### A Pluto.jl notebook ###
# v0.19.40

using Markdown
using InteractiveUtils

# ╔═╡ d928ee56-0f14-426c-b669-356627382f8f
using Braket, BraketSimulator

# ╔═╡ ab9771de-a268-4df8-811d-9310a12a7d79
using Plots

# ╔═╡ eefcfdb6-dc0f-4ec4-85a3-73bafc9e0177
md"
# Quantum Fourier Transform with Amazon Braket
"

# ╔═╡ e00258a8-e04e-4806-85ad-1ef5c4476dad
md"""
In this example, we show an implementation of the Quantum Fourier Transform (QFT) and its inverse. QFT is an important subroutine in many quantum algorithms, like Shor's algorithm and Quantum Phase Estimation. The QFT can be performed efficiently on a quantum computer, using only $\mathcal{O}(n^2)$ single-qubit Hadamard gates and two-qubit controlled phase shift gates, where $n$ is the number of qubits. First, we review the basics of the QFT circuit and how it relates to classical Fourier transforms. Then, we implement QFT in both recursive and non-recursive form, as well as its inverse.
"""

# ╔═╡ b5530b38-f3e1-4bd7-bf6e-b2052a44850b
md"""
## Background of QFT

The Quantum Fourier Transform is a quantum algorithm for applying a discrete Fourier transform to the amplitudes of a quantum state. It's the quantum analogue of the [classical discrete Fourier transform](https://en.wikipedia.org/wiki/Discrete_Fourier_transform). As a refresher, if we have a length $N$ vector $(x_0, x_1, \ldots, x_{N-1}) \in \mathbb{C}^N$, then the discrete Fourier transform of this vector is a length $N$ vector $(y_0, y_1, \ldots, y_{N-1}) \in \mathbb{C}^N$:

```math
y_j = \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1} x_k \exp{\left\{\frac{2\pi i}{N} j k \right\}}
```

To develop the *quantum* analogue to this, we can consider an arbitrary state $|\mathtt{input}\rangle = \sum_{j=0}^{N-1} x_j |j\rangle$, and map it to an output state $|\mathtt{output}\rangle = \sum_{k=0}^{N-1} y_k |k\rangle$ such that:

```math
y_k = \frac{1}{\sqrt{N}} \sum_{\ell = 0}^{N-1} x_\ell \exp{\left\{ \frac{2\pi i}{N} \ell k \right\}}
```

As an example, let's examine the QFT of state $|0\rangle$ on 3 qubits, which we could also write as $|000\rangle$. We can calculate from the above formula for the coefficients $y_k$ that the output will be an equal superposition over all states:

```math
\begin{align*}
y_0 &= \frac{1}{\sqrt{2^3}} \\
y_1 &= \frac{1}{\sqrt{2^3}} \\
\ldots & \\
y_{N-1} &= \frac{1}{\sqrt{2^3}} \\
|\mathtt{output}\rangle &= \frac{1}{\sqrt{2^3}}\sum_{m=0}^{N-1} |m\rangle \\
\end{align*}
```

This state can be factored (you can multiply it out to check):
```math
\begin{align*}
|\mathtt{output}\rangle &= \frac{1}{\sqrt{2^3}} \left(|0\rangle + |1\rangle\right) \otimes \left(|0\rangle + |1\rangle\right) \otimes \left(|0\rangle + |1\rangle\right) \\
&= \otimes_{q=0}^{N-1} \left| + \right\rangle \\
\left| + \right\rangle &= \frac{1}{\sqrt{2}} \left(|0\rangle + |1\rangle\right) \\
\end{align*}
```

We can also implement the *inverse* QFT, which restores the original $|\mathtt{input}\rangle$ state. We will show the code for this below.
"""

# ╔═╡ d18db1cb-58b8-4949-b80a-67f36768e59c
md"""
### Implementing recursive QFT

In this section, we will implement the QFT algorithm using a recursive approach. The function `qft_no_swap` will apply all gates up to the final `SWAP`s, and `qft_recursive` will call it recursively then apply the `SWAP` gates.
"""

# ╔═╡ 608cf437-7493-494b-a4d7-1e852bcae8d9
"""
	qft_no_swap(qubits) -> Circuit

Subroutine of the QFT excluding the final `SWAP` gates, applied to the `qubits` argument.

"""
function qft_no_swap(qubits)
    # On a single qubit, the QFT is just a Hadamard.
    length(qubits) == 1 && return Circuit([(H, qubits)])
    
    # For more than one qubit, we define the QFT recursively
    qftcirc = Circuit()
	# First add a Hadamard gate
	qftcirc(H, first(qubits))
	# Then apply the controlled rotations, with weights (angles) defined by the distance to the control qubit.
	for (k, qubit) in enumerate(qubits[2:end])
		qftcirc(CPhaseShift, qubit, first(qubits), 2π/2^(k+1))
	end
	# Now apply the above gates recursively to the rest of the qubits
	qftcirc(qft_no_swap(qubits[2:end]))
    return qftcirc
end

# ╔═╡ 2c612fa4-1a8d-404a-8b3e-5e1907e20d84
"""
	qft_recursive(qubits)

Construct a circuit object corresponding to the Quantum Fourier Transform (QFT)
algorithm, applied to the argument `qubits`.
"""
function qft_recursive(qubits)
    qftcirc = Circuit()
    
    # First add the QFT subroutine above
    qftcirc(qft_no_swap(qubits))
    
    # Then add SWAP gates to reverse the order of the qubits:
    for i in 1:div(length(qubits), 2)
        qftcirc(Swap, qubits[i], qubits[end-i-1])
	end
    return qftcirc
end

# ╔═╡ ac3b8437-7f21-4e36-97bd-5af10d0eaf40
md"""
### Implementing non-recursive QFT

In this section, we write a function `qft` which implements the QFT algorithm non-recursively.
"""

# ╔═╡ 3ccc5bb1-16ec-4fd5-b82f-31da2734223e
"""
	qft(qubits)

Construct a circuit object corresponding to the Quantum Fourier Transform (QFT)
algorithm, applied to the argument `qubits`.  Does not use recursion to generate the QFT.
"""
function qft(qubits)   

    qftcirc = Circuit()
    
    # get number of qubits
    num_qubits = length(qubits)
    
    for k in 1:num_qubits
        # First add a Hadamard gate
        qftcirc(H, qubits[k])
        # Then apply the controlled rotations, with weights (angles) defined by the distance to the control qubit.
        # Start on the qubit after qubit k, and iterate until the end.  When num_qubits==1, this loop does not run.
        for j in 1:(num_qubits - k - 1)
            angle = 2π/2^(j+1)
            qftcirc(CPhaseShift, qubits[k+j], qubits[k], angle)
		end
	end
            
    # Then add SWAP gates to reverse the order of the qubits:
    for i in 1:div(num_qubits, 2)
        qftcirc(Swap, qubits[i], qubits[end-i])
	end
        
    return qftcirc
end

# ╔═╡ 25274ab6-b0a2-485f-baee-3a8a23c23834
md"""
### Inverse QFT

Finally, we can implement the inverse QFT algorithm.
"""

# ╔═╡ 55a3b85b-6d3c-43b9-9c89-5af3f5c57629
"""
	inverse_qft(qubits)

Construct a circuit object corresponding to the inverse Quantum Fourier Transform (QFT)
algorithm, applied to the argument `qubits`.  Does not use recursion to generate the circuit.
"""
function inverse_qft(qubits)

    # instantiate circuit object
    qftcirc = Circuit()
    
    # get number of qubits
    num_qubits = length(qubits)
    
    # First add SWAP gates to reverse the order of the qubits:
    for i in 1:div(num_qubits, 2)
        qftcirc(Swap, qubits[i], qubits[end-i+1])
	end
        
    # Start on the last qubit and work to the first.
    for k in reverse(1:num_qubits)
        # Apply the controlled rotations, with weights (angles) defined by the distance to the control qubit.
        # These angles are the negative of the angle used in the QFT.
        # Start on the last qubit and iterate until the qubit after k.  
        # When num_qubits==1, this loop does not run.
        for j in reverse(1:num_qubits - k)
            angle = -2π/2^(j+1)
            qftcirc(CPhaseShift, qubits[k+j], qubits[k], angle)
		end
        # Then add a Hadamard gate
        qftcirc(H, qubits[k])
	end
    return qftcirc
end

# ╔═╡ 52bf519d-806c-4d4c-915a-fa28be9b2fc2
md"""
## Running QFT on a simulator

With these routines defined, we can run QFT and its inverse on a simulator to develop more intuition. We'll use the Braket local simulator, which runs on hardware you control and is free to use.
"""

# ╔═╡ b3a750d1-7a2b-4de6-9e6c-166336ba6cec
device = LocalSimulator("braket_sv_v2")

# ╔═╡ 7127d624-fb0d-41b9-a8cd-6cae7d34d55f
"""
	prettify_state_vector(state_vector)

Rounds down small amplitudes in a given `state_vector` make it easier to analyse.
"""
function prettify_state_vector(state_vector)
	state_vec_pretty = complex.(trunc.(real.(state_vector), digits=3), trunc.(imag.(state_vector), digits=3))

	state_vec_pretty = map(ampl->abs(ampl)>1e-5 ? ampl : zero(ampl), state_vec_pretty)
	return state_vec_pretty
end

# ╔═╡ 83b78b1a-3935-4ca7-88f6-7a15cc523591
"""
	run_and_plot_qft(c::Circuit)

Runs a given `Circuit` `c` and plots the resulting state vector probabilities.
"""
function run_and_plot_qft(c::Circuit)
	num_qubits = qubit_count(c)
	bitstring_keys = [prod(string.(digits(ii, base=2, pad=num_qubits))) for ii in 0:(2^num_qubits)-1]
	# specify desired result types
	c(Braket.StateVector)
	c(Probability)

	# Run the task
	task = simulate(device, c, shots=0)
	res  = result(task)
	state_vector = res.values[1]
	probs_values = res.values[2]
	p = bar(bitstring_keys, probs_values, xlabel="bitstrings", ylabel="probability", legend=false)
	return p
end

# ╔═╡ 335e537b-c6f8-4c1b-a9b0-5e14c4f7323b
md"""
### Examining the effect of QFT on various states

Let's first run the QFT on the $|0\rangle$ state, which, as we showed above, should be mapped to an equal superposition over all states by the QFT:
"""

# ╔═╡ 256bb268-e2fc-4483-a6f7-4a76028fcf80
# check output for input |0,0,0> -> expect uniform distribution
run_and_plot_qft(qft(collect(0:2)))

# ╔═╡ 9bfd35b5-82cf-4c5e-8006-a3b2de15598a
md"""
We can also prepare by hand (using the formula above) the result of QFT on a particular state, say $|2\rangle = |010\rangle$, and apply the inverse QFT to recover the input state:
"""

# ╔═╡ 20cfa5e8-d4ad-4720-9d27-f7c161ea09e3
begin
	nqs1 = 3
	qbs1 = collect(0:nqs1-1)
	circ2 = Circuit([(H, qbs1)])
	for ii in 1:nqs1 - 1
	    circ2(Rz, ii, π/2^(ii-1))
	end
	# add iQFT circuit
	circ2(inverse_qft(qbs1))
	run_and_plot_qft(circ2)
end

# ╔═╡ ac34f36b-9ede-4bf0-9120-a5e0253d6f64
md"And we can verify that our code works on qubit counts larger than 3:"

# ╔═╡ 0508acef-1de0-42e4-9bbc-9127c2a162b5
begin
	# test that the iQFT circuit works for larger qubit counts
	nqs2 = 4
	qbs2 = collect(0:nqs2-1)
	circ3 = Circuit(vcat((H, qbs2), [(Rz, ii, π/2^(ii-1)) for ii in 1:nqs2-1]))
	# add iQFT circuit   
	circ3(inverse_qft(qbs2))
	run_and_plot_qft(circ3)
end

# ╔═╡ f584f456-c803-4fa8-99f5-cb9f87b13097
md"""
If we apply QFT and then its inverse, we should recover the input state. Let's verify this:
"""

# ╔═╡ ae77c39f-9f7e-4c45-8823-a16a38c33e0d
begin
	#test that the QFT and iQFT circuits cancel each other
	nqs3 = 3
	qbs3 = collect(0:nqs3-1)
	circ4 = qft(qbs3)
	circ4(inverse_qft(qbs3))
	
	run_and_plot_qft(circ4)
end

# ╔═╡ cc46be99-41e5-4b30-a158-bd5c8c961247
md"""
## Conclusion

In this example, we gave a brief introduction to the quantum Fourier transform, which is the quantum analogue of the classical discrete Fourier transform. This algorithm is an essential building block of many higher level quantum algorithms. We verified that QFT and iQFT are inverses of each other and that we see the expected output when running on simple test cases.
"""

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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.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[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.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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+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.libsodium_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "848ab3d00fe39d6fbc2a8641048f8f272af1c51e"
uuid = "a9144af2-ca23-56d9-984f-0d03f7b5ccf8"
version = "1.0.20+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 = "1.52.0+1"

[[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:
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# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
