"Basis"->{{1,0},{0,1}}|>,Join[Table[{0,1},b],Table[{1,0},a]],t,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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FTGraph[{3,8},1]

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FTGraph[{3,8},4]

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Out[]=

FTGraphFull[{a_,b_},t_:1]:=QuantumMultiwayCompile[<|"Operator"->{{1,1},{1,-1}},"Basis"->{{1,0},{0,1}}|>,Join[Table[{0,1},b],Table[{1,0},a]],t,"EvolutionGraphFull","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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FTGraphFull[{3,8},3]

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Out[]=

FT[{a_,b_}]:=With[{g=QuantumMultiwayCompile[<|"Operator"->{{1,1},{1,-1}},"Basis"->{{1,0},{0,1}}|>,Join[Table[{0,1},b],Table[{1,0},a]],1,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]},Rule@@@Transpose@{Last/@Select[VertexList[g],VertexOutDegree[g,#]0&],With[{gx=IndexGraph[g]},AnnotationValue[{gx,#},VertexWeight]&/@Select[VertexList[gx],VertexOutDegree[gx,#]0&]]}]

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FT[{3,5}]

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{{0, -1}5,{0, 1}3,{1, 0}8}

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#### Generate the Bloch sphere :

Generate the Bloch sphere :

Use a non-commensurate rotation

Round[100RotationMatrix[17Degree]//N]

In[]:=

{{96,-29},{29,96}}

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QuantumMultiwayCompile[<|"Operator"->Round[100RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1}},2,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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Round[10RotationMatrix[17Degree]//N]

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{{10,-3},{3,10}}

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QuantumMultiwayCompile[<|"Operator"->Round[10RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1}},4,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

In[]:=

Out[]=

QuantumMultiwayCompile[<|"Operator"->Round[10RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1},{0,1},{1,0}},4,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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αKet[0]+Sqrt[1-α^2]Ket[1]

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α|0〉+

1-

|1〉2

α

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{1080-12000,10081-5400}

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{-10920,4681}

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### Two choices for initial condition preparation

Two choices for initial condition preparation

1. Explicitly insert at a particular generation in the cosmological rest frame

2. Thread your qof through what is “naturally being generated”

#2 is the only one people actually get to do; #1 is the fiction for a gedanken experiment (“fiction of free will”)

### Sampling a Bloch sphere

Sampling a Bloch sphere

Make a multiway system that gradually walks over the surface of the Bloch sphere with time

### Quantum phases

Quantum phases

Relative angle between different bundles of geodesics

ggx=;

In[]:=

ggx=QuantumMultiwayCompile[<|"Operator"->Round[10RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1},{0,1},{1,0}},4,"EvolutionGraphStructure","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]

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A non-possible geodesic:

HighlightGraph[IndexGraph[ggx],PathGraph[{1,4,7}]]

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A good geodesic:

HighlightGraph[IndexGraph[ggx],PathGraph[{1,4,6}]]

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## What is the multiway graph?

What is the multiway graph?

Answer: it is the pure quantum graph, independent of spacetime

## Principle of Equivalence

Principle of Equivalence

### Relativistic state preparation

Relativistic state preparation

Forget the masses; just pick an appropriate accelerating frame

[ This is about simultaneity; i.e. angle in spacetime ]

### Quantum state preparation

Quantum state preparation

[ This is about quantum phase; i.e. angle in branchtime ]

<A QoF slice is defining a surface of zero phase>

< Analog of inertial motion is to choose QoFs that simply follow the natural turnings of the geodesics >

You could have a flat multiway system [i.e. commutators are trivial] [changed variables to commuting variables] ... but with an appropriate qof you can reproduce

[[ A multiway system with trivial commutators is a multiway system with instant causal invariance : this is the analog of zero curvature ]]

< With trivial CI, there is no distant entanglement; only neighboring states get entangled > <I.e. there is no quantumness; by the time you could measure, it has converged down again>

#### Equivalence Principle

Equivalence Principle

By picking the right sequence of QoFs, with accelerating precision, you can kid yourself in any way you want

### Fundamental point of Equivalence Principle

Fundamental point of Equivalence Principle

You can get any answer you want if you modify your coordinate system at an increasing rate [you need an increasing rate because there is intrinsic dynamics, which you are fighting]

### Rulial space

Rulial space

You cannot distinguish intelligence in the actual behavior of the system from intelligence in your choice of RoFs...

### [Acceleration in P of E ]

[Acceleration in P of E ]

As time progresses, you have to run around in defining your frames faster and faster

### < Equivalence of gravitational and inertial mass >

< Equivalence of gravitational and inertial mass >

In GR case, it is simply that they are both represented by causal edge densities

#### < What is the analog in QM ? >

< What is the analog in QM ? >

The analog of inertial mass is amplitude magnitude density [ e.g. in the MW graph ] ?????

Gravitational mass might be amplitude magnitude density independent of

[[ In branchial space, amplitude magnitude density is related to energy/action, and is the source of curvature AKA turning of geodesics ]]

Gravitational mass might be amplitude magnitude density independent of

[[ In branchial space, amplitude magnitude density is related to energy/action, and is the source of curvature AKA turning of geodesics ]]

#### The eventual convergence of {causal, branch} pairs implies that the geodesics must be curved

The eventual convergence of {causal, branch} pairs implies that the geodesics must be curved

(otherwise they can’t all fit ) [ pair generation: divergence in the geodesic bundle ]

### Rulial space

Rulial space

If there are unresolved rule pairs, eventually things will diverge so much in rulial space, that simulation becomes impossible except by a hypercomputer; in effect you get a hypercomputer.

## Comparison to hypergraphs

Comparison to hypergraphs

There is a net angle to the bundle of 8 geodesics

finalState=ResourceFunction["WolframModel"][{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},13,"FinalState"];toHighlight=Function[r,Catenate@Through[{Join[List@@@#,List@@@Reverse/@#]&@*EdgeList,VertexList}[With[{graph=UndirectedGraph[Rule@@@finalState]},GraphUnion@@(NeighborhoodGraph[graph,#,r]&/@FindShortestPath[graph,200,900])]]]]/@{1,3,5};ResourceFunction["WolframModelPlot"][finalState,EdgeStyle<|Alternatives@@#Directive[Thick,Red]|>,VertexStyle<|Alternatives@@#Directive[Thick,Red]|>]&/@toHighlight

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Find angles between geodesics by “webbing construction” (AKA dropping a perpendicular)