[Measurements] Create workbook for part II tasks (#305)
Co-authored-by: Martin Roetteler <martinro@microsoft.com>
This commit is contained in:
Markus Buchberger
GitHub
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3e6633f967
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@ -561,7 +561,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Task 2.1*. $|0\\rangle$ or $|+\\rangle$?\n",
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"### <a name=\"task-21\"></a>Task 2.1*. $|0\\rangle$ or $|+\\rangle$?\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in either the $|0\\rangle$ or the $|+\\rangle$ state.\n",
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"\n",
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@ -569,7 +569,7 @@
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"\n",
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"In this task your solution will be called multiple times, with one of the states picked with equal probability every time. You have to get overall accuracy of at least 80%.\n",
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"\n",
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"This task is an example of quantum hypothesis testing, or state discrimination with minimum error."
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"> This task is an example of quantum hypothesis testing, or state discrimination with minimum error."
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]
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},
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{
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@ -589,7 +589,14 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Task 2.2**. $|0\\rangle$, $|+\\rangle$ or inconclusive?\n",
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"*Can't come up with a solution? See the explained solution in the [Measurements Workbook](./Workbook_Measurements_Part2.ipynb#task-21).*"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### <a name=\"task-22\"></a>Task 2.2**. $|0\\rangle$, $|+\\rangle$ or inconclusive?\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in either the $|0\\rangle$ or the $|+\\rangle$ state.\n",
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"\n",
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@ -609,11 +616,11 @@
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"\n",
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"The state of the qubit at the end of the operation does not matter.\n",
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"\n",
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"This task is an example of unambiguous state discrimination.\n",
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"> This task is an example of unambiguous state discrimination.\n",
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"\n",
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"<br/>\n",
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"<details>\n",
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" <summary>Need a hint? Click here</summary>\n",
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" <summary><b>Need a hint? Click here</b></summary>\n",
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" You can use extra qubit(s) is your solution.\n",
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"</details>"
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]
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@ -635,7 +642,14 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Task 2.3**. Peres/Wooters game\n",
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"*Can't come up with a solution? See the explained solution in the [Measurements Workbook](./Workbook_Measurements_Part2.ipynb#task-22).*"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### <a name=\"peres-wooters-game\"></a>Task 2.3**. Peres/Wooters game\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in one of the three states:\n",
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"\n",
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@ -668,6 +682,13 @@
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" // ...\n",
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"}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"*Can't come up with a solution? See the explained solution in the [Measurements Workbook](./Workbook_Measurements_Part2.ipynb#peres-wooters-game).*"
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]
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}
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],
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"metadata": {
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@ -680,7 +701,7 @@
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"file_extension": ".qs",
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"mimetype": "text/x-qsharp",
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"name": "qsharp",
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"version": "0.4"
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"version": "0.10"
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}
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},
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"nbformat": 4,
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@ -0,0 +1,416 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Measurements Kata Workbook, Part 2\n",
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"\n",
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"The [Measurements Kata Workbook, Part 1]() is still work in progress and will include the solutions of the tasks from Part I. This workbook continues the explanations for the tasks in Part II."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"To begin, first prepare this notebook for execution (if you skip the first step, you'll get \"Syntax does not match any known patterns\" error when you try to execute Q# code in the next cells; if you skip the second step, you'll get \"Invalid test name\" error):"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": false
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},
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"outputs": [],
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"source": [
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"%package Microsoft.Quantum.Katas::0.11.2003.3107"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"> The package versions in the output of the cell above should always match. If you are running the Notebooks locally and the versions do not match, please install the IQ# version that matches the version of the `Microsoft.Quantum.Katas` package.\n",
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"> <details>\n",
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"> <summary><u>How to install the right IQ# version</u></summary>\n",
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"> For example, if the version of `Microsoft.Quantum.Katas` package above is 0.1.2.3, the installation steps are as follows:\n",
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">\n",
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"> 1. Stop the kernel.\n",
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"> 2. Uninstall the existing version of IQ#:\n",
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"> dotnet tool uninstall microsoft.quantum.iqsharp -g\n",
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"> 3. Install the matching version:\n",
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"> dotnet tool install microsoft.quantum.iqsharp -g --version 0.1.2.3\n",
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"> 4. Reinstall the kernel:\n",
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"> dotnet iqsharp install\n",
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"> 5. Restart the Notebook.\n",
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"> </details>\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": false
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},
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"outputs": [],
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"source": [
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"%workspace reload"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Part II*. Discriminating Nonorthogonal States"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### <a name=\"task-21\"></a>Task 2.1*. $|0\\rangle$ or $|+\\rangle$?\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in either the $|0\\rangle$ or the $|+\\rangle$ state.\n",
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"\n",
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"**Output:** `true` if the qubit was in the $|0\\rangle$ state, or `false` if it was in the $|+\\rangle$ state. The state of the qubit at the end of the operation does not matter.\n",
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"\n",
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"In this task your solution will be called multiple times, with one of the states picked with equal probability every time. You have to get overall accuracy of at least 80%.\n",
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"\n",
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"> This task is an example of quantum hypothesis testing, or state discrimination with minimum error."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Solution\n",
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"\n",
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"Let ${|E_a\\rangle, |E_b\\rangle}$ be a measurement with two outcomes $a$ and $b$, which we identify with the answers, i.e., outcome \"a\" means we answer \"state was |0\\rangle\" and outcome \"b\" means we answer \"state was |+\\rangle\". Then we define\n",
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"\n",
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"* $P(a|0)$ = probability to observe first outcome given that the state was $|0\\rangle$\n",
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"* $P(b|0)$ = probability to observe second outcome given that the state was $|0\\rangle$\n",
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"* $P(a|+)$ = probability to observe first outcome given that the state was $|+\\rangle$\n",
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"* $P(b|+)$ = probability to observe second outcome given that the state was $|+\\rangle$\n",
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"\n",
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"The task is to maximize the probability to be correct on a single shot experiment, which is the same as to minimize the probability to be wrong on a single shot. \n",
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"\n",
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"Since the task promises uniform prior distribution of the inputs $|0\\rangle$ and $|+\\rangle$, i.e., $P(+) = P(0) = \\frac{1}{2}$, we get the following expression for the probability of giving a correct answer:\n",
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"\n",
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"$$P_{correct} = P(0) P(a|0) + P(+) P(b|+) = \\frac{1}{2} (P(a|0) + P(b|+))$$\n",
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"\n",
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"We can represent our measurement as a von Neumann measurement of the following form:\n",
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"\n",
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"$$|E_a\\rangle = R_y(2\\alpha) \\begin{bmatrix} 1\\\\0\\end{bmatrix} = \\begin{bmatrix}\\cos(\\alpha) \\\\ \\sin(\\alpha)\\end{bmatrix}, |E_b\\rangle = R_y(2\\alpha) \\begin{bmatrix} 0\\\\1\\end{bmatrix} = \\begin{bmatrix}-\\sin(\\alpha) \\\\ \\cos(\\alpha)\\end{bmatrix}$$\n",
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"\n",
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"<img src=\"./img/Task201_Rotation.png\" width=\"70%\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Using this representation, we can express our probabilities as follows:\n",
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"\n",
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"$$P(a|0) = |\\langle E_a|0\\rangle|^2 = \\cos^2(\\alpha)$$\n",
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"\n",
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"$$P(b|+) = |\\langle E_b|+\\rangle|^2 = \\frac{1}{2} - \\cos(\\alpha) \\sin(\\alpha)$$\n",
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"\n",
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"$$P_{correct} = \\frac{1}{2} (\\frac{1}{2} + \\cos^2(\\alpha) - \\cos(\\alpha) \\sin(\\alpha))$$\n",
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"\n",
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"Maximizing this for $\\alpha$, we get max $P_{success} = \\frac{1}{2} (1 + \\frac{1}{\\sqrt{2}}) = 0.8535...$, which is attained for $\\alpha = -\\pi/8$.\n",
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"\n",
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"This means that $|E_a\\rangle$ and $|E_b\\rangle$ are the result of rotating $|0\\rangle$ and $|1\\rangle$, respectively, by $-\\pi/8$. If we rotate the whole system by $-\\alpha = \\pi/8$, we will get $|E_a\\rangle=|0\\rangle$ and $|E_b\\rangle=|1\\rangle$, and a measurement in the computational basis will give us the correct result with a probability of 85%.\n",
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"\n",
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"> In Q#, rotating the input state by some angle $\\theta$ can be done by applying $Ry$ gate with angle parameter $2\\theta$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"%kata T201_IsQubitZeroOrPlus_Test\n",
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"\n",
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"open Microsoft.Quantum.Math;\n",
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"\n",
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"operation IsQubitPlusOrZero (q : Qubit) : Bool {\n",
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" Ry(0.25 * PI(), q);\n",
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" return M(q) == Zero;\n",
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"}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"[Return to task 2.1 of the Measurements kata.](./Measurements.ipynb#task-21)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### <a name=\"task-22\"></a>Task 2.2**. $|0\\rangle$, $|+\\rangle$ or inconclusive?\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in either the $|0\\rangle$ or the $|+\\rangle$ state.\n",
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"\n",
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"**Output:** \n",
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"\n",
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"* 0 if the qubit was in the $|0\\rangle$ state, \n",
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"* 1 if it was in the $|+\\rangle$ state,\n",
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"* -1 if you can't decide, i.e., an \"inconclusive\" result. \n",
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"\n",
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"Your solution:\n",
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"\n",
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"* should never give 0 or 1 answer incorrectly (i.e., identify $|0\\rangle$ as 1 or $|+\\rangle$ as 0),\n",
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"* will be called multiple times, with one of the states picked with equal probability every time,\n",
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"* may give an inconclusive (-1) answer in at most 80% of all the cases,\n",
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"* must correctly identify the $|0\\rangle$ state as 0 in at least 10% of all the cases,\n",
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"* must correctly identify the $|1\\rangle$ state as 1 in at least 10% of all the cases.\n",
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"\n",
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"The state of the qubit at the end of the operation does not matter.\n",
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"\n",
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"> This task is an example of unambiguous state discrimination.\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Solution\n",
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"\n",
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"A simple strategy that gives an inconclusive result with probability 0.75 and never errs in case it yields a conclusive result can be obtained from randomizing the choice of measurement basis between the computational basis (std) and the Hadamard basis (had).\n",
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"\n",
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"Notice that when measured in the standard basis, the state $|0\\rangle$ will always lead to the outcome \"0\", and the state $|+\\rangle$ will lead to outcomes \"0\" and \"1\" with probability $\\frac12$ each. This means that if we measure \"1\", we can with certainty conclude that the state was $|+\\rangle$.\n",
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"\n",
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"A similar argument applies to the scenario where we measure in the Hadamard basis, where $|0\\rangle$ can lead to both \"+\" and \"-\" outcomes, and $|+\\rangle$ always leads to \"+\". Then if we measured \"-\", we can with certainty conclude that the state was $|0\\rangle$.\n",
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"\n",
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"This leads to the following scenarios (shown are the conditional probabilities\n",
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"of the resulting answers in each of the above scenarios).\n",
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"\n",
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"\n",
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" State | Basis | P(0) | P(1) | P(-1)\n",
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"-------------|-------|-----------|-----------|----------\n",
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" $|0\\rangle$ | std | $0$ | $0$ | $1$\n",
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" $|+\\rangle$ | std | $0$ | $\\frac12$ | $\\frac12$\n",
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" $|0\\rangle$ | had | $\\frac12$ | $0$ | $\\frac12$\n",
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" $|+\\rangle$ | had | $0$ | $0$ | $1$\n",
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"\n",
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"> The easiest way to implement the measurement in the Hadamard basis in Q# is to apply an H gate followed by a regular measurement in computational basis."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"%kata T202_IsQubitZeroOrPlusSimpleUSD_Test\n",
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"\n",
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"open Microsoft.Quantum.Math;\n",
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"\n",
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"operation IsQubitPlusZeroOrInconclusiveSimpleUSD (q : Qubit) : Int {\n",
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" let basis = RandomInt(2);\n",
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"\n",
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" // randomize over std and had measurement bases\n",
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" if (basis == 0) {\n",
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" // use standard basis\n",
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" let result = M(q);\n",
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" // result can be One only if the state was |+⟩; if result is Zero we can't be certain\n",
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" return result == One ? 1 | -1;\n",
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" }\n",
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" else {\n",
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" // use Hadamard basis\n",
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" H(q);\n",
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" let result = M(q);\n",
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" // result can be One only if the state was |0⟩; if result is Zero we can't be certain\n",
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" return result == One ? 0 | -1;\n",
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" }\n",
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"}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"[Return to task 2.2 of the Measurements kata.](./Measurements.ipynb#task-22)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### <a name=\"peres-wooters-game\"></a>Task 2.3**. Peres/Wooters game\n",
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"\n",
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"**Input:** A qubit which is guaranteed to be in one of the three states:\n",
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"\n",
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"* $|A\\rangle = \\frac{1}{\\sqrt{2}} \\big( |0\\rangle + |1\\rangle \\big)$,\n",
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"* $|B\\rangle = \\frac{1}{\\sqrt{2}} \\big( |0\\rangle + \\omega |1\\rangle \\big)$,\n",
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"* $|C\\rangle = \\frac{1}{\\sqrt{2}} \\big( |0\\rangle + \\omega^2 |1\\rangle \\big)$,\n",
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"\n",
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"Here $\\omega = e^{2i \\pi/ 3}$.\n",
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"\n",
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"**Output:** \n",
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"\n",
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"* 1 or 2 if the qubit was in the $|A\\rangle$ state, \n",
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"* 0 or 2 if the qubit was in the $|B\\rangle$ state, \n",
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"* 0 or 1 if the qubit was in the $|C\\rangle$ state.\n",
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"\n",
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"You are never allowed to give an incorrect answer. Your solution will be called multiple times, with one of the states picked with equal probability every time.\n",
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"\n",
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"The state of the qubit at the end of the operation does not matter. "
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||||
]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Solution\n",
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"\n",
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"> The task is a simple game inspired by a quantum detection problem due to Holevo<sup><a href=\"#peres-wooters-game-ref1\">[1]</a></sup> and Peres/Wootters<sup><a href=\"#peres-wooters-game-ref2\">[2]</a></sup>. In the game, a player A thinks of a number (0,1 or 2) and the opponent, player B, tries to guess any number but the one chosen by player A. \n",
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">\n",
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"> Classically, if you just made a guess, you'd have to ask two questions to be right $100\\%$ of the time. If instead, player A prepares a qubit with 0, 1, or 2 encoded into three single qubit states that are at an angle of 120 degrees with respect to each other and then hands the state to the opponent, then player B can apply a Positive Operator Valued Measure (POVM) consisting of 3 states that are perpendicular to the states chosen by player A. \n",
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"> It can be shown that this allows B to be right $100\\%$ of the time with only 1 measurement, which is something that is not achievable with a von Neumann measurement on 1 qubit.\n",
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"See also Peres<sup><a href=\"#peres-wooters-game-ref3\">[3, chapter 9.6]</a></sup> for a nice description of the optimal POVM.\n",
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"\n",
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"Next, we address how we can implement the mentioned POVM by way of a von Neumann measurement, and then how to implement said von Neumann measurement in Q\\#. First, we note that the POVM elements are given by the columns of the following matrix: \n",
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" \n",
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"$$M = \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{rrr}\n",
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"1 & 1 & 1 \\\\ \n",
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"1 & \\omega & \\omega^2 \n",
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"\\end{array}\n",
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"\\right)$$\n",
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"\n",
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"where $\\omega = e^{2 \\pi i/3}$ denotes a primitive $3$rd root of unity. Our task will be to implement the rank 1 POVM given by the columns of $M$ via a von Neumann measurement. This can be done by \"embedding\" $M$ into a larger unitary matrix (taking complex conjugates and transposed):\n",
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||||
"\n",
|
||||
"$$M' = \\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cccc}\n",
|
||||
"1 & -1 & 1 & 0 \\\\ \n",
|
||||
"1 & -\\omega^2 & \\omega & 0 \\\\\n",
|
||||
"1 & -\\omega & \\omega^2 & 0 \\\\\n",
|
||||
"0 & 0 & 0 & -i \n",
|
||||
"\\end{array}\n",
|
||||
"\\right)$$\n",
|
||||
"\n",
|
||||
"Notice that applying $M'$ to input states given by column $i$ of $M$ (padded with two zeros to make it a vector of length $4$), where $i=0, 1, 2$ will never return the label $i$ as the corresponding vectors are perpendicular. \n",
|
||||
"\n",
|
||||
"We are therefore left with the problem of implementing $M'$ as a sequence of elementary quantum gates. Notice that \n",
|
||||
"\n",
|
||||
"$$M' \\cdot {\\rm diag}(1,-1,1,-1) = M' \\cdot (\\mathbf{1}_2 \\otimes Z) = \n",
|
||||
"\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cccc}\n",
|
||||
"1 & 1 & 1 & 0 \\\\ \n",
|
||||
"1 & \\omega^2 & \\omega & 0 \\\\\n",
|
||||
"1 & \\omega & \\omega^2 & 0 \\\\\n",
|
||||
"0 & 0 & 0 & i \n",
|
||||
"\\end{array}\n",
|
||||
"\\right)$$\n",
|
||||
"\n",
|
||||
"Using a technique used in the Rader (also sometimes called Rader-Winograd) decomposition of the discrete Fourier transform <sup><a href=\"#peres-wooters-game-ref4\">[4]</a></sup>, which reduces it to a cyclic convolution, we apply a $2\\times 2$ Fourier transform on the indices $i,j=1,2$ of this matrix (i.e. a block matrix which consists of a direct sum of blocks $\\mathbf{1}_1$, $H$, and $\\mathbf{1}_1$ which we abbreviate in short as ${\\rm diag}(1,H,1)$). This yields\n",
|
||||
"\n",
|
||||
"$${\\rm diag}(1, H, 1) \\cdot M' \\cdot (\\mathbf{1}_2 \\otimes Z) \\cdot {\\rm diag}(1, H, 1) = \n",
|
||||
"\\left(\\begin{array}{rrrr}\n",
|
||||
"\\frac{1}{\\sqrt3} & \\sqrt{\\frac23} & 0 & 0 \\\\ \n",
|
||||
"\\sqrt{\\frac23} & -\\frac{1}{\\sqrt3} & 0 & 0 \\\\\n",
|
||||
"0 & 0 & i & 0 \\\\\n",
|
||||
"0 & 0 & 0 & i \n",
|
||||
"\\end{array}\n",
|
||||
"\\right)$$\n",
|
||||
"\n",
|
||||
"This implies that after multiplication with the diagonal operator $(S^\\dagger \\otimes \\mathbf{1}_2)$, we are left with \n",
|
||||
"\n",
|
||||
"$${\\rm diag}(1, H, 1) \\cdot M' \\cdot (\\mathbf{1}_2 \\otimes Z) \\cdot {\\rm diag}(1, H, 1)\\cdot (S^\\dagger \\otimes \\mathbf{1}_2) = \n",
|
||||
"\\left(\\begin{array}{rrrr}\n",
|
||||
"\\frac{1}{\\sqrt3} & \\sqrt{\\frac23} & 0 & 0 \\\\ \n",
|
||||
"\\sqrt{\\frac23} & -\\frac{1}{\\sqrt3} & 0 & 0 \\\\\n",
|
||||
"0 & 0 & 1 & 0 \\\\\n",
|
||||
"0 & 0 & 0 & 1 \n",
|
||||
"\\end{array}\n",
|
||||
"\\right)$$\n",
|
||||
"\n",
|
||||
"which is a (zero-)controlled rotation $R$ around the $Y$-axis by an angle given by $\\arccos \\sqrt{\\frac23}$. Putting everything together, i.e., by applying the inverses of gates so that we find that \n",
|
||||
"\n",
|
||||
"$$M' = {\\rm diag}(1,H,1) \\cdot R \\cdot (S \\otimes \\mathbf{1}_2) \\cdot {\\rm diag}(1,H,1) \\cdot (\\mathbf{1}_2 \\otimes Z)$$\n",
|
||||
"\n",
|
||||
" Noting finally, that to apply this sequence of unitaries to a column vector, we have to apply it in reverse when writing it as a program (as actions on vectors are left-associative). After applying a circuit identity that simplifies the resulting circuit after implementing ${\\rm diag}(1,H,1)$ via controlled $H$, \n",
|
||||
"we arrive at the listing shown below.\n",
|
||||
"\n",
|
||||
"<a name=\"peres-wooters-game-ref1\"></a>[1] A. Holevo. Information-theoretical aspects of quantum measurement. Problems of Information Transmission, vol. 9, no. 2,pp. 110–118 (1973) \n",
|
||||
"<a name=\"peres-wooters-game-ref2\"></a>[2] A. Peres and W. K. Wootters. Optimal detection of quantum information. Phys. Rev. Lett., vol. 66, pp. 1119-1122, Mar.1991. \n",
|
||||
"<a name=\"peres-wooters-game-ref3\"></a>[3] A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 2002. \n",
|
||||
"<a name=\"peres-wooters-game-ref4\"></a>[4] C. M. Rader. Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56, 1107–1108 (1968"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"%kata T203_IsQubitNotInABC_Test\n",
|
||||
"\n",
|
||||
"open Microsoft.Quantum.Math;\n",
|
||||
"open Microsoft.Quantum.Measurement;\n",
|
||||
"\n",
|
||||
"operation IsQubitNotInABC (q : Qubit) : Int {\n",
|
||||
" \n",
|
||||
" let alpha = ArcCos(Sqrt(2.0 / 3.0));\n",
|
||||
"\n",
|
||||
" using (a = Qubit()) {\n",
|
||||
" Z(q);\n",
|
||||
" CNOT(a, q);\n",
|
||||
" Controlled H([q], a);\n",
|
||||
" S(a);\n",
|
||||
" X(q);\n",
|
||||
"\n",
|
||||
" (ControlledOnInt(0, Ry))([a], (-2.0 * alpha, q));\n",
|
||||
" CNOT(a, q);\n",
|
||||
" Controlled H([q], a);\n",
|
||||
" CNOT(a, q);\n",
|
||||
"\n",
|
||||
" // finally, measure in the standard basis\n",
|
||||
" let res0 = MResetZ(a);\n",
|
||||
" let res1 = M(q);\n",
|
||||
"\n",
|
||||
" // dispatch on the cases\n",
|
||||
" if (res0 == Zero and res1 == Zero) {\n",
|
||||
" return 0;\n",
|
||||
" }\n",
|
||||
" elif (res0 == One and res1 == Zero) {\n",
|
||||
" return 1;\n",
|
||||
" }\n",
|
||||
" elif (res0 == Zero and res1 == One) {\n",
|
||||
" return 2;\n",
|
||||
" }\n",
|
||||
" else {\n",
|
||||
" // this should never occur\n",
|
||||
" return 3;\n",
|
||||
" }\n",
|
||||
" }\n",
|
||||
"}"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"[Return to task 2.3 of the Measurements kata.](./Measurements.ipynb#peres-wooters-game)"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Q#",
|
||||
"language": "qsharp",
|
||||
"name": "iqsharp"
|
||||
},
|
||||
"language_info": {
|
||||
"file_extension": ".qs",
|
||||
"mimetype": "text/x-qsharp",
|
||||
"name": "qsharp",
|
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"version": "0.10"
|
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}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 2
|
||||
}
|
||||
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