Hadamard gate: π/2 FT

In[]:=
QuantumMultiwayCompile[<|"Operator"->{{1,1},{1,-1}},"Basis"->{{1,0},{0,1}}|>,{{1,0},{1,0},{0,1}},1,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]
Out[]=
The edges correspond to multiplications in the matrix multiply.
In[]:=
Fourier[{1,-1,2,2}]
Out[]=
{2.+0.,-0.5-1.5,1.+0.,-0.5+1.5}
In[]:=
Sqrt[2]Fourier[{2,1}]
Out[]=
{3.,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 like a vector; there is a reverse vector needed.

Arbitrary Fourier

In[]:=
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"]
In[]:=
FTGraph[{3,8},1]
Out[]=
In[]:=
FTGraph[{3,8},4]
Out[]=
In[]:=
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"]
In[]:=
FTGraphFull[{3,8},3]
Out[]=
In[]:=
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&]]}]
In[]:=
FT[{3,5}]
Out[]=
{{0, -1}5,{0, 1}3,{1, 0}8}

Generate the Bloch sphere :

Use a non-commensurate rotation
In[]:=
Round[100RotationMatrix[17Degree]//N]
Out[]=
{{96,-29},{29,96}}
In[]:=
QuantumMultiwayCompile[<|"Operator"->Round[100RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1}},2,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]
Out[]=
In[]:=
Round[10RotationMatrix[17Degree]//N]
Out[]=
{{10,-3},{3,10}}
In[]:=
QuantumMultiwayCompile[<|"Operator"->Round[10RotationMatrix[17Degree]//N],"Basis"->{{1,0},{0,1}}|>,{{0,1}},4,"EvolutionGraph","IncludeStatePathWeights"True,VertexLabels"VertexWeight"]
Out[]=

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

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

Quantum phases

Relative angle between different bundles of geodesics
A non-possible geodesic:
A good geodesic:

What is the multiway graph?

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

Principle of Equivalence

Relativistic state preparation

Forget the masses; just pick an appropriate accelerating frame
[ This is about simultaneity; i.e. angle in spacetime ]

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

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

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

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

[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 >

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

< 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 ]]

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

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

There is a net angle to the bundle of 8 geodesics
Find angles between geodesics by “webbing construction” (AKA dropping a perpendicular)