diff --git a/Exercises/Session 4 - Quantum Information.ipynb b/Exercises/Session 4 - Quantum Information.ipynb new file mode 100644 index 0000000..5cbe3bd --- /dev/null +++ b/Exercises/Session 4 - Quantum Information.ipynb @@ -0,0 +1,174 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "e4f44eb7", + "metadata": {}, + "source": [ + "

TMQS Workshop 2024 @ Zuse Institute Berlin

\n", + "

Summer School on Tensor Methods for Quantum Simulation

\n", + "

June 3 - 5, 2024

\n", + "$\\def\\tcoreleft{\\textcolor{MidnightBlue}{\\Huge⦸}}$\n", + "$\\def\\tcorecenter{\\textcolor{RedOrange}{\\Huge⦿}}$\n", + "$\\def\\tcoreright{\\textcolor{MidnightBlue}{\\Huge\\oslash}}$\n", + "
\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "
$\\tcoreleft$$-$$\\tcoreleft$$-$$\\tcoreleft$$-$$\\cdots$$-$$\\tcorecenter$$-$$\\cdots$$-$$\\tcoreright$$-$$\\tcoreright$
$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$
\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "6f6cc702", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "74ce99d5-268a-4ad8-980b-de19d60b6be6", + "metadata": {}, + "source": [ + "## **Session 4 - Quantum Information**" + ] + }, + { + "cell_type": "markdown", + "id": "e8441c54-ffd4-45b4-ab04-ff8cf85e6474", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "a781c1c8", + "metadata": {}, + "source": [ + "## Exercise 4.1\n", + "\n", + "Which of the following are valid quantum states?\n", + "\n", + "$\\hspace{1cm}$$\\begin{pmatrix} 0 \\\\ 1\\end{pmatrix}$, $\\quad \\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}$, $\\quad \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ -i\\end{pmatrix}$, $\\quad \\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ 2\\end{pmatrix}$, $\\quad \\begin{pmatrix} \\sqrt{2/3} \\\\ i/\\sqrt{3}\\end{pmatrix}$\n", + "\n", + "What is the probability to measure $0$ and $1$ for the valid quantum states?" + ] + }, + { + "cell_type": "markdown", + "id": "6e1b9898-10a5-422d-8cdb-a8f42bdc01c4", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "0e8243a2-a54e-4aaf-b460-f1f7f9e369b0", + "metadata": {}, + "source": [ + "## Exercise 4.2\n", + "\n", + "**a)**$\\quad$Write down the state vector of two quantum states \n", + "\n", + "$\\hspace{1cm}$$\\displaystyle|\\Psi_1\\rangle = \\alpha_1 |0\\rangle + \\beta_1 |1\\rangle \\quad $ and $\\quad |\\Psi_2\\rangle = \\alpha_2 |0\\rangle + \\beta_2 |1\\rangle$, \n", + "\n", + "$\\hspace{0.35cm}$$\\quad$i.e. the tensor product, in the computational basis. Write down the basis vectors of the composite system.\n", + "\n", + "**b)**$\\quad$Consider the $2$-qubit state \n", + "\n", + "$\\hspace{1cm}$$\\displaystyle|\\Psi\\rangle = \\frac{1}{\\sqrt{2}} |00\\rangle + \\frac{1}{2}|01\\rangle + \\frac{1}{2} |11\\rangle$. \n", + "\n", + "$\\hspace{0.35cm}$$\\quad$What is the state after a measurement of the first qubit where you obtain $|0\\rangle$?\n", + "\n", + "$\\hspace{0.35cm}$$\\quad$Is this an entangled state?\n", + "\n", + "$\\hspace{0.35cm}$$\\quad$*Hint:* Quantum states are normalized!" + ] + }, + { + "cell_type": "markdown", + "id": "818bf57f-be0c-408a-9776-414578b7763a", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "bed85201-1bfb-45a2-a57a-44332f370276", + "metadata": {}, + "source": [ + "## Exercise 4.3\n", + "\n", + "Suppose you have $n + 1$ qubits. We will write $|\\vec{x}\\rangle$ to mean the $n$-qubit classical state given by the number $x$ in binary. For instance, if $n = 2$ then:\n", + "\n", + "$\\hspace{0.5cm}$$|\\vec{0}\\rangle = |00\\rangle , \\quad |\\vec{1}\\rangle = |01\\rangle, \\quad |\\vec{2}\\rangle = |10\\rangle, \\quad |\\vec{3}\\rangle = |11\\rangle$.\n", + "\n", + "Assume the qubits are in the state\n", + "\n", + "$\\hspace{0.5cm}$$\\displaystyle |\\Psi\\rangle = \\frac{1}{\\sqrt{2^n}} \\sum_{x=0}^{2^n -1} | \\vec{x} \\rangle \\otimes | x~\\text{mod}~2 \\rangle$.\n", + "\n", + "**a)**$\\quad$What is the resulting state if we measure the last qubit and obtain $|0\\rangle$?\n", + "\n", + "**b)**$\\quad$What is the resulting state if we measure the last qubit and obtain $|1\\rangle$?" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.8" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/Exercises/Session 5 - Quantum Computing.ipynb b/Exercises/Session 5 - Quantum Computing.ipynb new file mode 100644 index 0000000..ec708cb --- /dev/null +++ b/Exercises/Session 5 - Quantum Computing.ipynb @@ -0,0 +1,328 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "e4f44eb7", + "metadata": {}, + "source": [ + "

TMQS Workshop 2024 @ Zuse Institute Berlin

\n", + "

Summer School on Tensor Methods for Quantum Simulation

\n", + "

June 3 - 5, 2024

\n", + "$\\def\\tcoreleft{\\textcolor{MidnightBlue}{\\Huge⦸}}$\n", + "$\\def\\tcorecenter{\\textcolor{RedOrange}{\\Huge⦿}}$\n", + "$\\def\\tcoreright{\\textcolor{MidnightBlue}{\\Huge\\oslash}}$\n", + "
\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "
$\\tcoreleft$$-$$\\tcoreleft$$-$$\\tcoreleft$$-$$\\cdots$$-$$\\tcorecenter$$-$$\\cdots$$-$$\\tcoreright$$-$$\\tcoreright$
$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$
\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "6f6cc702", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "74ce99d5-268a-4ad8-980b-de19d60b6be6", + "metadata": {}, + "source": [ + "## **Session 5 - Quantum Computing**" + ] + }, + { + "cell_type": "markdown", + "id": "e8441c54-ffd4-45b4-ab04-ff8cf85e6474", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "352b5d1e-a082-49c0-a29a-3c73d819a5a5", + "metadata": {}, + "source": [ + "## Exercise 5.1\n", + "\n", + "The qubit polar form is given by $|\\psi\\rangle = \\cos{\\frac{\\theta}{2}} |0\\rangle + e^{i \\varphi} \\sin{\\frac{\\theta}{2}} |1\\rangle$. \n", + "\n", + "The first two quantum logic gates, we consider in this exercise are the Hadamard gate $H$ and the phase shift gate $T$:\n", + "\n", + "$\\hspace{0.5cm}$$\\displaystyle H = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1\\end{bmatrix} \\qquad $ and $ \\qquad \\displaystyle T = \\begin{bmatrix}1 & 0 \\\\ 0 & e^{i \\frac{\\pi}{4}}\\end{bmatrix}$\n", + "\n", + "When applied to a basis state $|0\\rangle$ or $|1\\rangle$, the Hadamard gate transforms the state as follows:\n", + "\n", + "$\\hspace{0.5cm}$$\\displaystyle H | 0 \\rangle = \\frac{1}{\\sqrt{2}} \\left( |0\\rangle + |1\\rangle\\right) \\qquad$ and $\\qquad \\displaystyle H | 1 \\rangle = \\frac{1}{\\sqrt{2}} \\left( |0\\rangle - |1\\rangle\\right)$\n", + "\n", + "The $T$ gate applies a phase factor to the state $|1\\rangle$, while leaving the state $|0\\rangle$ unchanged:\n", + "\n", + "$\\hspace{0.5cm}$$\\displaystyle T | 0 \\rangle = | 0 \\rangle \\qquad$ and $\\qquad \\displaystyle T | 1 \\rangle = e^{i \\frac{\\pi}{4}} | 1 \\rangle$\n", + "\n", + "**a)**$\\quad$ Import all necessary packages:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "4f9f119a-5785-4aaa-a65b-bed55c8656d0", + "metadata": {}, + "outputs": [], + "source": [ + "from qiskit.visualization import *\n", + "from qiskit.quantum_info import Statevector\n", + "from qiskit import QuantumCircuit, transpile\n", + "from qiskit_aer import AerSimulator" + ] + }, + { + "cell_type": "markdown", + "id": "47fc839a", + "metadata": {}, + "source": [ + "$\\hspace{0.8cm}$and create a quantum circuit consisting of one qubit by \n", + "\n", + "> qc = QuantumCircuit(1)\n", + "\n", + "$\\hspace{0.8cm}$Apply a Hadamard gate and a T gate:\n", + "\n", + "> qc.h(0), qc.t(0)\n", + "\n", + "$\\hspace{0.8cm}$Draw the quantum circuit using \n", + "\n", + "> qc.draw() \n", + "\n", + "$\\hspace{0.8cm}$and plot the Bloch vector of the quantum state at each stage using\n", + "\n", + "> plot_bloch_multivector(Statevector(qc))\n", + "\n", + "$\\hspace{0.8cm}$Write down the polar form of the quantum state at each step of the circuit.\n", + "\n", + "**b)**$\\quad$Now, we want to measure our qubit using\n", + "\n", + "> qc.measure_all()\n", + "\n", + "$\\hspace{0.8cm}$To do this, we choose a simulator backend, which is responsible for simulating the execution of the circuit. \n", + "\n", + "$\\hspace{0.8cm}$The simulator plays a crucial role in simulating the behavior of a quantum circuit. \n", + "\n", + "$\\hspace{0.8cm}$It allows you to run quantum circuits on a classical computer and obtain the expected outcomes of the circuit without requiring a physical quantum device. \n", + "\n", + "$\\hspace{0.8cm}$Here, we choose the qasm_simulator, which simulates the execution of the circuit using a classical computer and provides the measurement statistics:\n", + "\n", + "> backend = AerSimulator()\n", + "\n", + "$\\hspace{0.8cm}$We then transpile, i.e., optimize a quantum circuit for a specific backend, and execute the circuit on the simulator by passing the circuit to the run function.\n", + "\n", + "$\\hspace{0.8cm}$We also specify the number of shots to simulate, which determines the number of times the circuit is executed to obtain statistical results.\n", + "\n", + "> qc = transpile(qc, backend)\n", + "> \n", + "> job = backend.run(qc, shots=10000)\n", + "\n", + "$\\hspace{0.8cm}$Finally, we get the results of the simulation using the result method, and extract the measurement statistics using the get_counts method:\n", + "\n", + "> result = job.result()\n", + "\n", + "> counts = result.get_counts(qc)\n", + "\n", + "$\\hspace{0.8cm}$The measurement statistics represent the probabilities of observing each possible measurement outcome.\n", + "\n", + "$\\hspace{0.8cm}$They are obtained by running the circuit multiple times and counting the number of times each outcome is observed.\n", + "\n", + "$\\hspace{0.8cm}$The histogram of the counts data can be plotted by using\n", + "\n", + "> plot_histogram(counts)\n", + "\n", + "$\\hspace{0.8cm}$What happens if we measure the quantum state after the H gate, what if we measure after the T gate. Explain!\n", + "\n", + "**c)**$\\quad$How can we measure the relative phase $\\varphi$?\n", + "\n", + "$\\hspace{0.8cm}$*Hint:* We need just one additional gate.\n", + "\n", + "**d)**$\\quad$*extra task:* Show that any state $|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1 \\rangle$ with $|\\alpha|^2 + |\\beta|^2 = 1$ can be written in the polar form (up to a global phase)." + ] + }, + { + "cell_type": "markdown", + "id": "d980f769-6854-4d71-a713-e219ddea1f65", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "ddf38779", + "metadata": {}, + "source": [ + "### Exercise 5.2\n", + "\n", + "Now, we want to construct some specific states. The quantum logic gates which come now into play are the Pauli-$Z$ gate, the Pauli-$X$ (NOT) gate, and the CNOT gate:\n", + "\n", + "$\\hspace{0.5cm}$$\\displaystyle Z = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1\\end{bmatrix}$, $\\qquad \\displaystyle \\text{NOT} = \\begin{bmatrix} 0& 1\\\\ 1 & 0 \\end{bmatrix}$, $\\qquad \\displaystyle \\text{CNOT} = \\begin{bmatrix}1 & 0 & 0 & 0\\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0& 1\\\\ 0 & 0 & 1 & 0 \\end{bmatrix}$\n", + "\n", + "The $Z$ gate flips the phase of the basis state $|1\\rangle$ while the NOT gate (sometimes called bit-flip) maps \n", + "$|0\\rangle$ to $|1\\rangle$ and $|1\\rangle$ to $|0\\rangle$.\n", + "\n", + "The CNOT (controlled NOT) gate acts on 2 qubits and performs the NOT operation on the second qubit only when the first qubit is $|1\\rangle$.\n", + "\n", + "
\n", + "\n", + "**a)**$\\quad$Construct the Bell state $\\displaystyle |\\Phi^+\\rangle = \\frac{|00\\rangle + |11\\rangle}{\\sqrt{2}} $. To do so, create a quantum circuit with two qubits and and use one Hadamard gate and one CNOT gate:\n", + "\n", + "> qc.cx(0,1)\n", + "\n", + "$\\hspace{0.8cm}$where the first argument is the control qubit. Draw the circuit, measure the final state, and plot the histogram\n", + "\n", + "**b)**$\\quad$How can we construct the other Bell states?\n", + "\n", + "$\\hspace{1cm}$$\\displaystyle|\\Phi^-\\rangle = \\frac{|00\\rangle - |11\\rangle}{\\sqrt{2}} $, $\\qquad \\displaystyle|\\Psi^+\\rangle = \\frac{|01\\rangle + |10\\rangle}{\\sqrt{2}} $, $\\qquad \\displaystyle|\\Psi^-\\rangle = \\frac{|01\\rangle - |10\\rangle}{\\sqrt{2}} $\n", + "\n", + "**c)**$\\quad$In quantum computing, oracles play a crucial role in many quantum algorithms. \n", + "\n", + "$\\hspace{0.8cm}$An oracle is a black box subroutine used to perform a specific computation on a quantum state. \n", + "\n", + "$\\hspace{0.8cm}$Find an oracle $U$, i.e., a unitary operation, acting on three qubits corresponding to the following truth table:\n", + "\n", + "| | Input | Output |\n", + "| -- | ----- | ------ |\n", + "| 0 | 000 | 000 |\n", + "| 1 | 001 | 001 |\n", + "| 2 | 010 | 011 |\n", + "| 3 | 011 | 010 |\n", + "| 4 | 100 | 111 |\n", + "| 5 | 101 | 110 |\n", + "| 6 | 110 | 100 |\n", + "| 7 | 111 | 101 |\n", + "\n", + "\n", + "$\\hspace{0.8cm}$Write down the matrix representation! What does the circuit look like?\n", + "\n", + "**d)**$\\quad$*extra task:* What kind of gate is expressed by the following circuit? Here, 'tdg' denotes the adjoint of the T gate." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "b5856bd8-fced-4218-bdc3-df61b56e5fdf", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
                                                               ┌───┐       \n",
+       "q_0: ───────────────────■─────────────────────■───────────■────┤ T ├────■──\n",
+       "                        │                     │   ┌───┐ ┌─┴─┐ ┌┴───┴┐ ┌─┴─┐\n",
+       "q_1: ───────■───────────┼─────────■───────────┼───┤ T ├─┤ X ├─┤ Tdg ├─┤ X ├\n",
+       "     ┌───┐┌─┴─┐┌─────┐┌─┴─┐┌───┐┌─┴─┐┌─────┐┌─┴─┐ ├───┤ ├───┤ └─────┘ └───┘\n",
+       "q_2: ┤ H ├┤ X ├┤ Tdg ├┤ X ├┤ T ├┤ X ├┤ Tdg ├┤ X ├─┤ T ├─┤ H ├──────────────\n",
+       "     └───┘└───┘└─────┘└───┘└───┘└───┘└─────┘└───┘ └───┘ └───┘
" + ], + "text/plain": [ + " ┌───┐ \n", + "q_0: ───────────────────■─────────────────────■───────────■────┤ T ├────■──\n", + " │ │ ┌───┐ ┌─┴─┐ ┌┴───┴┐ ┌─┴─┐\n", + "q_1: ───────■───────────┼─────────■───────────┼───┤ T ├─┤ X ├─┤ Tdg ├─┤ X ├\n", + " ┌───┐┌─┴─┐┌─────┐┌─┴─┐┌───┐┌─┴─┐┌─────┐┌─┴─┐ ├───┤ ├───┤ └─────┘ └───┘\n", + "q_2: ┤ H ├┤ X ├┤ Tdg ├┤ X ├┤ T ├┤ X ├┤ Tdg ├┤ X ├─┤ T ├─┤ H ├──────────────\n", + " └───┘└───┘└─────┘└───┘└───┘└───┘└─────┘└───┘ └───┘ └───┘ " + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "circuit = QuantumCircuit(3)\n", + "circuit.h(2)\n", + "circuit.cx(1,2)\n", + "circuit.tdg(2)\n", + "circuit.cx(0,2)\n", + "circuit.t(2)\n", + "circuit.cx(1,2)\n", + "circuit.tdg(2)\n", + "circuit.cx(0,2)\n", + "circuit.barrier()\n", + "circuit.t(1)\n", + "circuit.t(2)\n", + "circuit.barrier()\n", + "circuit.h(2)\n", + "circuit.cx(0,1)\n", + "circuit.barrier()\n", + "circuit.t(0)\n", + "circuit.tdg(1)\n", + "circuit.barrier()\n", + "circuit.cx(0,1)\n", + "circuit.draw(plot_barriers = False) " + ] + }, + { + "cell_type": "markdown", + "id": "c072e544-570f-4949-bd2b-d9c13df9bb23", + "metadata": {}, + "source": [ + "$\\hspace{0.8cm}$*Hint:* For states of the form $|11x\\rangle$, we have $|110\\rangle \\mapsto |111\\rangle$ and $|111\\rangle \\mapsto |110\\rangle$. What happens with quantum states of different form?" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.8" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/Exercises/Session 6 - Quantum Simulation.ipynb b/Exercises/Session 6 - Quantum Simulation.ipynb new file mode 100644 index 0000000..952c935 --- /dev/null +++ b/Exercises/Session 6 - Quantum Simulation.ipynb @@ -0,0 +1,314 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "e4f44eb7", + "metadata": {}, + "source": [ + "

TMQS Workshop 2024 @ Zuse Institute Berlin

\n", + "

Summer School on Tensor Methods for Quantum Simulation

\n", + "

June 3 - 5, 2024

\n", + "$\\def\\tcoreleft{\\textcolor{MidnightBlue}{\\Huge⦸}}$\n", + "$\\def\\tcorecenter{\\textcolor{RedOrange}{\\Huge⦿}}$\n", + "$\\def\\tcoreright{\\textcolor{MidnightBlue}{\\Huge\\oslash}}$\n", + "
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$\\tcoreleft$$-$$\\tcoreleft$$-$$\\tcoreleft$$-$$\\cdots$$-$$\\tcorecenter$$-$$\\cdots$$-$$\\tcoreright$$-$$\\tcoreright$
$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$$\\tiny\\mid$
\n", + "
" + ] + }, + { + "cell_type": "markdown", + "id": "6f6cc702", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "74ce99d5-268a-4ad8-980b-de19d60b6be6", + "metadata": {}, + "source": [ + "## **Session 6 - Quantum Simulation**" + ] + }, + { + "cell_type": "markdown", + "id": "e8441c54-ffd4-45b4-ab04-ff8cf85e6474", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "id": "352b5d1e-a082-49c0-a29a-3c73d819a5a5", + "metadata": {}, + "source": [ + "## Exercise 6.1\n", + "\n", + "Let us examine the quantum counterpart of a standard classical circuit. The *quantum full adder* (QFA) is the quantum analogue of a full-adder circuit used in classical computers to add up to three bits. Due to reversibility requirements, the QFA acts on four qubits: The input qubits are $\\ket{C_\\mathrm{in}}$, $\\ket{A}$, $\\ket{B}$, and $\\ket{0}$ and the output qubits are $\\ket{S}$, $\\ket{A}$, $\\ket{B}$, and $\\ket{C_\\mathrm{out}}$. The qubit $\\ket{C_\\mathrm{in}}$ is carried in from the previous (less-significant) stage of a multi-digit addition. The circuit produces the sum of the input qubits including a carry-out signal for the overflow into the next digit. \n", + "\n", + "The QFA algorithm is given by the following circuit:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "eb600b78-178c-4ab2-b25b-17a629e98a44", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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+       "   A: ──■────■────┼────┼────■──\n",
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+       "   0: ┤ X ├─────┤ X ├──────────\n",
+       "      └───┘     └───┘
" + ], + "text/plain": [ + " ┌───┐ \n", + "C_in: ────────────■──┤ X ├─────\n", + " │ └─┬─┘ \n", + " A: ──■────■────┼────┼────■──\n", + " │ ┌─┴─┐ │ │ ┌─┴─┐\n", + " B: ──■──┤ X ├──■────■──┤ X ├\n", + " ┌─┴─┐└───┘┌─┴─┐ └───┘\n", + " 0: ┤ X ├─────┤ X ├──────────\n", + " └───┘ └───┘ " + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from qiskit import QuantumRegister, QuantumCircuit\n", + "\n", + "C_in = QuantumRegister(1, 'C_in')\n", + "A = QuantumRegister(1, 'A')\n", + "B = QuantumRegister(1, 'B')\n", + "Zero = QuantumRegister(1, '0')\n", + "\n", + "qc = QuantumCircuit(C_in, A, B, Zero)\n", + "\n", + "qc.ccx(1,2,3)\n", + "qc.cx(1,2)\n", + "qc.ccx(0,2,3)\n", + "qc.cx(2,0)\n", + "qc.cx(1,2)\n", + "qc.draw() " + ] + }, + { + "cell_type": "markdown", + "id": "9bfa9c5f-b45b-4c3d-9d8c-f07d41ce2a35", + "metadata": {}, + "source": [ + "**a)**$\\quad$Write down the corresponding truth table!\n", + "\n", + "**b**)$\\quad$The unitary operator of the QFA can be written as an MPO of the following form:\n", + "\n", + "$\\hspace{1cm}$$\\mathbf{G} = \\begin{bmatrix} \\sigma_x C_0 & I & \\sigma_x C_1\\end{bmatrix} \\otimes \\begin{bmatrix} C_0 & C_1 & 0 & 0 \\\\ 0 & C_0 & C_1 & 0 \\\\ 0 & 0 & C_0 & C_1 \\end{bmatrix} \\otimes \\begin{bmatrix} C_1 & 0 \\\\ C_0 & 0 \\\\ 0 & C_1 \\\\ 0 & C_0 \\end{bmatrix} \\otimes \\begin{bmatrix} I \\\\ \\sigma_x \\end{bmatrix}$,\n", + "\n", + "$\\hspace{0.8cm}$where the matrices in the core elements are given by\n", + "\n", + "$\\hspace{1cm}$$I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}, \\quad \\sigma_x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\quad C_0 = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix}, \\quad C_1 = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix}$.\n", + "\n", + "$\\hspace{0.8cm}$Derive a canonical representation of the operator by contracting theMPO cores!\n", + "\n", + "**c**)$\\quad$*extra task:* Show that this representation is unitary. Show that $\\mathbf{G}$ actually represents the QFA!\n", + "\n", + "**d**)$\\quad$Implement a *quantum full ader network* (QFAN), i.e., a quantum circuit to add $n$-bit numbers, in Qiskit by coupling several QFAs:" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "6b986e09-4215-440f-a577-8b84592a1b86", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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+       "q_0: ────────────■──┤ X ├──────░───────────────────────────░───────────────────────────░─\n",
+       "                 │  └─┬─┘      ░                           ░                           ░ \n",
+       "q_1: ──■────■────┼────┼────■───░───────────────────────────░───────────────────────────░─\n",
+       "       │  ┌─┴─┐  │    │  ┌─┴─┐ ░                           ░                           ░ \n",
+       "q_2: ──■──┤ X ├──■────■──┤ X ├─░───────────────────────────░───────────────────────────░─\n",
+       "     ┌─┴─┐└───┘┌─┴─┐     └───┘ ░                ┌───┐      ░                           ░ \n",
+       "q_3: ┤ X ├─────┤ X ├───────────░─────────────■──┤ X ├──────░───────────────────────────░─\n",
+       "     └───┘     └───┘           ░             │  └─┬─┘      ░                           ░ \n",
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+       "                               ░   │  ┌─┴─┐  │    │  ┌─┴─┐ ░                           ░ \n",
+       "q_5: ──────────────────────────░───■──┤ X ├──■────■──┤ X ├─░───────────────────────────░─\n",
+       "                               ░ ┌─┴─┐└───┘┌─┴─┐     └───┘ ░                ┌───┐      ░ \n",
+       "q_6: ──────────────────────────░─┤ X ├─────┤ X ├───────────░─────────────■──┤ X ├──────░─\n",
+       "                               ░ └───┘     └───┘           ░             │  └─┬─┘      ░ \n",
+       "q_7: ──────────────────────────░───────────────────────────░───■────■────┼────┼────■───░─\n",
+       "                               ░                           ░   │  ┌─┴─┐  │    │  ┌─┴─┐ ░ \n",
+       "q_8: ──────────────────────────░───────────────────────────░───■──┤ X ├──■────■──┤ X ├─░─\n",
+       "                               ░                           ░ ┌─┴─┐└───┘┌─┴─┐     └───┘ ░ \n",
+       "q_9: ──────────────────────────░───────────────────────────░─┤ X ├─────┤ X ├───────────░─\n",
+       "                               ░                           ░ └───┘     └───┘           ░
" + ], + "text/plain": [ + " ┌───┐ ░ ░ ░ \n", + "q_0: ────────────■──┤ X ├──────░───────────────────────────░───────────────────────────░─\n", + " │ └─┬─┘ ░ ░ ░ \n", + "q_1: ──■────■────┼────┼────■───░───────────────────────────░───────────────────────────░─\n", + " │ ┌─┴─┐ │ │ ┌─┴─┐ ░ ░ ░ \n", + "q_2: ──■──┤ X ├──■────■──┤ X ├─░───────────────────────────░───────────────────────────░─\n", + " ┌─┴─┐└───┘┌─┴─┐ └───┘ ░ ┌───┐ ░ ░ \n", + "q_3: ┤ X ├─────┤ X ├───────────░─────────────■──┤ X ├──────░───────────────────────────░─\n", + " └───┘ └───┘ ░ │ └─┬─┘ ░ ░ \n", + "q_4: ──────────────────────────░───■────■────┼────┼────■───░───────────────────────────░─\n", + " ░ │ ┌─┴─┐ │ │ ┌─┴─┐ ░ ░ \n", + "q_5: ──────────────────────────░───■──┤ X ├──■────■──┤ X ├─░───────────────────────────░─\n", + " ░ ┌─┴─┐└───┘┌─┴─┐ └───┘ ░ ┌───┐ ░ \n", + "q_6: ──────────────────────────░─┤ X ├─────┤ X ├───────────░─────────────■──┤ X ├──────░─\n", + " ░ └───┘ └───┘ ░ │ └─┬─┘ ░ \n", + "q_7: ──────────────────────────░───────────────────────────░───■────■────┼────┼────■───░─\n", + " ░ ░ │ ┌─┴─┐ │ │ ┌─┴─┐ ░ \n", + "q_8: ──────────────────────────░───────────────────────────░───■──┤ X ├──■────■──┤ X ├─░─\n", + " ░ ░ ┌─┴─┐└───┘┌─┴─┐ └───┘ ░ \n", + "q_9: ──────────────────────────░───────────────────────────░─┤ X ├─────┤ X ├───────────░─\n", + " ░ ░ └───┘ └───┘ ░ " + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "n = 3\n", + "qc = QuantumCircuit(3*n+1)\n", + "for i in range(n):\n", + " qc.ccx(3*i+1,3*i+2,3*i+3)\n", + " qc.cx(3*i+1,3*i+2)\n", + " qc.ccx(3*i+0,3*i+2,3*i+3)\n", + " qc.cx(3*i+2,3*i+0)\n", + " qc.cx(3*i+1,3*i+2)\n", + " qc.barrier()\n", + "qc.draw(fold=-1) " + ] + }, + { + "cell_type": "markdown", + "id": "50a02475-3438-436c-b904-62da8707cc51", + "metadata": {}, + "source": [ + "$\\hspace{0.8cm}$Simulate the circuit for $n = 1, \\dots, 8$ with 1.000.000 shots after applying Hadamard gates to the qubits $|A_i\\rangle $ and $| B_i \\rangle$ of each QFA. Plot the relative computation time!\n", + "\n", + "**e**)$\\quad$Load the corresponding MPO representation ```scikit_tt.models``` by\n", + "\n", + "> import scikit_tt.models as mdl\n", + ">\n", + "> G = mdl.qfan(n)\n", + "\n", + "$\\hspace{0.8cm}$where ```n``` is the number of QFAs.\n", + "\n", + "$\\hspace{0.8cm}$The initial quantum state can be expressed as a rank-one tensor. Can you explain why? \n", + "\n", + "$\\hspace{0.8cm}$Use the following routine for construction." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "3b5ab505-6a59-4059-9d28-3c4b71247710", + "metadata": {}, + "outputs": [], + "source": [ + "from scikit_tt.tensor_train import TT\n", + "\n", + "def initial_state(n):\n", + " cores = [np.array([1,0]).reshape([1,2,1,1]) for _ in range(3*n+1)]\n", + " for i in range(n):\n", + " cores[3*i+1] = np.sqrt(0.5)*np.array([1,1]).reshape([1,2,1,1])\n", + " init = TT(cores)\n", + " return init" + ] + }, + { + "cell_type": "markdown", + "id": "04dfd8d2-d44c-4db1-9791-04de8fec3b39", + "metadata": {}, + "source": [ + "$\\hspace{0.8cm}$Repeat the above experiment on a tensor-based level by computing the state tensor $\\mathbf{G} \\mathbf{\\Psi}$, where $\\mathbf{\\Psi}$ is the initial state.\n", + "\n", + "$\\hspace{0.8cm}$Use the sampling routine from ```scikit_tt.quantum_computation```:\n", + "\n", + "> from scikit_tt.quantum_computation import sampling\n", + ">\n", + "> samples, probabilities = sampling(G@init, list(np.arange(3*i+1)), 1000000)\n", + "\n", + "$\\hspace{0.8cm}$What do you observe?" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.8" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +}