## Quantum Operation Counts

Quantum Operation Counts

E.g. quantum Fourier transforms

Operation count is affected by size n of the transform, and the precision p (measured by the maximum denominator):

classical ops: 2^n * 4 * p quantum: 1 (or p)

Operation count is affected by size n of the transform, and the precision p (measured by the maximum denominator):

classical ops: 2^n * 4 * p quantum: 1 (or p)

## Two Approaches to Experiments

Two Approaches to Experiments

#### 1. You explicitly (conceptually) set up a state [imagine the experimenter has free will]

1. You explicitly (conceptually) set up a state [imagine the experimenter has free will]

Analogous to generational multiway system where you insist on having done all the updates necessary to reach a classical state

#### 2. You wait for the universe to deliver the state you want to study

2. You wait for the universe to deliver the state you want to study

You wait for causal invariance to deliver you the classical state

### By inserting completions, you are going to case #1

By inserting completions, you are going to case #1

Analogous to adding completions in theorem proving

(i.e. adding lemmas; but adding lemmas is an exponential speedup)

(i.e. adding lemmas; but adding lemmas is an exponential speedup)

## Visualizing Branchial Space

Visualizing Branchial Space

In[]:=

GridGraph[{2,2,2},VertexLabelsAutomatic]

Out[]=

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HighlightGraph[GridGraph[{2,2,2}],Subgraph[GridGraph[{2,2,2}],{1,3,5}]]

Out[]=

In[]:=

HighlightGraph[GridGraph[{2,2,2,2}],Subgraph[GridGraph[{2,2,2,2}],{1,3,5}]]

Out[]=

In[]:=

GridGraph[Table[2,7]]

Out[]=

In[]:=

Graph3D[%]

Out[]=

In[]:=

ResourceFunction["MultiwaySystem"][{"A""AB"},"AA",3,"StatesGraph"]

Out[]=

In[]:=

GridGraph[{2,2,2},VertexLabels->Automatic]

Out[]=

Take the ancestry distance, and the further back they branch, the further apart they should be.

Ancestry distance k between nodes; 1 - k/(total history)

## Enumerate SSs

Enumerate SSs

#### [[[ Of possible substitution systems, how many correspond to named quantum operators? ]]]

[[[ Of possible substitution systems, how many correspond to named quantum operators? ]]]

## Analog of Event Horizons in Branchial / Multiway Space

Analog of Event Horizons in Branchial / Multiway Space

Simple event horizon: superselection rule

Disconnection in branchial graph

#### Superselection sector

Superselection sector

Validation for a quantum circuit

#### Geometrization of correctness proofs

Geometrization of correctness proofs

Computability is topology of branchial space; computational complexity of rulial space

## Systematic Measurement

Systematic Measurement