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[RippleCaryAdder] Add workbook for tasks 1.1-1.3 (microsoft#794)
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Ripple Carry Adder Workbok\n", | ||
| "**What is this workbook?** \n", | ||
| "A workbook is a collection of problems, accompanied by solutions to them. The explanations focus on the logical steps required to solve a problem; they illustrate the concepts that need to be applied to come up with a solution to the problem, explaining the mathematical steps required.\n", | ||
| "\n", | ||
| "Note that a workbook should not be the primary source of knowledge on the subject matter; it assumes that you've already read a tutorial or a textbook and that you are now seeking to improve your problem-solving skills. You should attempt solving the tasks of the respective kata first, and turn to the workbook only if stuck or for reinforcement. While a textbook emphasizes knowledge acquisition, a workbook emphasizes skill acquisition.\n", | ||
| "\n", | ||
| "This workbook describes the solutions to the problems offered in the [Ripple Carry Adder Kata](./RippleCarryAdder.ipynb). \n", | ||
| "Since the tasks are offered as programming problems, the explanations also cover some elements of Q# that might be non-obvious for a novice." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "**What you should know for this workbook**\n", | ||
| "\n", | ||
| "You should be familiar with the following concepts and associated techniques **prior to** beginning work on the Basic Gates Quantum Kata.\n", | ||
| "\n", | ||
| "1. [Basic Gates Kata](../BasicGates/BasicGates.ipynb). \n", | ||
| "2. [Single-qubit gates](../tutorials/SingleQubitGates/SingleQubitGates.ipynb).\n", | ||
| "3. List of basic gates available in Q# can be found at [Microsoft.Quantum.Intrinsic](https://docs.microsoft.com/qsharp/api/qsharp/microsoft.quantum.intrinsic).\n", | ||
| "4. [Syntax of flow control statements in Q#](https://docs.microsoft.com/azure/quantum/user-guide/language/statements/iterations) \n", | ||
| "5. [Q# conditional branching](https://docs.microsoft.com/azure/quantum/user-guide/language/statements/conditionalbranching) documentation.\n", | ||
| "\n", | ||
| "You can also consult the [complete Quantum Katas learning path](https://github.com/microsoft/QuantumKatas#learning-path)." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Part I. Simple Adder Outputting to Empty Qubits\n", | ||
| "\n", | ||
| "\n", | ||
| "### Theory\n", | ||
| "\n", | ||
| "* [Classical binary adder on Wikipedia](https://en.wikipedia.org/wiki/Adder_(electronics)).\n", | ||
| "* Part 2 of the [paper on quantum binary addition](https://arxiv.org/pdf/quant-ph/0008033.pdf) by Thomas G. Draper explains how to adapt the classical adder to a quantum environment." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Task 1.1. Summation of two bits\n", | ||
| "\n", | ||
| "**Inputs:**\n", | ||
| "\n", | ||
| " 1. qubit `a` in an arbitrary state $|\\phi\\rangle$, \n", | ||
| " 2. qubit `b` in an arbitrary state $|\\psi\\rangle$, \n", | ||
| " 3. qubit `sum` in state $|0\\rangle$.\n", | ||
| "\n", | ||
| "**Goal:** Transform the `sum` qubit into the lowest bit of the binary sum of $\\phi$ and $\\psi$.\n", | ||
| "\n", | ||
| "* $|0\\rangle + |0\\rangle \\to |0\\rangle$\n", | ||
| "* $|0\\rangle + |1\\rangle \\to |1\\rangle$\n", | ||
| "* $|1\\rangle + |0\\rangle \\to |1\\rangle$\n", | ||
| "* $|1\\rangle + |1\\rangle \\to |0\\rangle$\n", | ||
| "\n", | ||
| "Any superposition should map appropriately. \n", | ||
| "\n", | ||
| "**Example:** (Recall that $|+\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle)$, $|-\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle - |1\\rangle)$)\n", | ||
| "\n", | ||
| "$|+\\rangle \\otimes |-\\rangle \\otimes |0\\rangle \\to \\frac{1}{2}(|000\\rangle + |101\\rangle - |011\\rangle - |110\\rangle)$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Solution\n", | ||
| "\n", | ||
| "> In this workbook we will reason about the computations acting on basis states, as opposed to arbitrary superposition states. \n", | ||
| "> (Since all computations will be implemented using quantum gates, they'll be linear, and behave correctly on superpositions if they are correct on all basis states.) \n", | ||
| "> This will allow us to use qubit states and the classical values stored in them interchangeably.\n", | ||
| "\n", | ||
| "Clearly, the expression we're looking for is $a \\oplus b$ - sum of bits $a$ and $b$ modulo 2.\n", | ||
| "\n", | ||
| "If we apply $CNOT(a,sum)$, then $a=a$, $b=b$, and $sum$ becomes $0 \\oplus a = a$.\n", | ||
| "\n", | ||
| "Again applying $CNOT(b,sum)$, we get $a=a$, $b=b$, and $sum$ becomes $sum \\oplus b = a \\oplus b$. That's exactly what we're looking for!" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "%kata T11_LowestBitSum\n", | ||
| "\n", | ||
| "operation LowestBitSum (a : Qubit, b : Qubit, sum : Qubit) : Unit is Adj {\n", | ||
| " CNOT(a, sum);\n", | ||
| " CNOT(b, sum);\n", | ||
| "}" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Alternate Solution\n", | ||
| "\n", | ||
| "If we apply $CNOT(a,b)$ , then $a = a$, $b = a \\oplus b$, and $sum = 0$.\n", | ||
| "\n", | ||
| "Again applying $CNOT(b,sum)$, we get $a = a$, $b = a \\oplus b$, and $sum = sum \\oplus b = a \\oplus b$.\n", | ||
| "\n", | ||
| "Now by applying $CNOT(a,b)$, we get $a = a$, $b = a \\oplus (a \\oplus b) = b$, $sum = a \\oplus b$, and thus we will restore the previous value of $b$." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "%kata T11_LowestBitSum\n", | ||
| "\n", | ||
| "operation LowestBitSum (a : Qubit, b : Qubit, sum : Qubit) : Unit is Adj {\n", | ||
| " CNOT(a, b);\n", | ||
| " CNOT(b, sum);\n", | ||
| " CNOT(a, b);\n", | ||
| "}" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "[Return to Task 1.1 of the Ripple Carry Adder kata.](./RippleCarryAdder.ipynb#Task-1.1.-Summation-of-two-bits)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Task 1.2. Carry of two bits\n", | ||
| "\n", | ||
| "**Inputs:**\n", | ||
| "\n", | ||
| " 1. qubit `a` in an arbitrary state $|\\phi\\rangle$,\n", | ||
| " 2. qubit `b` in an arbitrary state $|\\psi\\rangle$,\n", | ||
| " 3. qubit `carry` in state $|0\\rangle$.\n", | ||
| "\n", | ||
| "**Goal:** Set the `carry` qubit to $|1\\rangle$ if the binary sum of $\\phi$ and $\\psi$ produces a carry.\n", | ||
| "\n", | ||
| "* $|0\\rangle$ and $|0\\rangle \\to |0\\rangle$\n", | ||
| "* $|0\\rangle$ and $|1\\rangle \\to |0\\rangle$\n", | ||
| "* $|1\\rangle$ and $|0\\rangle \\to |0\\rangle$\n", | ||
| "* $|1\\rangle$ and $|1\\rangle \\to |1\\rangle$\n", | ||
| "\n", | ||
| "Any superposition should map appropriately. \n", | ||
| "\n", | ||
| "**Example:**\n", | ||
| "\n", | ||
| "$|+\\rangle \\otimes |-\\rangle \\otimes |0\\rangle \\to \\frac{1}{2}(|000\\rangle + |100\\rangle - |010\\rangle - |111\\rangle)$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Solution\n", | ||
| "Clearly, $carry = a \\text{ AND } b$.\n", | ||
| "\n", | ||
| "If we apply $CCNOT(a, b, carry)$, then $a=a$, $b=b$, and $carry$ becomes $(a \\text{ AND } b) \\oplus carry = (a \\text{ AND } b) \\oplus 0 = (a \\text{ AND } b)$." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "%kata T12_LowestBitCarry\n", | ||
| "\n", | ||
| "operation LowestBitCarry (a : Qubit, b : Qubit, carry : Qubit) : Unit is Adj {\n", | ||
| " CCNOT(a, b, carry);\n", | ||
| "}" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "[Return to Task 1.2 of the Ripple Carry Adder kata.](./RippleCarryAdder.ipynb#Task-1.2.-Carry-of-two-bits)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Task 1.3. One-bit adder\n", | ||
| "\n", | ||
| "**Inputs:**\n", | ||
| "\n", | ||
| " 1. qubit `a` in an arbitrary state $|\\phi\\rangle$,\n", | ||
| " 2. qubit `b` in an arbitrary state $|\\psi\\rangle$,\n", | ||
| " 3. two qubits `sum` and `carry` in state $|0\\rangle$.\n", | ||
| "\n", | ||
| "**Goals:**\n", | ||
| "\n", | ||
| "* Transform the `sum` qubit into the lowest bit of the binary sum of $\\phi$ and $\\psi$.\n", | ||
| "* Transform the `carry` qubit into the carry bit produced by that sum." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "### Solution\n", | ||
| "This task is just a combination of the two previous ones. Using [task 1.1](./Workbook_RippleCarryAdder.ipynb#Task-1.1.-Summation-of-two-bits) and [task 1.2](./Workbook_RippleCarryAdder.ipynb#Task-1.2.-Carry-of-two-bits), \n", | ||
| "we can get the solution below." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "%kata T13_OneBitAdder\n", | ||
| "\n", | ||
| "operation OneBitAdder (a : Qubit, b : Qubit, sum : Qubit, carry : Qubit) : Unit is Adj {\n", | ||
| " LowestBitSum(a, b, sum);\n", | ||
| " LowestBitCarry(a, b, carry);\n", | ||
| "}" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "[Return to Task 1.3 of the Ripple Carry Adder kata.](./RippleCarryAdder.ipynb#Task-1.3.-One-bit-adder)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "More solutions comming soon.." | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Q#", | ||
| "language": "qsharp", | ||
| "name": "iqsharp" | ||
| }, | ||
| "language_info": { | ||
| "file_extension": ".qs", | ||
| "mimetype": "text/x-qsharp", | ||
| "name": "qsharp", | ||
| "version": "0.14" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 2 | ||
| } |