{ "cells": [ { "cell_type": "markdown", "id": "947f670f", "metadata": {}, "source": [ "# 6.2 作为量子仿真的量子搜索" ] }, { "cell_type": "markdown", "id": "e54cb3d9", "metadata": {}, "source": [ "## 练习 6.7" ] }, { "cell_type": "markdown", "id": "3a35ef5d", "metadata": {}, "source": [ ":::{admonition} 练习 6.7\n", "\n", "验证图 6.4 和图 6.5 所示线路分别实现运算 $e^{- i |x\\rangle \\langle x| \\Delta t}$ 和 $e^{- i |\\psi\\rangle \\langle\\psi| \\Delta t}$，其中 $|\\psi\\rangle$ 如式 (6.24) 所示：\n", "\n", "$$\n", "|\\psi\\rangle = \\frac{\\sum_a |a\\rangle}{\\sqrt{N}}\n", "\\tag{6.24}\n", "$$\n", "\n", "图 6.4\n", "\n", "![ex-6.7.1](assets/ex-6.7.1.svg)\n", "\n", "图 6.5\n", "\n", "![ex-6.7.2](assets/ex-6.7.2.svg)\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "9bec1ae0", "metadata": {}, "source": [ ":::{warning}\n", "\n", "这道题可能有一些表述不当或错误。\n", "\n", "- 式 (6.24) 的求和对象是 $a$ 而非原书中的 $x$；因为在上下文中，$|x\\rangle$ 已经用来表示解了，因此用于表示均匀叠加态的求和指标应当用其它记号比较好。\n", "- 图中的相位门的符号可能是错的，这在很多讨论中已经发现 ([serab.net 的解答](https://serab.net/docs/qcqi/chapter6/#67)、Stack Exchange 的讨论 [1](https://quantumcomputing.stackexchange.com/a/11965/14843) 与 [2](https://quantumcomputing.stackexchange.com/a/11860/14843))。因此，我们这里也采用经过修正的相位。\n", "- 我们修改了代入线路的初始状态为 $|\\phi\\rangle |0\\rangle$。原来的书中是 $|y\\rangle |0\\rangle$，但我认为不太合适，因为 $|y\\rangle$ 已经用来表示非解了。\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "84f65929", "metadata": {}, "source": [ "**图 6.4 证明 (1) 指数算符的简化**" ] }, { "cell_type": "markdown", "id": "791bf9f2", "metadata": {}, "source": [ "以 $\\Delta t$ 作为小量进行 Taylor 展开：\n", "\n", "$$\n", "\\begin{align*}\n", "e^{- i |x\\rangle \\langle x| \\Delta t} &= \\sum_{k=0}^\\infty \\frac{1}{k!} (- i |x\\rangle \\langle x| \\Delta t)^k \\\\\n", "&= \\sum_{k=0}^\\infty \\frac{1}{k!} (-i \\Delta t)^k (|x\\rangle\\langle x|)^k\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "b40e6b0f", "metadata": {}, "source": [ "我们回顾到$|x\\rangle \\langle x|$、$|\\psi\\rangle \\langle\\psi|$ 等外积算符是投影算符。因此，它满足幂等性，即 $( |x\\rangle \\langle x| )^2 = |x\\rangle \\langle x|$。幂等性的推广可以是，若 $k \\geqslant 1$ 且为整数，那么 $(|x\\rangle\\langle x|)^k = |x\\rangle\\langle x|$。因此，上式的求和可以化为\n", "\n", "$$\n", "\\begin{align*}\n", "e^{- i |x\\rangle \\langle x| \\Delta t}\n", "&= |x\\rangle\\langle x| \\sum_{k=1}^\\infty \\frac{1}{k!} (-i \\Delta t)^k + I \\\\\n", "&= |x\\rangle\\langle x| \\sum_{k=0}^\\infty \\frac{1}{k!} (-i \\Delta t)^k + I - |x\\rangle \\langle x| \\\\\n", "&= I - |x\\rangle \\langle x| + e^{- i \\Delta t} |x\\rangle \\langle x|\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "0c686af3", "metadata": {}, "source": [ "最后，由于 $|x\\rangle$ 与 $|y\\rangle$ 在整个空间互补，因此 $|x\\rangle \\langle x| + |y\\rangle \\langle y| = I$。进而，\n", "\n", "$$\n", "e^{- i |x\\rangle \\langle x| \\Delta t} = |y\\rangle \\langle y| + e^{- i \\Delta t} |x\\rangle \\langle x|\n", "$$" ] }, { "cell_type": "markdown", "id": "9a938ec1", "metadata": {}, "source": [ "我们最后考察一下 $e^{- i |x\\rangle \\langle x| \\Delta t}$ 算符作用在 $|\\phi\\rangle = \\alpha |x\\rangle + \\beta |y\\rangle$ 上的结果 (留意到因为 $|x\\rangle$ 是解态而 $|y\\rangle$ 是非解态，因此两者正交即 $\\langle x|y\\rangle = 0$)：\n", "\n", "$$\n", "\\begin{align*}\n", "e^{- i |x\\rangle \\langle x| \\Delta t} |\\phi\\rangle &= \\big( |y\\rangle \\langle y| + e^{- i \\Delta t} |x\\rangle \\langle x| \\big) \\big( \\alpha |x\\rangle + \\beta |y\\rangle \\big) \\\\\n", "&= \\alpha e^{- i \\Delta t} |x\\rangle + \\beta |y\\rangle\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "04442414", "metadata": {}, "source": [ "**图 6.4 证明 (2) 线路图**" ] }, { "cell_type": "markdown", "id": "16bffd0a", "metadata": {}, "source": [ "![ex-6.7.3](assets/ex-6.7.3.svg)" ] }, { "cell_type": "markdown", "id": "7b559398", "metadata": {}, "source": [ "我们仍旧与以前一样，通过每个门路之后，考察线路的态的行为。\n", "\n", "$$\n", "|\\psi_0\\rangle = \\big( \\alpha |x\\rangle + \\beta |y\\rangle \\big) |0\\rangle\n", "$$" ] }, { "cell_type": "markdown", "id": "de6535ce", "metadata": {}, "source": [ "对于 orcale 过程，若代入的是问题的解 (即 $|x\\rangle$)，则互换 orcale 工作空间比特的 $|0\\rangle$ 与 $|1\\rangle$；但若不是问题的解 (即 $|y\\rangle$)，那么不作变换。因此，\n", "\n", "$$\n", "|\\psi_1\\rangle = \\alpha |x\\rangle |1\\rangle + \\beta |y\\rangle |0\\rangle\n", "$$" ] }, { "cell_type": "markdown", "id": "7c4c6266", "metadata": {}, "source": [ "对于最末的相位过程，若代入的是 $|0\\rangle$ 则不作变换；若代入的是 $|1\\rangle$ 则加以相位 $e^{i \\Delta t}$。因此，\n", "\n", "$$\n", "|\\psi_2\\rangle = \\alpha e^{-i \\Delta t} |x\\rangle |1\\rangle + \\beta |y\\rangle |0\\rangle\n", "$$" ] }, { "cell_type": "markdown", "id": "47d25a7a", "metadata": {}, "source": [ "最后再用一次 orcale 过程：\n", "\n", "$$\n", "|\\psi_3\\rangle = \\alpha e^{-i \\Delta t} |x\\rangle |0\\rangle + \\beta |y\\rangle |0\\rangle = \\big( \\alpha e^{-i \\Delta t} |x\\rangle + \\beta |y\\rangle \\big) |0\\rangle\n", "$$\n", "\n", "我们发现这个结果中，除去 orcale 工作量子比特，其余部分与 $e^{- i |x\\rangle \\langle x| \\Delta t} |\\phi\\rangle$ 结果一致。" ] }, { "cell_type": "markdown", "id": "559fcc2a", "metadata": {}, "source": [ "**图 6.5 证明 (1) 指数算符的简化**" ] }, { "cell_type": "markdown", "id": "9a38c8e0", "metadata": {}, "source": [ "与之前的计算过程非常类似地，我们可以得到图 6.5 待证的算符是\n", "\n", "$$\n", "e^{- i |\\psi\\rangle \\langle\\psi| \\Delta t} = I + (e^{- i \\Delta t} - 1) |\\psi\\rangle \\langle\\psi|\n", "$$" ] }, { "cell_type": "markdown", "id": "0c97da51", "metadata": {}, "source": [ "但在图 6.5 的证明中，比较方便的做法是直接使用数学推演，而非使用线路结果。我们提前需要表明，作为均匀叠加态，\n", "\n", "$$\n", "H^{\\otimes n} |0\\rangle = |\\psi\\rangle\n", "$$" ] }, { "cell_type": "markdown", "id": "21230e09", "metadata": {}, "source": [ "**图 6.5 证明 (2) 线路图**" ] }, { "cell_type": "markdown", "id": "f7c93b54", "metadata": {}, "source": [ "我们令相位门 $P_{- i \\Delta t} = \\begin{bmatrix} 1 & 0 \\\\ 0 & e^{- i \\Delta t} \\end{bmatrix}$，那么图 6.5 的线路中存有三种门路：Hadamard 门 $H^{\\otimes n} \\otimes I$、受控非门 $I^{\\otimes n} \\otimes I - |0\\rangle\\langle0| \\otimes (I-X)$、相位门 $I^{\\otimes n} \\otimes P_{-i \\Delta t}$。其中，受控非门的 $|0\\rangle\\langle0|$ 表示的是 $n$ 量子比特态的投影算符，而非单量子比特投影算符。我们依次作计算；首先是第一个 (Hadamard 门)、第二个 (受控非门) 算符作用：" ] }, { "cell_type": "markdown", "id": "10b06594", "metadata": {}, "source": [ "$$\n", "\\begin{align*}\n", "&\\quad \\big( I^{\\otimes n} \\otimes I - |0\\rangle\\langle0| \\otimes (I-X) \\big) \\big( H^{\\otimes n} \\otimes I \\big) \\\\\n", "&= H^{\\otimes n} \\otimes I - |0\\rangle\\langle0| H^{\\otimes n} \\otimes (I-X) \\\\\n", "&= H^{\\otimes n} \\otimes I - |0\\rangle\\langle\\psi| \\otimes (I-X)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "0629b270", "metadata": {}, "source": [ "再作用第三个算符 (相位门)：\n", "\n", "$$\n", "\\begin{align*}\n", "&\\quad \\big( I^{\\otimes n} \\otimes P_{-i \\Delta t} \\big) \\big( H^{\\otimes n} \\otimes I - |0\\rangle\\langle\\psi| \\otimes (I-X) \\big) \\\\\n", "&= H^{\\otimes n} \\otimes P_{-i \\Delta t} - |0\\rangle\\langle\\psi| \\otimes P_{-i \\Delta t} (I-X)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "f5fd73c7", "metadata": {}, "source": [ "再作用第四个算符 (受控非门)：\n", "\n", "$$\n", "\\begin{align*}\n", "&\\quad \\big( I^{\\otimes n} \\otimes I - |0\\rangle\\langle0| \\otimes (I-X) \\big) \\big( H^{\\otimes n} \\otimes P_{-i \\Delta t} - |0\\rangle\\langle\\psi| \\otimes P_{-i \\Delta t} (I-X) \\big) \\\\\n", "&= H^{\\otimes n} \\otimes P_{-i \\Delta t} - |0\\rangle\\langle\\psi| \\otimes P_{-i \\Delta t} (I-X) \\\\\n", "&\\quad - |0\\rangle\\langle0| H^{\\otimes n} \\otimes (I-X) P_{-i \\Delta t} + |0\\rangle\\langle0|0\\rangle\\langle\\psi| \\otimes (I-X) P_{-i \\Delta t} (I-X) \\\\\n", "&= H^{\\otimes n} \\otimes P_{-i \\Delta t} - |0\\rangle\\langle\\psi| \\otimes P_{-i \\Delta t} (I-X) \\\\\n", "&\\quad - |0\\rangle\\langle\\psi| \\otimes (I-X) P_{-i \\Delta t} + |0\\rangle\\langle\\psi| \\otimes (I-X) P_{-i \\Delta t} (I-X)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "044eab47", "metadata": {}, "source": [ "再作用第五个算符 (Hadamard 门) 得到" ] }, { "cell_type": "markdown", "id": "90e0ff49", "metadata": {}, "source": [ "$$\n", "\\begin{align*}\n", "&\\quad I^{\\otimes n} \\otimes P_{-i \\Delta t} - |\\psi\\rangle\\langle\\psi| \\otimes P_{-i \\Delta t} (I-X) \\\\\n", "&- |\\psi\\rangle\\langle\\psi| \\otimes (I-X) P_{-i \\Delta t} + |\\psi\\rangle\\langle\\psi| \\otimes (I-X) P_{-i \\Delta t} (I-X)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "0baf59ef", "metadata": {}, "source": [ "最后我们注意到，最末的量子比特被规定是 $|0\\rangle$，因此\n", "\n", "$$\n", "\\begin{align*}\n", "P_{-i \\Delta t} |0\\rangle &= |0\\rangle \\\\\n", "P_{-i \\Delta t} (I-X) |0\\rangle &= |0\\rangle - e^{-i \\Delta t} |1\\rangle \\\\\n", "(I-X) P_{-i \\Delta t} |0\\rangle &= |0\\rangle - |1\\rangle \\\\\n", "(I-X) P_{-i \\Delta t} (I-X) |0\\rangle &= (1 + e^{-i \\Delta t}) (|0\\rangle - |1\\rangle)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "32997e9f", "metadata": {}, "source": [ "因此，该线路作用在 $|\\phi\\rangle |0\\rangle$ 后会得到\n", "\n", "$$\n", "\\begin{align*}\n", "&\\quad I^{\\otimes n} |\\phi\\rangle \\otimes |0\\rangle - |\\psi\\rangle\\langle\\psi|\\phi\\rangle \\otimes \\big( |0\\rangle - e^{-i \\Delta} |1\\rangle \\big) \\\\\n", "&- |\\psi\\rangle\\langle\\psi|\\phi\\rangle \\otimes \\big( |0\\rangle - |1\\rangle \\big) + |\\psi\\rangle\\langle\\psi|\\phi\\rangle \\otimes (1 + e^{-i \\Delta t}) (|0\\rangle - |1\\rangle)\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "115d6359", "metadata": {}, "source": [ "整理上式得到\n", "\n", "$$\n", "\\begin{align*}\n", "&\\quad I^{\\otimes n} |\\phi\\rangle \\otimes |0\\rangle + |\\psi\\rangle\\langle\\psi|\\phi\\rangle \\otimes (e^{-i \\Delta t} - 1) |0\\rangle \\\\\n", "&= \\big( I^{\\otimes n} + (e^{-i \\Delta t} - 1) |\\psi\\rangle\\langle\\psi| \\big) |\\phi\\rangle \\otimes |0\\rangle \\\\\n", "&= e^{- i |\\psi\\rangle \\langle\\psi| \\Delta t} |\\phi\\rangle \\otimes |0\\rangle\n", "\\end{align*}\n", "$$\n", "\n", "这就证明完毕了。" ] }, { "cell_type": "markdown", "id": "f4af9cb4", "metadata": {}, "source": [ "## 练习 6.8" ] }, { "cell_type": "markdown", "id": "a8112c0e", "metadata": {}, "source": [ ":::{admonition} 练习 6.8\n", "\n", "设仿真步的精度可达 $O(\\Delta t^{r})$，证明以合理的精度模拟 $H$ 所需要的 oracle 调用次数是 $O(N^\\frac{r}{2 (r-1)})$。注意当 $r$ 增大时，$N$ 的指数接近 $1/2$。\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "1c0310cc", "metadata": {}, "source": [ "首先，总步数始终没有变化，为 $t / \\Delta t = O (\\sqrt{N} / \\Delta t)$。累积的误差是 $O(\\Delta t^r \\times \\sqrt{N} / \\Delta t) = O(\\Delta t^{r-1} \\times \\sqrt{N})$。为了要让这个误差在 $O(1)$ 级别，因此\n", "\n", "$$\n", "\\Delta t = \\Theta \\big( (1/\\sqrt{N})^{1/(r-1)} \\big) = O \\big( N^{-\\frac{1}{2(r-1)}} \\big)\n", "$$" ] }, { "cell_type": "markdown", "id": "86fcfd68", "metadata": {}, "source": [ "算法调用步数则是\n", "\n", "$$\n", "\\frac{t}{\\Delta t} = O (\\sqrt{N} / \\Delta t) = O \\big( N^{\\frac{1}{2} + \\frac{1}{2(r-1)}} \\big) = O(N^\\frac{r}{2 (r-1)})\n", "$$" ] }, { "cell_type": "markdown", "id": "a827794e", "metadata": {}, "source": [ "容易验证\n", "\n", "$$\n", "\\lim_{r \\rightarrow \\infty} \\frac{r}{2 (r-1)} = \\frac{1}{2}\n", "$$" ] }, { "cell_type": "markdown", "id": "f0b274a2", "metadata": {}, "source": [ "## 练习 6.9" ] }, { "cell_type": "markdown", "id": "7d1aae0f", "metadata": {}, "source": [ ":::{admonition} 练习 6.9\n", "\n", "验证式 (6.25) (提示：参看练习 4.15)。\n", "\n", "$$\n", "\\begin{align*}\n", "U(\\Delta t) &= e^{-i |\\psi\\rangle\\langle\\psi| \\Delta t} e^{-i |x\\rangle\\langle x| \\Delta t} \\\\\n", "&= \\left( \\cos^2 \\left( \\frac{\\Delta t}{2} \\right) - \\sin^2 \\left( \\frac{\\Delta t}{2} \\right) \\vec \\psi \\cdot \\hat z \\right) I \\\\\n", "&\\quad - 2 i \\sin \\left( \\frac{\\Delta t}{2} \\right) \\left( \\cos \\left( \\frac{\\Delta t}{2} \\right) \\frac{\\vec \\psi + \\hat z}{2} + \\sin \\left( \\frac{\\Delta t}{2} \\right) \\frac{\\vec \\psi \\times \\hat z}{2} \\right) \\cdot \\vec \\sigma\n", "\\tag{6.25}\n", "\\end{align*}\n", "$$\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "d3b3504f", "metadata": {}, "source": [ "此题实在是太直观了，没办法给出过程。\n", "\n", "首先，利用练习 4.4 的结论，知道 $e^{-i |\\psi\\rangle\\langle\\psi| \\Delta t}$ 与 $e^{-i |x\\rangle\\langle x| \\Delta t}$ 都是旋转算符；旋转轴分别是该题目在书中上文所定义的 $\\vec \\psi = (2 \\alpha \\beta, 0, \\alpha^2 - \\beta^2)$ 和 $\\hat z = (0, 0, 1)$；旋转角都是 $\\Delta t$。然后直接套用练习 4.15 的结论就行了。" ] }, { "cell_type": "markdown", "id": "64d4d507", "metadata": {}, "source": [ "## 练习 6.10" ] }, { "cell_type": "markdown", "id": "600e48c9", "metadata": {}, "source": [ ":::{admonition} 练习 6.10\n", "\n", "证明通过适当选取 $\\Delta t$，可以得到用 $O(\\sqrt{N})$ 次调用的量子搜索算法，并且最终状态恰好是 $|x\\rangle$。该算法成功概率为 $1$，而不是以稍小一些的概率成功。\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "96b13d28", "metadata": {}, "source": [ "我们使用倒推法。重新考察式 (6.28)，\n", "\n", "$$\n", "\\cos \\left( \\frac{\\theta}{2} \\right) = 1 - \\frac{2}{N} \\sin^2 \\left( \\frac{\\Delta t}{2} \\right)\n", "\\tag{6.28}\n", "$$" ] }, { "cell_type": "markdown", "id": "25575c17", "metadata": {}, "source": [ "因此，\n", "\n", "$$\n", "\\Delta t = 2 \\arcsin \\left( \\sqrt{\\frac{N}{2} \\left( 1 - \\cos \\left( \\frac{\\theta}{2} \\right) \\right)} \\right)\n", "$$\n", "\n", "若我们令 $\\theta = 4 / \\sqrt{N}$，则该式符合反三角函数定义域要求，且该值也不需要事先对 $|x\\rangle$ 有所了解。我们不妨绘制一下 $\\Delta t$ 随 $N$ 的变化情况 (下图的纵坐标是关于 $\\pi$ 的倍数)。" ] }, { "cell_type": "code", "execution_count": 9, "id": "037589f0", "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "from matplotlib import pyplot as plt\n", "import numpy as np" ] }, { "cell_type": "code", "execution_count": 10, "id": "6719426b", "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "/* global mpl */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function () {\n", " if (typeof WebSocket !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof MozWebSocket !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert(\n", " 'Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.'\n", " );\n", " }\n", "};\n", "\n", "mpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = this.ws.binaryType !== undefined;\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById('mpl-warnings');\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent =\n", " 'This browser does not support binary websocket messages. ' +\n", " 'Performance may be slow.';\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = document.createElement('div');\n", " this.root.setAttribute('style', 'display: inline-block');\n", " this._root_extra_style(this.root);\n", "\n", " parent_element.appendChild(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message('supports_binary', { value: fig.supports_binary });\n", " fig.send_message('send_image_mode', {});\n", " if (fig.ratio !== 1) {\n", " fig.send_message('set_dpi_ratio', { dpi_ratio: fig.ratio });\n", " }\n", " fig.send_message('refresh', {});\n", " };\n", "\n", " this.imageObj.onload = function () {\n", " if (fig.image_mode === 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function () {\n", " fig.ws.close();\n", " };\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "};\n", "\n", "mpl.figure.prototype._init_header = function () {\n", " var titlebar = document.createElement('div');\n", " titlebar.classList =\n", " 'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n", " var titletext = document.createElement('div');\n", " titletext.classList = 'ui-dialog-title';\n", " titletext.setAttribute(\n", " 'style',\n", " 'width: 100%; text-align: center; padding: 3px;'\n", " );\n", " titlebar.appendChild(titletext);\n", " this.root.appendChild(titlebar);\n", " this.header = titletext;\n", "};\n", "\n", "mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n", "\n", "mpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n", "\n", "mpl.figure.prototype._init_canvas = function () {\n", " var fig = this;\n", "\n", " var canvas_div = (this.canvas_div = document.createElement('div'));\n", " canvas_div.setAttribute(\n", " 'style',\n", " 'border: 1px solid #ddd;' +\n", " 'box-sizing: content-box;' +\n", " 'clear: both;' +\n", " 'min-height: 1px;' +\n", " 'min-width: 1px;' +\n", " 'outline: 0;' +\n", " 'overflow: hidden;' +\n", " 'position: relative;' +\n", " 'resize: both;'\n", " );\n", "\n", " function on_keyboard_event_closure(name) {\n", " return function (event) {\n", " return fig.key_event(event, name);\n", " };\n", " }\n", "\n", " canvas_div.addEventListener(\n", " 'keydown',\n", " on_keyboard_event_closure('key_press')\n", " );\n", " canvas_div.addEventListener(\n", " 'keyup',\n", " on_keyboard_event_closure('key_release')\n", " );\n", "\n", " this._canvas_extra_style(canvas_div);\n", " this.root.appendChild(canvas_div);\n", "\n", " var canvas = (this.canvas = document.createElement('canvas'));\n", " canvas.classList.add('mpl-canvas');\n", " canvas.setAttribute('style', 'box-sizing: content-box;');\n", "\n", " this.context = canvas.getContext('2d');\n", "\n", " var backingStore =\n", " this.context.backingStorePixelRatio ||\n", " this.context.webkitBackingStorePixelRatio ||\n", " this.context.mozBackingStorePixelRatio ||\n", " this.context.msBackingStorePixelRatio ||\n", " this.context.oBackingStorePixelRatio ||\n", " this.context.backingStorePixelRatio ||\n", " 1;\n", "\n", " this.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n", " 'canvas'\n", " ));\n", " rubberband_canvas.setAttribute(\n", " 'style',\n", " 'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n", " );\n", "\n", " // Apply a ponyfill if ResizeObserver is not implemented by browser.\n", " if (this.ResizeObserver === undefined) {\n", " if (window.ResizeObserver !== undefined) {\n", " this.ResizeObserver = window.ResizeObserver;\n", " } else {\n", " var obs = _JSXTOOLS_RESIZE_OBSERVER({});\n", " this.ResizeObserver = obs.ResizeObserver;\n", " }\n", " }\n", "\n", " this.resizeObserverInstance = new this.ResizeObserver(function (entries) {\n", " var nentries = entries.length;\n", " for (var i = 0; i < nentries; i++) {\n", " var entry = entries[i];\n", " var width, height;\n", " if (entry.contentBoxSize) {\n", " if (entry.contentBoxSize instanceof Array) {\n", " // Chrome 84 implements new version of spec.\n", " width = entry.contentBoxSize[0].inlineSize;\n", " height = entry.contentBoxSize[0].blockSize;\n", " } else {\n", " // Firefox implements old version of spec.\n", " width = entry.contentBoxSize.inlineSize;\n", " height = entry.contentBoxSize.blockSize;\n", " }\n", " } else {\n", " // Chrome <84 implements even older version of spec.\n", " width = entry.contentRect.width;\n", " height = entry.contentRect.height;\n", " }\n", "\n", " // Keep the size of the canvas and rubber band canvas in sync with\n", " // the canvas container.\n", " if (entry.devicePixelContentBoxSize) {\n", " // Chrome 84 implements new version of spec.\n", " canvas.setAttribute(\n", " 'width',\n", " entry.devicePixelContentBoxSize[0].inlineSize\n", " );\n", " canvas.setAttribute(\n", " 'height',\n", " entry.devicePixelContentBoxSize[0].blockSize\n", " );\n", " } else {\n", " canvas.setAttribute('width', width * fig.ratio);\n", " canvas.setAttribute('height', height * fig.ratio);\n", " }\n", " canvas.setAttribute(\n", " 'style',\n", " 'width: ' + width + 'px; height: ' + height + 'px;'\n", " );\n", "\n", " rubberband_canvas.setAttribute('width', width);\n", " rubberband_canvas.setAttribute('height', height);\n", "\n", " // And update the size in Python. We ignore the initial 0/0 size\n", " // that occurs as the element is placed into the DOM, which should\n", " // otherwise not happen due to the minimum size styling.\n", " if (fig.ws.readyState == 1 && width != 0 && height != 0) {\n", " fig.request_resize(width, height);\n", " }\n", " }\n", " });\n", " this.resizeObserverInstance.observe(canvas_div);\n", "\n", " function on_mouse_event_closure(name) {\n", " return function (event) {\n", " return fig.mouse_event(event, name);\n", " };\n", " }\n", "\n", " rubberband_canvas.addEventListener(\n", " 'mousedown',\n", " on_mouse_event_closure('button_press')\n", " );\n", " rubberband_canvas.addEventListener(\n", " 'mouseup',\n", " on_mouse_event_closure('button_release')\n", " );\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband_canvas.addEventListener(\n", " 'mousemove',\n", " on_mouse_event_closure('motion_notify')\n", " );\n", "\n", " rubberband_canvas.addEventListener(\n", " 'mouseenter',\n", " on_mouse_event_closure('figure_enter')\n", " );\n", " rubberband_canvas.addEventListener(\n", " 'mouseleave',\n", " on_mouse_event_closure('figure_leave')\n", " );\n", "\n", " canvas_div.addEventListener('wheel', function (event) {\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " on_mouse_event_closure('scroll')(event);\n", " });\n", "\n", " canvas_div.appendChild(canvas);\n", " canvas_div.appendChild(rubberband_canvas);\n", "\n", " this.rubberband_context = rubberband_canvas.getContext('2d');\n", " this.rubberband_context.strokeStyle = '#000000';\n", "\n", " this._resize_canvas = function (width, height, forward) {\n", " if (forward) {\n", " canvas_div.style.width = width + 'px';\n", " canvas_div.style.height = height + 'px';\n", " }\n", " };\n", "\n", " // Disable right mouse context menu.\n", " this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n", " event.preventDefault();\n", " return false;\n", " });\n", "\n", " function set_focus() {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "};\n", "\n", "mpl.figure.prototype._init_toolbar = function () {\n", " var fig = this;\n", "\n", " var toolbar = document.createElement('div');\n", " toolbar.classList = 'mpl-toolbar';\n", " this.root.appendChild(toolbar);\n", "\n", " function on_click_closure(name) {\n", " return function (_event) {\n", " return fig.toolbar_button_onclick(name);\n", " };\n", " }\n", "\n", " function on_mouseover_closure(tooltip) {\n", " return function (event) {\n", " if (!event.currentTarget.disabled) {\n", " return fig.toolbar_button_onmouseover(tooltip);\n", " }\n", " };\n", " }\n", "\n", " fig.buttons = {};\n", " var buttonGroup = document.createElement('div');\n", " buttonGroup.classList = 'mpl-button-group';\n", " for (var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " /* Instead of a spacer, we start a new button group. */\n", " if (buttonGroup.hasChildNodes()) {\n", " toolbar.appendChild(buttonGroup);\n", " }\n", " buttonGroup = document.createElement('div');\n", " buttonGroup.classList = 'mpl-button-group';\n", " continue;\n", " }\n", "\n", " var button = (fig.buttons[name] = document.createElement('button'));\n", " button.classList = 'mpl-widget';\n", " button.setAttribute('role', 'button');\n", " button.setAttribute('aria-disabled', 'false');\n", " button.addEventListener('click', on_click_closure(method_name));\n", " button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n", "\n", " var icon_img = document.createElement('img');\n", " icon_img.src = '_images/' + image + '.png';\n", " icon_img.srcset = '_images/' + image + '_large.png 2x';\n", " icon_img.alt = tooltip;\n", " button.appendChild(icon_img);\n", "\n", " buttonGroup.appendChild(button);\n", " }\n", "\n", " if (buttonGroup.hasChildNodes()) {\n", " toolbar.appendChild(buttonGroup);\n", " }\n", "\n", " var fmt_picker = document.createElement('select');\n", " fmt_picker.classList = 'mpl-widget';\n", " toolbar.appendChild(fmt_picker);\n", " this.format_dropdown = fmt_picker;\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = document.createElement('option');\n", " option.selected = fmt === mpl.default_extension;\n", " option.innerHTML = fmt;\n", " fmt_picker.appendChild(option);\n", " }\n", "\n", " var status_bar = document.createElement('span');\n", " status_bar.classList = 'mpl-message';\n", " toolbar.appendChild(status_bar);\n", " this.message = status_bar;\n", "};\n", "\n", "mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', { width: x_pixels, height: y_pixels });\n", "};\n", "\n", "mpl.figure.prototype.send_message = function (type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "};\n", "\n", "mpl.figure.prototype.send_draw_message = function () {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_save = function (fig, _msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "};\n", "\n", "mpl.figure.prototype.handle_resize = function (fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1], msg['forward']);\n", " fig.send_message('refresh', {});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_rubberband = function (fig, msg) {\n", " var x0 = msg['x0'] / fig.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n", " var x1 = msg['x1'] / fig.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0,\n", " 0,\n", " fig.canvas.width / fig.ratio,\n", " fig.canvas.height / fig.ratio\n", " );\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "};\n", "\n", "mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "};\n", "\n", "mpl.figure.prototype.handle_cursor = function (fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch (cursor) {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "};\n", "\n", "mpl.figure.prototype.handle_message = function (fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "};\n", "\n", "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "};\n", "\n", "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "};\n", "\n", "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n", " for (var key in msg) {\n", " if (!(key in fig.buttons)) {\n", " continue;\n", " }\n", " fig.buttons[key].disabled = !msg[key];\n", " fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n", " if (msg['mode'] === 'PAN') {\n", " fig.buttons['Pan'].classList.add('active');\n", " fig.buttons['Zoom'].classList.remove('active');\n", " } else if (msg['mode'] === 'ZOOM') {\n", " fig.buttons['Pan'].classList.remove('active');\n", " fig.buttons['Zoom'].classList.add('active');\n", " } else {\n", " fig.buttons['Pan'].classList.remove('active');\n", " fig.buttons['Zoom'].classList.remove('active');\n", " }\n", "};\n", "\n", "mpl.figure.prototype.updated_canvas_event = function () {\n", " // Called whenever the canvas gets updated.\n", " this.send_message('ack', {});\n", "};\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function (fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = 'image/png';\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src\n", " );\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data\n", " );\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " } else if (\n", " typeof evt.data === 'string' &&\n", " evt.data.slice(0, 21) === 'data:image/png;base64'\n", " ) {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig['handle_' + msg_type];\n", " } catch (e) {\n", " console.log(\n", " \"No handler for the '\" + msg_type + \"' message type: \",\n", " msg\n", " );\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\n", " \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n", " e,\n", " e.stack,\n", " msg\n", " );\n", " }\n", " }\n", " };\n", "};\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function (e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e) {\n", " e = window.event;\n", " }\n", " if (e.target) {\n", " targ = e.target;\n", " } else if (e.srcElement) {\n", " targ = e.srcElement;\n", " }\n", " if (targ.nodeType === 3) {\n", " // defeat Safari bug\n", " targ = targ.parentNode;\n", " }\n", "\n", " // pageX,Y are the mouse positions relative to the document\n", " var boundingRect = targ.getBoundingClientRect();\n", " var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n", " var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n", "\n", " return { x: x, y: y };\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys(original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object') {\n", " obj[key] = original[key];\n", " }\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function (event, name) {\n", " var canvas_pos = mpl.findpos(event);\n", "\n", " if (name === 'button_press') {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * this.ratio;\n", " var y = canvas_pos.y * this.ratio;\n", "\n", " this.send_message(name, {\n", " x: x,\n", " y: y,\n", " button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event),\n", " });\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "};\n", "\n", "mpl.figure.prototype._key_event_extra = function (_event, _name) {\n", " // Handle any extra behaviour associated with a key event\n", "};\n", "\n", "mpl.figure.prototype.key_event = function (event, name) {\n", " // Prevent repeat events\n", " if (name === 'key_press') {\n", " if (event.which === this._key) {\n", " return;\n", " } else {\n", " this._key = event.which;\n", " }\n", " }\n", " if (name === 'key_release') {\n", " this._key = null;\n", " }\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which !== 17) {\n", " value += 'ctrl+';\n", " }\n", " if (event.altKey && event.which !== 18) {\n", " value += 'alt+';\n", " }\n", " if (event.shiftKey && event.which !== 16) {\n", " value += 'shift+';\n", " }\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n", " return false;\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function (name) {\n", " if (name === 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message('toolbar_button', { name: name });\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "\n", "///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n", "// prettier-ignore\n", "var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";/* global mpl */\n", "\n", "var comm_websocket_adapter = function (comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function () {\n", " comm.close();\n", " };\n", " ws.send = function (m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function (msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data']);\n", " });\n", " return ws;\n", "};\n", "\n", "mpl.mpl_figure_comm = function (comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = document.getElementById(id);\n", " var ws_proxy = comm_websocket_adapter(comm);\n", "\n", " function ondownload(figure, _format) {\n", " window.open(figure.canvas.toDataURL());\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element;\n", " fig.cell_info = mpl.find_output_cell(\"
\");\n", " if (!fig.cell_info) {\n", " console.error('Failed to find cell for figure', id, fig);\n", " return;\n", " }\n", " fig.cell_info[0].output_area.element.on(\n", " 'cleared',\n", " { fig: fig },\n", " fig._remove_fig_handler\n", " );\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function (fig, msg) {\n", " var width = fig.canvas.width / fig.ratio;\n", " fig.cell_info[0].output_area.element.off(\n", " 'cleared',\n", " fig._remove_fig_handler\n", " );\n", " fig.resizeObserverInstance.unobserve(fig.canvas_div);\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable();\n", " fig.parent_element.innerHTML =\n", " '';\n", " fig.close_ws(fig, msg);\n", "};\n", "\n", "mpl.figure.prototype.close_ws = function (fig, msg) {\n", " fig.send_message('closing', msg);\n", " // fig.ws.close()\n", "};\n", "\n", "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n", " // Turn the data on the canvas into data in the output cell.\n", " var width = this.canvas.width / this.ratio;\n", " var dataURL = this.canvas.toDataURL();\n", " this.cell_info[1]['text/html'] =\n", " '';\n", "};\n", "\n", "mpl.figure.prototype.updated_canvas_event = function () {\n", " // Tell IPython that the notebook contents must change.\n", " IPython.notebook.set_dirty(true);\n", " this.send_message('ack', {});\n", " var fig = this;\n", " // Wait a second, then push the new image to the DOM so\n", " // that it is saved nicely (might be nice to debounce this).\n", " setTimeout(function () {\n", " fig.push_to_output();\n", " }, 1000);\n", "};\n", "\n", "mpl.figure.prototype._init_toolbar = function () {\n", " var fig = this;\n", "\n", " var toolbar = document.createElement('div');\n", " toolbar.classList = 'btn-toolbar';\n", " this.root.appendChild(toolbar);\n", "\n", " function on_click_closure(name) {\n", " return function (_event) {\n", " return fig.toolbar_button_onclick(name);\n", " };\n", " }\n", "\n", " function on_mouseover_closure(tooltip) {\n", " return function (event) {\n", " if (!event.currentTarget.disabled) {\n", " return fig.toolbar_button_onmouseover(tooltip);\n", " }\n", " };\n", " }\n", "\n", " fig.buttons = {};\n", " var buttonGroup = document.createElement('div');\n", " buttonGroup.classList = 'btn-group';\n", " var button;\n", " for (var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " /* Instead of a spacer, we start a new button group. */\n", " if (buttonGroup.hasChildNodes()) {\n", " toolbar.appendChild(buttonGroup);\n", " }\n", " buttonGroup = document.createElement('div');\n", " buttonGroup.classList = 'btn-group';\n", " continue;\n", " }\n", "\n", " button = fig.buttons[name] = document.createElement('button');\n", " button.classList = 'btn btn-default';\n", " button.href = '#';\n", " button.title = name;\n", " button.innerHTML = '';\n", " button.addEventListener('click', on_click_closure(method_name));\n", " button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n", " buttonGroup.appendChild(button);\n", " }\n", "\n", " if (buttonGroup.hasChildNodes()) {\n", " toolbar.appendChild(buttonGroup);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = document.createElement('span');\n", " status_bar.classList = 'mpl-message pull-right';\n", " toolbar.appendChild(status_bar);\n", " this.message = status_bar;\n", "\n", " // Add the close button to the window.\n", " var buttongrp = document.createElement('div');\n", " buttongrp.classList = 'btn-group inline pull-right';\n", " button = document.createElement('button');\n", " button.classList = 'btn btn-mini btn-primary';\n", " button.href = '#';\n", " button.title = 'Stop Interaction';\n", " button.innerHTML = '';\n", " button.addEventListener('click', function (_evt) {\n", " fig.handle_close(fig, {});\n", " });\n", " button.addEventListener(\n", " 'mouseover',\n", " on_mouseover_closure('Stop Interaction')\n", " );\n", " buttongrp.appendChild(button);\n", " var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n", " titlebar.insertBefore(buttongrp, titlebar.firstChild);\n", "};\n", "\n", "mpl.figure.prototype._remove_fig_handler = function (event) {\n", " var fig = event.data.fig;\n", " if (event.target !== this) {\n", " // Ignore bubbled events from children.\n", " return;\n", " }\n", " fig.close_ws(fig, {});\n", "};\n", "\n", "mpl.figure.prototype._root_extra_style = function (el) {\n", " el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n", "};\n", "\n", "mpl.figure.prototype._canvas_extra_style = function (el) {\n", " // this is important to make the div 'focusable\n", " el.setAttribute('tabindex', 0);\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " } else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "};\n", "\n", "mpl.figure.prototype._key_event_extra = function (event, _name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager) {\n", " manager = IPython.keyboard_manager;\n", " }\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which === 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "};\n", "\n", "mpl.figure.prototype.handle_save = function (fig, _msg) {\n", " fig.ondownload(fig, null);\n", "};\n", "\n", "mpl.find_output_cell = function (html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i = 0; i < ncells; i++) {\n", " var cell = cells[i];\n", " if (cell.cell_type === 'code') {\n", " for (var j = 0; j < cell.output_area.outputs.length; j++) {\n", " var data = cell.output_area.outputs[j];\n", " if (data.data) {\n", " // IPython >= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] === html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "};\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel !== null) {\n", " IPython.notebook.kernel.comm_manager.register_target(\n", " 'matplotlib',\n", " mpl.mpl_figure_comm\n", " );\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N_list = np.arange(1, 10, 0.1)\n", "Dt_list = 2 * np.arcsin(np.sqrt(N_list / 2 * (1 - np.cos(2 / np.sqrt(N_list))))) / np.pi\n", "fig, ax = plt.subplots(figsize=(4, 3))\n", "ax.plot(N_list, Dt_list)\n", "ax.set_xlabel(\"$N$\"); ax.set_ylabel(\"$\\Delta t / \\pi$\")\n", "ax.set_title(r\"When $\\theta = 4/\\sqrt{N}$\")\n", "fig.tight_layout()" ] }, { "cell_type": "markdown", "id": "da59439e", "metadata": {}, "source": [ "## 练习 6.11 (连续量子搜索的多重解)" ] }, { "cell_type": "markdown", "id": "3b34070d", "metadata": {}, "source": [ ":::{admonition} 练习 6.11\n", "\n", "猜测一个 Hamiltonian 量，用以求解有 $M$ 个解的连续时间搜索问题。\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "57ece75c", "metadata": {}, "source": [ "我们回顾到之前的 Hamiltonian 量是 $H = |x\\rangle \\langle x| + |\\psi\\rangle \\langle\\psi|$。之前讨论的 $|x\\rangle$ 是单解态。但 $|x\\rangle$ 也可以像上一节一样是多解态，即每个解 $|m\\rangle$ 的总和 (不考虑归一化系数) $|x\\rangle = \\sum_m |m\\rangle$。在这种情况下，如果量子仿真的每一步精度都足够高，那么 $O(\\sqrt{N/M})$ 步可以完成一次解的搜索，但每次无法确定是其中哪一个解。" ] }, { "cell_type": "markdown", "id": "ee04b22a", "metadata": {}, "source": [ "## 练习 6.12 (量子搜索的不同 Hamiltonian 量)" ] }, { "cell_type": "markdown", "id": "21591aa4", "metadata": {}, "source": [ ":::{admonition} 练习 6.12\n", "\n", "设\n", "\n", "$$\n", "H = |x\\rangle \\langle\\psi| + |\\psi\\rangle \\langle x|\n", "\\tag{6.29}\n", "$$\n", "\n", "1. 给定按 Hamiltonian 量的 $H$ 演化，证明从状态 $|\\psi\\rangle$ 到状态 $|x\\rangle$ 用 $O(1)$ 次旋转；\n", "2. 说明如何进行 Hamiltonian 量 $H$ 的量子仿真，并确定以高概率得到该解该仿真技术需要的 orcale 调用次数。\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "f1f7e51c", "metadata": {}, "source": [ "**第一问**" ] }, { "cell_type": "markdown", "id": "9e3ae567", "metadata": {}, "source": [ "我们仿照这一节开头的讨论。将 $|x\\rangle$ 与 $|y\\rangle$ 作为基，表示 Hamiltonian 量 $H$。我们令 $|\\psi\\rangle = \\alpha |x\\rangle + \\beta |y\\rangle$，那么\n", "\n", "$$\n", "H = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\begin{bmatrix} \\alpha & \\beta \\end{bmatrix} + \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix} \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 2 \\alpha & \\beta \\\\ \\beta & 0 \\end{bmatrix} = \\alpha I + \\beta X + \\alpha Z\n", "$$" ] }, { "cell_type": "markdown", "id": "3c816bbc", "metadata": {}, "source": [ "因此，依据式 (4.8)，\n", "\n", "$$\n", "e^{-i H t} |\\psi\\rangle = e^{-i t \\alpha} \\big( \\cos(t) |\\psi\\rangle - i \\sin (t) (\\beta X + \\alpha Z) |\\psi\\rangle \\big)\n", "$$" ] }, { "cell_type": "markdown", "id": "ffc68eca", "metadata": {}, "source": [ "我们已经知道 $(\\beta X + \\alpha Z) |\\psi\\rangle = |x\\rangle$，因此\n", "\n", "$$\n", "e^{-i H t} |\\psi\\rangle = e^{-i t \\alpha} \\big( \\cos(t) |\\psi\\rangle - i \\sin (t) |x\\rangle \\big)\n", "$$\n", "\n", "为了让变换 $e^{-i H t}$ 保证将 $|\\psi\\rangle$ 变换到 $|x\\rangle$，只要让 $\\cos(t) = 0$ 即可。因此可以选用 $t = \\pi/2$。" ] }, { "cell_type": "markdown", "id": "ffc59ed2", "metadata": {}, "source": [ "---" ] }, { "cell_type": "markdown", "id": "0e3399b0", "metadata": {}, "source": [ "**第二问**" ] }, { "cell_type": "markdown", "id": "2191bf91", "metadata": {}, "source": [ "我们发现 $H = |x\\rangle \\langle\\psi| + |\\psi\\rangle \\langle x|$ 与 $H = |x\\rangle \\langle x| + |\\psi\\rangle \\langle\\psi|$，两者在 $e^{- i H t}$ 算符上唯二的区别在于常数相位与仿真时间间隔。这前者是不关键的；后者是在实际量子门路中可以调整的。因此，这两者在仿真实现上是等价的。" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 5 }