### A Pluto.jl notebook ###
# v0.19.16

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 4a35ff54-5d10-11ed-3969-83c1ef512684
begin
	using Plots
	using QuantumInformation
	using LinearAlgebra
	using SparseArrays
	using HypertextLiteral
	using PlutoReport
	using PlutoUI
	using Luxor
	using StatsBase
	theme(:dark)
end

# ╔═╡ d0ad6f88-c906-44a5-908a-9996bd33bf0e
TableOfContents()

# ╔═╡ f9c42e32-8ec9-4bc7-ab27-d3260d587f5c
@htl("""
<script id="first">

	var editor = document.querySelector("pluto-editor")
	var prev = document.querySelector("button.changeslide.prev")
	var next = document.querySelector("button.changeslide.next")
	
	const click_background = (e => {
		// debugger;
		if (editor != e.target) return;
		e.preventDefault();		
		console.log(e.button);
		if (e.button === 2 && prev) {
			prev.click();
		} else if (e.button === 0 && next) {
			next.click();
		} 
	})
	editor.addEventListener("click", click_background)
	editor.addEventListener("contextmenu", click_background)

	invalidation.then(() => { 
		editor.removeEventListener("click", click_background);
		editor.removeEventListener("contextmenu", click_background);
	})
	
	return true;
</script>
""")

# ╔═╡ b3a75a22-3700-4a4a-ab13-d064715f2a08
html"<style>
.markdown{
	font-size: 20px;
}
main {
	max-width: 80%;
}
pluto-output>div>img {
	margin: auto;
	display: block;
}
</style>"

# ╔═╡ 504843f6-33ab-475c-9936-10992be19386
@bind pcon presentation_controls(aside=true)

# ╔═╡ 7de512e2-f43a-42aa-92bf-49a75170b3ed
@htl "$(apply_css_fixes()) $(presentation_ui(pcon))"

# ╔═╡ 62adfcd1-e110-40ae-a0bf-f2b2625b037e
Title("Quantum Resetting of Walks", "An exploration of the effect of stochastic and quantum resetting on classical and quantum walks", "Dhruva Sambrani, Supervised by Manabendra Nath Bera", "Dept of Physics, IISER - Mohali") # TODO Add image

# ╔═╡ 9bc0dc54-4e12-4d4d-b162-4ef738e6fd73
md"# Classical Walks and Stochastic Reset

Let us first define classical random walks and Stochastic resetting on classical random walks.
"

# ╔═╡ a20a8ab6-2f1d-4118-8135-e68d7882b285
md"""
## Classical Walks

Consider an infinite 1 D chain, with nodes marked by $\mathbb{Z}$.
"""

# ╔═╡ bced9c5d-5cbf-4d4c-b6d1-b500d76d4145
let
	d = Drawing(800, 200)
	origin()
	for i in -5:4
		setcolor("yellow")
		arc2r(Point(i*60+30, -20), Point(i*60, -30), Point(i*60+60, -30), action=:stroke)
		setcolor("lightgreen")
		arc2r(Point(i*60+30, -40), Point(i*60+60, -30), Point(i*60, -30), action=:stroke)
	end
	setcolor("yellow")
	arc2r(Point(-6*60+30, -20), Point(-6*60+30, -50), Point(-6*60+60, -30), action=:stroke)
	setcolor("lightgreen")
	arc2r(Point(-6*60+30, -40), Point(-6*60+60, -30), Point(-6*60+30, -20), action=:stroke)
	setcolor("yellow")
	arc2r(Point(5*60+30, -20), Point(5*60, -30), Point(5*60+30, -50), action=:stroke)
	setcolor("lightgreen")
	arc2r(Point(5*60+30, -40), Point(5*60+30, -10), Point(5*60, -30), action=:stroke)
	setfont("Helvetica", 24)
	for i in -5:5
		setcolor("white")
		settext("$(i)", Point(i*60, 20))
		setcolor("red")
		circle(Point(i*60, -30), 10, action=:fill)
	end
	setcolor("cyan")
	setfont("Helvetica", 16)
	settext("0.5", Point(20, -51))
	settext("0.5", Point(20, -10))
	finish()
	d
end

# ╔═╡ ae30bd04-5325-4f53-9f4f-c62751d528fd
md"""

Define the probability of hopping from node $i$ to node $j$

$$p_{ij} = \begin{cases}
	1/2 & |i - j| = 1 \\
	0 & \text{otherwise}
\end{cases}$$
"""

# ╔═╡ c06f87ae-3a73-4640-969a-c6d09d7b97bf
_walk = cumsum(rand([1, -1], 40));

# ╔═╡ d44250f8-9808-45fe-9a5c-b5eeb1c3f202
md"""
t = $(@bind _t1 Slider(0:40, show_value=true))
"""

# ╔═╡ 1ed2f6a3-c55a-428b-90aa-8fe44e8ab965
let
	d = Drawing(800, 170)
	origin()
	active = _t1==0 ? 0 : _walk[_t1]
	for i in -5:5
		setcolor("white")
		settext("$(i)", Point(i*60, 20))
		setcolor(active == i ? "cyan" : "red")
		circle(Point(i*60, -30), 10, action=:fill)
		setcolor("white")
	end
	setfont("Helvetica", 24)
	settext("Time = $(_t1)", Point(-50, 70))
	finish()
	d
end

# ╔═╡ 13a3f0c1-4ec6-4c1b-a1f5-de4c03ebc3a6
html"<hr>"

# ╔═╡ af257d96-60e4-46af-ba6b-2a9c3f761036
bm = cumsum(rand([1, -1], (200, 10)), dims=1);

# ╔═╡ 4fee1c6d-1d11-4211-b159-987a5db90a26
plot(bm, xlabel="\$t\$", ylabel="\$n\$", label=reshape(map(i->"W $(i)", (1:10)), 1, :), legend=:outerright, size=(700, 400), ylims=(-30, 30))

# ╔═╡ b2603222-0616-4674-b26e-cec578a91691
html"<hr>"

# ╔═╡ d27b1c85-6d45-45fa-afdf-a408e5a11511
cps = let 
	U = SymTridiagonal(fill(0., 31), fill(0.5, 30))
	init = fill(0., 31)
	init[16] = 1
	ps = accumulate(1:40, init=init) do old, _
		U * old
	end
	map(0:40) do t
		if t == 0
			init = fill(0., 31)
			init[16] = 1
			plot(-15:15, init, ylims=(0, 1), ylabel="\$P_t(n)\$", xlabel="\$n\$", title="\$P_t(n)\$ vs \$n\$ at \$t=$(t)\$", legend=false)
		else
			plot(-15:15, ps[t], ylims=(0, 1), ylabel="\$P_t(n)\$", xlabel="\$n\$", title="\$P_t(n)\$ vs \$n\$ at \$t=$(t)\$", legend=false)
		end
	end
end;

# ╔═╡ 90fd60dd-bb0c-4da5-8fe8-b4e6f1200340
md"""
t = $(@bind _t2 Slider(0:40, show_value=true))
"""

# ╔═╡ 0a9fdd6f-fd72-4fc3-b290-e82865fcb31e
cps[_t2+1]

# ╔═╡ cf36b7f7-65e7-4936-9603-528d84cca003
md"""
- Mean - $\mu(t) = 0$
- Std Dev - $\sigma(t) \propto \sqrt t$
- Walk is irreducible, null recurrent $\implies$ no steady state
"""

# ╔═╡ 48091c30-f8e4-4fcb-a48c-334e06e1809a
html"<hr>"

# ╔═╡ 11c81fba-0142-454c-9595-d8bdd8bcceec
md"""
## Stochastic Resetting
"""

# ╔═╡ ba15db0b-d58a-47b4-a7bd-c54c472e5989
md"""
Allow at each step, a jump to node $r_0$ with a probability $\gamma$.

$$p_{ij} = \begin{cases}
	(1-\gamma) \cdot 1/2 & |i - j| = 1 \\
	\gamma & j = r_0 \\
	0 & \text{otherwise}
\end{cases}$$

"""

# ╔═╡ 4ae4e125-dce7-4d7e-a010-712f91401bb3
_walk_r = accumulate(sample([-1, 0, 1], Weights([0.45, 0.1, 0.45]), 40), dims=1) do old, move
	move == 0 ? 0 : old+move
end;

# ╔═╡ 527011b4-3eeb-44ac-9ac7-b95c0c5cbf09
md"""
t = $(@bind _t3 Slider(0:40, show_value=true))
"""

# ╔═╡ a681ebcf-d356-4f4d-bfa2-5f035593dfe9
let
	d = Drawing(800, 170)
	origin()
	active = _t3==0 ? 0 : _walk_r[_t3]
	for i in -5:5
		setcolor("white")
		settext("$(i)", Point(i*60, 20))
		setcolor(active == i ? "cyan" : "red")
		circle(Point(i*60, -30), 10, action=:fill)
		setcolor("white")
	end
	setfont("Helvetica", 24)
	settext("Time = $(_t3)", Point(-50, 70))
	finish()
	d
end

# ╔═╡ 45310b40-e290-4223-9736-d1c05065093f
html"<hr>"

# ╔═╡ 5b8d5924-7b1f-4301-b567-1f3fc2fdc5ab
md"""
Reset location $r_0$ = $(@bind _r₀ Slider(-20:5:20, show_value=true, default=0))
"""

# ╔═╡ 4120652b-3af6-4828-8192-7c1bd1291170
bmr = accumulate(sample([-1, 0, 1], Weights([0.425, 0.05, 0.425]), (200, 10)), dims=1) do old, move
	move == 0 ? _r₀ : old+move
end;

# ╔═╡ 818b36c9-c7cc-43e2-a93f-a035887a302c
begin
	p = plot(bmr, xlabel="\$t\$", ylabel="\$n\$", label=reshape(map(i->"W $(i)", (1:10)), 1, :), legend=:outertopright, size=(690, 400), title="γ = 0.05, r₀=$(_r₀)", ylims=(-30,30))
	p
end

# ╔═╡ b62884e9-b4a2-42df-8890-a50266d3eb49
html"<hr>"

# ╔═╡ 1662bb71-5e85-44e3-bbd4-3e5c9f354969
cprs = let 
	U1 = sparse(SymTridiagonal(fill(0., 31), fill(0.5, 30)))
	R = fill(0., (31, 31))
	R[21,:] .= 1
	U = sparse(0.8 * U1 + 0.2 * R)
	init = fill(0., 31)
	init[16] = 1.
	ps = accumulate(1:100, init=init) do old, _
		U * old
	end
	map(0:100) do t
		if t == 0
			init = fill(0., 31)
			init[16] = 1
			plot(-15:15, init, ylims=(0, 1), ylabel="\$P_t(n)\$", xlabel="\$n\$", title="\$P_t(n)\$ vs \$n\$ at \$t=$(t)\$, \$\\gamma=0.2\$, \$r_0=5\$", legend=false)
		else
			plot(-15:15, ps[t], ylims=(0, 1), ylabel="\$P_t(n)\$", xlabel="\$n\$", title="\$P_t(n)\$ vs \$n\$ at \$t=$(t)\$, \$\\gamma=0.2\$, \$r_0=5\$", legend=false)
		end
	end
end;

# ╔═╡ cffc5d0e-32a5-4a40-956e-cdc62a7c0fce
md"""
t = $(@bind _t4 Slider(0:40, show_value=true))
"""

# ╔═╡ af2f2bb4-126a-41ef-ad09-8789dd4a22a4
cprs[_t4+1]

# ╔═╡ 9cc709f7-15c1-4731-93be-f97c3ad98fa5
md"""
- Mean - $\mu(\infty) = r_0$
- Std Dev depends only on $\gamma$
- Walk is irreducible, positive recurrent $\iff$ steady state exists, which in the continuous limit is an exponential decay on either side of the reset location
"""

# ╔═╡ 9c099f8e-a0f5-43e3-b0b2-9e8b3af1fc17
html"<hr>"

# ╔═╡ 116415bd-b2cc-4bff-9ddd-8c004555e248
LocalResource("crcw.gif")

# ╔═╡ 6159b286-c8b9-4dad-b6c3-9d6c1380987b
html"<hr>"

# ╔═╡ 9992a1e9-aece-41de-a1a2-9fd4c8f14a01
md"""
# Quantum Walks
"""

# ╔═╡ dffac009-e2d0-435c-9761-9f3405d5762f
md"""
## Quantum Walks

Encode the state of the walker in a discrete, infinite level system, and the walk dynamics encoded by a unitary matrix U.

$$|\psi(t)\rangle = U(t)|0\rangle$$

### Coined Quantum Walks

Also called **Szegedy Quantum Walks**

1. Attach a two level coin - states denoted by $|0\rangle$ and $|1\rangle$. Resulting state lies in $\mathcal{H}_C\otimes\mathcal{H}_W$
    - Let capital $\Psi$ denote the coin-walker system, and small $\psi$ denote $\text{Tr}_C(\Psi)$
2. Define Left shift $\left(\text{denoted by }L\right)$ and a Right shift $\left(\text{denoted by }R\right)$ operator -

$$L |i\rangle = |i-1\rangle$$
$$R |i\rangle = |i+1\rangle$$

!!! note "Unitarity of L and R"
	Operators $L, R$ are unitary, and $L^\dagger = R$.

Thus, the resulting walk unitary $U$ can be written as the shift operations controlled on the coin.

$U = |0\rangle\langle 0|\otimes L + |1\rangle\langle 1|\otimes R$

- Define a coin operator $H_C\otimes I_W$, $H_C$ is a hadamard operation on the coin
- Apply this after every application of $U$ to get a superposition in the coin.
"""

# ╔═╡ 12ee2c63-d92b-476d-a7bf-2a570b95388f
html"<hr>"

# ╔═╡ a02d89a0-f33b-466e-8326-7e28b5f32b72
begin
	R = collect(Tridiagonal(fill(1., 60), zeros(61), zeros(60)))
	R[1, end] = 1
	L = collect(Tridiagonal(zeros(60), zeros(61), fill(1., 60)))
	L[end, 1] = 1
	U = KrausOperators([sparse(proj(ket(1, 2)) ⊗ L + proj(ket(2, 2)) ⊗ R)])
	init_coin = 1/√2 * (ket(1,2) - 1im * ket(2,2))
	H = KrausOperators([sparse(hadamard(2)⊗I(61))])
end;

# ╔═╡ 534ab6dc-16d5-473e-94d8-8741402f5d6c
qps = let
	init_state = proj(init_coin ⊗ ket(31,61))
	ψ = [[init_state]; accumulate(1:40, init=init_state) do old, _
		H(U(old))
	end]
	map(real.(diag.(ptrace.(ψ, Ref([2, 61]), Ref([1]))))) do ps
		plot(-30:30, ps, ylims=(0, 1), xlabel="\$i\$", ylabel="\$\\langle i|\\psi\\rangle\$", legend=false)
	end
end;

# ╔═╡ 8e3af979-2fcb-4833-bef4-b5749ee46eea
md"""
t = $(@bind _t6 Slider(0:30, show_value=true))
"""

# ╔═╡ 6d154d71-fd01-4d74-a813-39937cf88046
qps[_t6+1]

# ╔═╡ 81d77c7d-6fb5-4974-8f75-e2584db3b9ab
md"""
Note, this is a single walker, not an ensemble of walkers as is the case in classical random walks. Quantum walks (without measurement), are completely deterministic.
"""

# ╔═╡ fc22a2c3-7005-4859-9f5c-62c3bc61be05
md"""
- Mean - $\mu(t) = 0$
- Std Dev - $\sigma(t) \propto t$
- Walk is irreducible, transient $\implies$ no steady state
"""

# ╔═╡ 70b96e73-a918-403f-8b15-e36e35645bad
html"<hr>"

# ╔═╡ 12aa128d-9603-4131-bc4f-2869a890fd8a
md"""
## Stochastic Reset of Quantum Walks

$$O_{SR}(\rho) = (1-\gamma)\left(U\rho U^\dagger\right) + \gamma \left(c_{\text{init}}\otimes|r_0\rangle\langle r_0|\right)$$
"""

# ╔═╡ e6d007a7-2b6c-4566-8dae-bf22a34431c2
function swqw(n, γ)
	R = collect(Tridiagonal(fill(1., n), zeros(n+1), zeros(n)))
	R[1, end] = 1
	L = collect(Tridiagonal(zeros(n), zeros(n+1), fill(1., n)))
	L[end, 1] = 1
	U = KrausOperators([sparse(proj(ket(1, 2)) ⊗ L + proj(ket(2, 2)) ⊗ R)])
	init_coin = 1/√2 * (ket(1,2) - 1im * ket(2,2))
	H = KrausOperators([sparse(hadamard(2)⊗I(n+1))])
	init_state = proj(init_coin ⊗ ket(n÷2 + 1, n+1))
	ψ = [[init_state]; accumulate(1:40, init=init_state) do old, _
		(1-γ)*H(U(old)) + γ*proj(init_coin ⊗ ket(n÷2+6, n+1))
	end]
	map(real.(diag.(ptrace.(ψ, Ref([2, n+1]), Ref([1]))))) do ps
		plot(-n÷2:n÷2, ps, ylims=(0, 1), xlabel="\$i\$", ylabel="\$\\langle i|\\psi\\rangle\$", legend=false, title="\$\\gamma\$ = $(γ) r = \$|5\\rangle\$")
	end
end;

# ╔═╡ 94b8bfe2-cf3e-46c5-932c-f948f8693094
qpsrs = swqw(60, 0.1);

# ╔═╡ 078e87db-352a-48d6-8d0e-e45b4b5fae96
md"""
t = $(@bind _t7 Slider(0:30, show_value=true))
"""

# ╔═╡ faa02f93-d318-438e-af27-704c1b792662
qpsrs[_t7+1]

# ╔═╡ 8710669a-4850-4af9-b1f3-6796d9703457
html"<hr>"

# ╔═╡ b3916731-9722-4613-896a-083c8e30634e
LocalResource("./srqw.gif", :style=>"max-height: 80vh;")

# ╔═╡ 016dfe25-fe8e-432e-9435-f1c4064df090
md"""
# Hit rates

Main question - **Optimizing hit times**. 

Final requirement of any sort of search algorithm - system is measured, maximize probability of a particular readout

However, we first solve the simpler problem of mean hit time, shortest hit time and success probability for the multiple variations of the walk.

For the classical case, this poses no problem, as measurement does not disturb the system. In the quantum case however, we need to be a bit more careful.
"""

# ╔═╡ 684e065f-ab4f-42d5-96e6-b4ba159e3c2b
html"<hr>"

# ╔═╡ 762aa4f9-0913-4f2f-add9-27681d6d1f42
md"""
## Measurement in the Quantum walk
- Constantly measured walker will freeze dynamics - **Quantum Zeno effect**
- **Solution:** Measure after every $\tau$ steps.

!!! warning "τ=1"
	The $\tau = 1$ case reduces the discrete time quantum walk to a classical random walk

"""

# ╔═╡ 90ea1cb7-905b-4599-9807-9e36bdd0d167
md"""
### Measured Quantum Walk
"""

# ╔═╡ dcc71d75-5083-427a-98f9-1fc3568868bb
lrs, mqps, τ = let
	τ = 9
	init_state = proj(init_coin ⊗ ket(31,61))
	k = 200
	ψ = [[(31, init_state)]; accumulate(1:k, init=(31, init_state)) do (lr, old), t
		ψ′ = H(U(old))
		if t % τ == 0
			coin = ptrace(ψ′, [2, 61], [2])
			lr = sample(1:61, Weights(real(diag(ptrace(ψ′, [2, 61], [1])))))
			ψ′ = coin ⊗ proj(ket(lr, 61))
		end
		(lr, ψ′)
	end]
	ρs = real.(diag.(ptrace.(last.(ψ), Ref([2, 61]), Ref([1]))))
	lrs = first.(ψ)[1:τ:k+1] .- 31
	lrs, map(1:length(ρs)) do p
		plot(-30:30, ρs[p], ylims=(0, 1), xlabel="\$i\$", ylabel="\$\\langle i|\\psi\\rangle\$", title="With the coin operator, τ=$(τ)", legend=false)
		plot!(lrs[1:(p-1)÷τ+1], fill(0.05, (p-1)÷τ+1), st=:scatter, markersize=6, color=theme_palette(:dark)[5])
	end, τ
end;

# ╔═╡ 677f3a62-08f4-4bc5-939c-29ebe14c80dd
md"""
t = $(@bind _t8 Slider(0:200, show_value=true))
"""

# ╔═╡ c4f79c6b-6412-4a95-8040-7ba1d55c28bd
mqps[_t8+1]

# ╔═╡ 903c2923-1325-4620-a24f-1c51c197bfcc
html"<hr>"

# ╔═╡ 5e6af01a-2001-4433-b8e4-e81485f8ac34
mqwts, mqwpaths = let
	init_state = proj(init_coin ⊗ ket(31,61))
	k = 200
	τ = 6
	ys = map(1:10) do _
		ψ = [[(31, init_state)]; accumulate(1:k, init=(31, init_state)) do (lr, old), t
			ψ′ = H(U(old))
			if t % τ == 0
				coin = ptrace(ψ′, [2, 61], [2])
				lr = sample(1:61, Weights(real(diag(ptrace(ψ′, [2, 61], [1])))))
				ψ′ = coin ⊗ proj(ket(lr, 61))
			end
			(lr, ψ′)
		end]
		first.(ψ)[1:τ:k+1] .- 31
	end
	0:τ:τ*(length(first(ys))-1), ys
end;

# ╔═╡ fa8bab93-c9e6-4e66-ad20-0e40025c78b6
plot(
	mqwts, mqwpaths, 
	ylims=(-30,30), 
	ylabel="\$n\$", xlabel="t", title="Readout Path \$\\tau=6\$", 
	labels=reshape(map(i->"W $(i)", 1:length(mqwpaths)), 1, :), legend=:outerright,
	size=(700, 400)
)

# ╔═╡ bc10f9b5-70f0-47cf-9596-819755a6e687
html"<hr>"

# ╔═╡ c4348577-8caf-443b-9a88-f72dbf382fc4
md"""
## Survival probability

Define the following
- Denote the readout at the nᵗʰ measurement (at $t=n\tau$) as $X_n$.
- Select a target node $\delta$
- Probability of first hit $F_n = P(X_n = \delta | X_i \ne \delta \forall i \in [0, n-1])$
- Success probability $S_n = \sum_{i=1}^{n}F_i = P(\exists i \in [0, n]| X_i = \delta)$
- Survival probability $=$ Failure probability $= \mathcal{S}_n = 1 - S_n$
- Asymptomatic versions of these terms are given by taking $n\to \infty$
"""

# ╔═╡ c1ccabda-d3cd-4e60-ab03-95fd262afa0d
md"""
!!! danger "Continuous Walks"
	The following results are for continous walks. We intend to do similar analysis for discrete walks.
"""

# ╔═╡ b31dba15-91e0-476f-9611-07e64a492ee1
html"<hr>"

# ╔═╡ f5c5f906-7044-4a59-9732-f850dd7dffd2
LocalResource("./11-09-13-03.png", "style"=>"width: 75%")

# ╔═╡ cf9456a7-f475-406c-b580-a186005606a4
md"""
### Observations
- The quantum walk has a fast rise in the initial phase but saturates at ~0.1
- The classical walk has a slow rise, but eventually reaches 1

So while the quantum walker is faster, the walk is now transient, which leads to a non zero asymptomatic failure rate.
"""

# ╔═╡ 1438ce70-8f44-46f1-8a20-07a1827973e5
html"<hr>"

# ╔═╡ b76f796d-9180-49dd-a9fd-7c6cee304003
md"""
## The Solution
"""

# ╔═╡ b1b1592f-d6f3-4bb7-8d21-29f48eaad864
md"""
If the walk is fast in the initial phase, but saturates, is it possible to restart the walk when it starts saturating? 

- Allows the initial speedup, but the saturation will not occur

Thus we restart after $r$ measurement events (at $t = r\tau$).
"""

# ╔═╡ 368b3dc0-584e-4484-bd0a-210a02839218
LocalResource("./11-09-15-08.png", "style"=>"width: 75%")

# ╔═╡ 98793912-08f9-400f-9a4c-ac68038d149e
@htl """
<details>
<summary>$(md"Legend")</summary>
$(md"1. Target $\delta: 10$
2. Measurement $\tau = 0.25$
3. Quantum: $r = 35$
4. Classical-1: $r = 35$
5. Classical-2: $r = 191$")
</details>"""

# ╔═╡ 836fdba8-dae9-46fe-accc-eb49bcadafb8
html"<hr>"

# ╔═╡ 149cb9d9-e942-4b6e-90b9-1651cf743589
md"""
## Observations
- Deterministic restart leads to zero asymptomatic failure rate
- Eager restarting leads to walker never reaching $\delta$, reducing success rates drastically
- Cautious restart reduces the effect of restart, reducing success rates.
- There exists an optimal $r$, but needs to be optimized, which is non trivial for general parameters
"""

# ╔═╡ 589993bf-d6c5-4ddc-a290-139ea03641cf
LocalResource("./11-09-15-51.png", "style"=>"width: 55%")

# ╔═╡ b4ac675b-947a-4e8a-8ab3-e13750edeee8
md"""
**Can stochastic restarting be better than sharp reset?**

No, because even if $\langle r\rangle = r_{\text{optimal}}$, $\langle t_f\rangle > \langle t_f\rangle_{\text{optimal}}$ due to the non-monotonic nature of the curve
"""

# ╔═╡ 05902f82-f995-4ce3-b831-ba2ec8939c72
html"<hr>"

# ╔═╡ e3a99564-0c3a-4e66-aee0-070b3319861b
md"## REDACTED"

# ╔═╡ 01a9105c-e6ba-44d2-93f1-d598621f1100
html"<hr>"

# ╔═╡ 93f19125-ed16-4869-8197-a26610bb9472
md"""
# REDACTED
"""

# ╔═╡ dbf63c5f-066d-4e00-9fb3-33bd010ce16e
md"""
## REDACTED
"""

# ╔═╡ 3f20db60-d209-4b08-baa9-c932d5a0b815
@bind ref References()

# ╔═╡ 7303daab-5934-4b56-b43e-a834d18cb5a5
display_bibliography("./MidwayThesis.bib", ref)

# ╔═╡ fae1b25d-3386-498e-a540-4f6e5355c3e8
md"""
# Appendix

## Classical Reset Flow analysis

Stable distribution can be found using Flow analysis at stability -

Let $A = \{-\infty, a\}, A^C = \{a+1, \infty\}, a>0$ 

$$F(A, A^C) = \frac{1-\gamma}{2}\pi_a$$

$$F(A^C, A) = \frac{1-\gamma}{2}\pi_{a+1} + \gamma\sum_{i=a+1}^{\infty} \pi_{i}$$

At stability, 

$$F(A, A^C) = \frac{1-\gamma}{2}\pi_a = \frac{1-\gamma}{2}\pi_{a+1} + \gamma\sum_{i=a+1}^{\infty} \pi_{i} = F(A^C, A)$$

$$\implies \pi_{a+1} - \pi_{a} = -2 \frac{\gamma}{1 - \gamma}\sum_{i=a+1}^{\infty} \pi_{i}$$

$$\implies \pi_{a+1} - \pi_{a} = -\frac{\gamma}{1 - \gamma} \left(2\sum_{i=a+1}^{\infty} \pi_{i}\right)$$

$$\implies \pi_{a+1} - \pi_{a} = -\frac{\gamma}{1 - \gamma} \left(1 - 2\sum_{i=0}^{a} \pi_{i}\right)$$

$$\implies \pi_{a+1} = \pi_{a} -\frac{\gamma}{1 - \gamma} \left(1 - 2\sum_{i=0}^{a} \pi_{i}\right)$$

Normalization of $\pi$ sets $\pi_0$ of the solution.
"""

# ╔═╡ 346839a2-eaf2-43a6-83c2-6ff461368cb7
md"""
## Why the coin operator is necessary
"""

# ╔═╡ e69b65e5-d761-4cd8-9be7-7f56f56cc58a
md"""
On the first application of $U$, the state of the coin and the shifted states get entangled. Therefore, no superposition exists after the first step.
"""

# ╔═╡ 20df1bef-fd64-438f-9724-1409ebe6e684
pqps = let
	init_state = proj(init_coin ⊗ ket(31,61))
	ψ = [[init_state]; accumulate(1:30, init=init_state) do old, _
		U(old)
	end]
	map(real.(diag.(ptrace.(ψ, Ref([2, 61]), Ref([1]))))) do ps
		plot(-30:30, ps, ylims=(0, 1), xlabel="\$i\$", ylabel="\$\\langle i|\\psi\\rangle\$", title="No Coin Operator", legend=false)
	end
end;

# ╔═╡ def9138d-facf-4fd3-b836-0bef4bb037fa
md"""
t = $(@bind _t5 Slider(0:30, show_value=true))
"""

# ╔═╡ 5b75467c-bd44-4a03-ad95-b0270f2b7326
pqps[_t5+1]

# ╔═╡ 0caa18a4-f269-4e11-90ab-5a239bfe2338


# ╔═╡ 68915b43-d635-40e9-a4a4-035ccb5ef3b9
md"""
## REDACTED
"""

# ╔═╡ f2ed0af8-29f9-4532-858a-34ac6cfb5dd7
style_citations(ref, "./midway-report/MidwayThesis.bib", css=JustBold, content=By_Number)

# ╔═╡ c18a5537-caef-480f-b411-e24d51ecac97
md"""
Refs: 
$(cite"yin_restart_2022")
$(cite"goldsmith_link_2022")
$(cite"bonnetain_finding_2022")
$(cite"ostrovsky_simple_2014")
$(cite"ortega_generalized_2022")
$(cite"qiskit_quantum")
$(cite"breloff_plotsjl_2022")
$(cite"sambrani_plutoreportjl")
$(cite"van_der_plas_fonspplutojl_2022")
$(cite"juliagraphics_luxorjl")
$(cite"griffin_investigation_2019")
$(cite"friedman_quantum_2017")
$(cite"bonomo_first_2021")
$(cite"psaltis_markov_2022")
$(cite"kempe_quantum_2003")
$(cite"metropolis_beginning")
$(cite"bae_source_2021")
$(cite"mukherjee_grover_2022")
$(cite"luo_yaojl_2020")
$(cite"bezanson_julia_2017")
$(cite"gawron_quantuminformationjljulia_2018")
$(cite"norris_markov_1998")
$(cite"glasserman_monte_2004")
$(cite"portugal_quantum_2018")
"""

# ╔═╡ e099a79a-33ef-43ad-8890-c134a1cbc053
begin
	centerslide = html"<h1></h1>"
end;

# ╔═╡ 9d801bc0-42b0-44bb-a2cd-9a43d7c80f17
md"""$(centerslide) ### Multiple Walks"""

# ╔═╡ cf279915-1766-4008-bbce-c96e1f2c0745
md"""$(centerslide) ### Probability mass of walkers"""

# ╔═╡ f06f0a1e-d112-4f0d-89e0-539c0ea5a9b6
md"""$(centerslide) ### Multiple Walks"""

# ╔═╡ 8eec20a2-dca6-4001-a9ee-19e5313c6893
md"""$(centerslide) ### Probability mass of walkers"""

# ╔═╡ 59b20b4b-4971-4172-a621-c1907ade0bf0
centerslide

# ╔═╡ 94602660-6f76-4110-b7a0-1b1be770d69f
md"""
$(centerslide) ### Probability of readout
"""

# ╔═╡ 0337a9eb-cdf0-4013-bba1-b2bcd7ec6f3f
centerslide

# ╔═╡ 9d673662-cbcb-470f-9148-c7f06a8091df
md"""
$(centerslide) ### Readout paths
"""

# ╔═╡ a7fb6d92-a9ad-481d-983b-b34eafcb03f3
centerslide

# ╔═╡ 63165326-7d47-4c24-b2d7-012ceb3b777a
centerslide

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Luxor = "ae8d54c2-7ccd-5906-9d76-62fc9837b5bc"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoReport = "ab5eb977-4f23-42a0-954d-2743fb6218c4"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
QuantumInformation = "3c0b384b-479c-5684-b2ef-9d7a46dd931e"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

[compat]
HypertextLiteral = "~0.9.4"
Luxor = "~3.5.0"
Plots = "~1.36.6"
PlutoReport = "~0.4.0"
PlutoUI = "~0.7.49"
QuantumInformation = "~0.4.9"
StatsBase = "~0.33.21"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.YAML]]
deps = ["Base64", "Dates", "Printf", "StringEncodings"]
git-tree-sha1 = "dbc7f1c0012a69486af79c8bcdb31be820670ba2"
uuid = "ddb6d928-2868-570f-bddf-ab3f9cf99eb6"
version = "0.4.8"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.gdk_pixbuf_jll]]
deps = ["Artifacts", "Glib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Xorg_libX11_jll", "libpng_jll"]
git-tree-sha1 = "e9190f9fb03f9c3b15b9fb0c380b0d57a3c8ea39"
uuid = "da03df04-f53b-5353-a52f-6a8b0620ced0"
version = "2.42.8+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─4a35ff54-5d10-11ed-3969-83c1ef512684
# ╟─d0ad6f88-c906-44a5-908a-9996bd33bf0e
# ╟─7de512e2-f43a-42aa-92bf-49a75170b3ed
# ╟─f9c42e32-8ec9-4bc7-ab27-d3260d587f5c
# ╟─b3a75a22-3700-4a4a-ab13-d064715f2a08
# ╟─504843f6-33ab-475c-9936-10992be19386
# ╟─62adfcd1-e110-40ae-a0bf-f2b2625b037e
# ╟─9bc0dc54-4e12-4d4d-b162-4ef738e6fd73
# ╟─a20a8ab6-2f1d-4118-8135-e68d7882b285
# ╟─bced9c5d-5cbf-4d4c-b6d1-b500d76d4145
# ╟─ae30bd04-5325-4f53-9f4f-c62751d528fd
# ╟─c06f87ae-3a73-4640-969a-c6d09d7b97bf
# ╟─d44250f8-9808-45fe-9a5c-b5eeb1c3f202
# ╟─1ed2f6a3-c55a-428b-90aa-8fe44e8ab965
# ╟─13a3f0c1-4ec6-4c1b-a1f5-de4c03ebc3a6
# ╟─9d801bc0-42b0-44bb-a2cd-9a43d7c80f17
# ╟─af257d96-60e4-46af-ba6b-2a9c3f761036
# ╟─4fee1c6d-1d11-4211-b159-987a5db90a26
# ╟─b2603222-0616-4674-b26e-cec578a91691
# ╟─cf279915-1766-4008-bbce-c96e1f2c0745
# ╟─d27b1c85-6d45-45fa-afdf-a408e5a11511
# ╟─90fd60dd-bb0c-4da5-8fe8-b4e6f1200340
# ╟─0a9fdd6f-fd72-4fc3-b290-e82865fcb31e
# ╟─cf36b7f7-65e7-4936-9603-528d84cca003
# ╟─48091c30-f8e4-4fcb-a48c-334e06e1809a
# ╟─11c81fba-0142-454c-9595-d8bdd8bcceec
# ╟─ba15db0b-d58a-47b4-a7bd-c54c472e5989
# ╟─4ae4e125-dce7-4d7e-a010-712f91401bb3
# ╟─527011b4-3eeb-44ac-9ac7-b95c0c5cbf09
# ╟─a681ebcf-d356-4f4d-bfa2-5f035593dfe9
# ╟─45310b40-e290-4223-9736-d1c05065093f
# ╟─f06f0a1e-d112-4f0d-89e0-539c0ea5a9b6
# ╟─4120652b-3af6-4828-8192-7c1bd1291170
# ╟─5b8d5924-7b1f-4301-b567-1f3fc2fdc5ab
# ╟─818b36c9-c7cc-43e2-a93f-a035887a302c
# ╟─b62884e9-b4a2-42df-8890-a50266d3eb49
# ╟─8eec20a2-dca6-4001-a9ee-19e5313c6893
# ╟─1662bb71-5e85-44e3-bbd4-3e5c9f354969
# ╟─cffc5d0e-32a5-4a40-956e-cdc62a7c0fce
# ╟─af2f2bb4-126a-41ef-ad09-8789dd4a22a4
# ╟─9cc709f7-15c1-4731-93be-f97c3ad98fa5
# ╟─9c099f8e-a0f5-43e3-b0b2-9e8b3af1fc17
# ╟─59b20b4b-4971-4172-a621-c1907ade0bf0
# ╟─116415bd-b2cc-4bff-9ddd-8c004555e248
# ╟─6159b286-c8b9-4dad-b6c3-9d6c1380987b
# ╟─9992a1e9-aece-41de-a1a2-9fd4c8f14a01
# ╟─dffac009-e2d0-435c-9761-9f3405d5762f
# ╟─12ee2c63-d92b-476d-a7bf-2a570b95388f
# ╟─a02d89a0-f33b-466e-8326-7e28b5f32b72
# ╟─94602660-6f76-4110-b7a0-1b1be770d69f
# ╟─534ab6dc-16d5-473e-94d8-8741402f5d6c
# ╟─8e3af979-2fcb-4833-bef4-b5749ee46eea
# ╟─6d154d71-fd01-4d74-a813-39937cf88046
# ╟─81d77c7d-6fb5-4974-8f75-e2584db3b9ab
# ╟─fc22a2c3-7005-4859-9f5c-62c3bc61be05
# ╟─70b96e73-a918-403f-8b15-e36e35645bad
# ╟─12aa128d-9603-4131-bc4f-2869a890fd8a
# ╟─e6d007a7-2b6c-4566-8dae-bf22a34431c2
# ╟─94b8bfe2-cf3e-46c5-932c-f948f8693094
# ╟─078e87db-352a-48d6-8d0e-e45b4b5fae96
# ╟─faa02f93-d318-438e-af27-704c1b792662
# ╟─8710669a-4850-4af9-b1f3-6796d9703457
# ╟─0337a9eb-cdf0-4013-bba1-b2bcd7ec6f3f
# ╟─b3916731-9722-4613-896a-083c8e30634e
# ╟─016dfe25-fe8e-432e-9435-f1c4064df090
# ╟─684e065f-ab4f-42d5-96e6-b4ba159e3c2b
# ╟─762aa4f9-0913-4f2f-add9-27681d6d1f42
# ╟─90ea1cb7-905b-4599-9807-9e36bdd0d167
# ╟─dcc71d75-5083-427a-98f9-1fc3568868bb
# ╟─677f3a62-08f4-4bc5-939c-29ebe14c80dd
# ╟─c4f79c6b-6412-4a95-8040-7ba1d55c28bd
# ╟─903c2923-1325-4620-a24f-1c51c197bfcc
# ╟─9d673662-cbcb-470f-9148-c7f06a8091df
# ╟─5e6af01a-2001-4433-b8e4-e81485f8ac34
# ╟─fa8bab93-c9e6-4e66-ad20-0e40025c78b6
# ╟─bc10f9b5-70f0-47cf-9596-819755a6e687
# ╟─c4348577-8caf-443b-9a88-f72dbf382fc4
# ╟─c1ccabda-d3cd-4e60-ab03-95fd262afa0d
# ╟─b31dba15-91e0-476f-9611-07e64a492ee1
# ╟─a7fb6d92-a9ad-481d-983b-b34eafcb03f3
# ╟─f5c5f906-7044-4a59-9732-f850dd7dffd2
# ╟─cf9456a7-f475-406c-b580-a186005606a4
# ╟─1438ce70-8f44-46f1-8a20-07a1827973e5
# ╟─b76f796d-9180-49dd-a9fd-7c6cee304003
# ╟─b1b1592f-d6f3-4bb7-8d21-29f48eaad864
# ╟─368b3dc0-584e-4484-bd0a-210a02839218
# ╟─98793912-08f9-400f-9a4c-ac68038d149e
# ╟─836fdba8-dae9-46fe-accc-eb49bcadafb8
# ╟─149cb9d9-e942-4b6e-90b9-1651cf743589
# ╟─589993bf-d6c5-4ddc-a290-139ea03641cf
# ╟─b4ac675b-947a-4e8a-8ab3-e13750edeee8
# ╟─05902f82-f995-4ce3-b831-ba2ec8939c72
# ╟─e3a99564-0c3a-4e66-aee0-070b3319861b
# ╟─01a9105c-e6ba-44d2-93f1-d598621f1100
# ╟─93f19125-ed16-4869-8197-a26610bb9472
# ╟─dbf63c5f-066d-4e00-9fb3-33bd010ce16e
# ╟─3f20db60-d209-4b08-baa9-c932d5a0b815
# ╟─7303daab-5934-4b56-b43e-a834d18cb5a5
# ╟─fae1b25d-3386-498e-a540-4f6e5355c3e8
# ╟─346839a2-eaf2-43a6-83c2-6ff461368cb7
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