{
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"source": [
"# HHL 算法\n",
"\n",
"[](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/notebook/r2.5.0/mindquantum/zh_cn/case_library/mindspore_hhl_algorithm.ipynb)\n",
"[](https://mindspore-website.obs.cn-north-4.myhuaweicloud.com/notebook/r2.5.0/mindquantum/zh_cn/case_library/mindspore_hhl_algorithm.py)\n",
"[](https://gitee.com/mindspore/docs/blob/r2.5.0/docs/mindquantum/docs/source_zh_cn/case_library/hhl_algorithm.ipynb)"
]
},
{
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"source": [
"## 概述\n",
"\n",
"**HHL算法的目的：** 给定一个 Hermitian $A$ 和一个单位向量 $\\vec{b}$，\n",
"求解方程 $A \\vec{x} = \\vec{b}$ 。\n",
"\n",
"令 $\\left| b \\right\\rangle = \\sum_{j=1}^{N} b_{j } \\left| j \\right\\rangle$，\n",
"算法的关键在于在 $\\left|b \\right\\rangle$ 上模拟 $e^{i A \\Delta t }$ 。\n",
"\n",
"设 $A$ 的谱分解如下\n",
"\n",
"$$A = \\sum_{j=1}^{N} \\lambda_{j } \\left| u_{j} \\right\\rangle \\left\\langle u_{j} \\right|$$\n",
"\n",
"那么\n",
"\n",
"$$e^{i A \\Delta t } = \\sum_{j=1}^{N} e^{i \\lambda_{j} \\Delta t } \\left| u_{j} \\right\\rangle \\left\\langle u_{j} \\right|$$\n",
"\n",
"将 $\\left| b \\right\\rangle$ 用基 $\\left\\lbrace \\left| u_{j} \\right\\rangle \\right\\rbrace$ 表示 ： $\\left| b \\right\\rangle = \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle$ 。\n",
"\n",
"使用量子相位估计 QPE（Quantum Phase Estimation），其中 $U = e^{i A \\Delta t}$，作用结果如下：\n",
"\n",
"$$\\left| b \\right\\rangle \\xrightarrow{\\text{QPE}} \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle | \\widetilde{ \\lambda_j } \\rangle$$\n",
"\n",
"其中 $| \\widetilde{\\lambda_j} \\rangle$ 是特征值 $\\lambda_{j}$ 的估计（波浪线表示估计值，下文同理）。\n",
"\n",
"很容易看出方程的解就是\n",
"\n",
"$$\n",
"\\left| x \\right\\rangle = A^{-1 } \\left| b \\right\\rangle = \\sum_{j}^{ } \\beta_{j } (\\lambda_{j})^{-1 } \\left| u_{j} \\right\\rangle\n",
"$$\n",
"\n",
"所以我们只需要将信息从量子态 $| \\widetilde{\\lambda_j} \\rangle$ 中提取出来即可。"
]
},
{
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"source": [
"## HHL 算法步骤\n",
"\n",
"下面详细介绍 HHL 算法的详细步骤，包含详细的数学推导。"
]
},
{
"attachments": {},
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"source": [
"### 数据预处理\n",
"\n",
"前面提到 $A$ 需要是 Hermitian 的，也就是 $A^\\dagger = A$。其实这个条件可以放宽，如果 $A$ 不是 Hermitian 的，构造 $\\tilde{A}$ 如下：\n",
"\n",
"$$\n",
"\\tilde{A} = \\begin{pmatrix}\n",
"0 & A \\\\\n",
"A^{\\dagger } & 0 \\\\\n",
"\\end{pmatrix}\n",
"$$\n",
"\n",
"然后求解方程\n",
"\n",
"$$\n",
"\\tilde{A} \\vec{y} = \\begin{pmatrix}\n",
"\\vec{b} \\\\\n",
"0 \\\\\n",
"\\end{pmatrix}\n",
"$$\n",
"\n",
"很容易验证方程的解 $\\vec{y}$ 一定具有如下的形式\n",
"\n",
"$$\n",
"\\vec{y} = \\begin{pmatrix}\n",
"0 \\\\\n",
"\\vec{x} \\\\\n",
"\\end{pmatrix}\n",
"$$\n",
"\n",
"其中 $A \\vec{x} = \\vec{b}$ 。\n",
"\n",
"对于 $\\vec{b}$，因为 HHL 是量子算法，所以需要输入的是量子态 $|b\\rangle = \\sum_{j=1}^N b_j |j\\rangle$ 。关于如何制备这样量子态，如何将经典信息编码到量子信息，不是本算法关心的重点，这里只假设制备好了这样的量子态。"
]
},
{
"attachments": {},
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"source": [
"### 制备算子 $U = e^{i A \\Delta t}$\n",
"\n",
"如何高效的制备算子 $U = e^{i A \\Delta t}$ 是核心问题。后面将会看到，整个 HHL 算法的复杂度主要取决于算子 $U$ ，所以需要高效的模拟 $U$ 。\n",
"\n",
"制备这个时间演化算子 $U = e^{i A \\Delta t}$ 属于量子系统模拟（哈密顿量模拟）问题，本身是一个重要的量子问题，这里并不展开讨论。\n",
"\n",
"在原论文中，对矩阵 $A$ 作出了**稀疏性**的限制，其目的就是能够高效的模拟 $e^{i A \\Delta t}$。对于一般的稠密矩阵 $A$，模拟其演化复杂度可能很高。"
]
},
{
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"source": [
"这里不去展开证明 QPE 的正确性，读者只需要知道 QPE 的输入是一个酉算子 $U$ 和其特征向量 $|u\\rangle$，设 $U |u\\rangle = e^{2 \\pi i \\varphi} |u\\rangle$ ，输出是 $\\varphi$ 的估计 $\\tilde{\\varphi}$ 。\n",
"\n",
"具体来说，使用 $t$ 个辅助量子比特, QPE 的作用如下\n",
"\n",
"$$\\left| b \\right\\rangle \\left| 0 \\right\\rangle^{\\otimes t} \\left| 0 \\right\\rangle = \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| 0 \\right\\rangle^{\\otimes t} \\left| 0 \\right\\rangle\n",
"\\xrightarrow{\\text{QPE}} \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| \\widetilde{\\varphi_j} \\right\\rangle \\left| 0 \\right\\rangle$$\n",
"\n",
"其中 $\\left| \\widetilde{\\varphi_j} \\right\\rangle$ 是相位 $\\varphi_{j}$ 的估计。\n",
"这里在 $t$ 个辅助比特后面还增加了一个辅助比特，是用于后面的旋转步骤。\n",
"\n",
"为了更好的理解，这里举一个例子：\n",
"取\n",
"\n",
"$$A = Z = \\left| 0 \\right\\rangle \\left\\langle 0 \\right| - \\left| 1 \\right\\rangle \\left\\langle 1 \\right|$$\n",
"\n",
"那么\n",
"\n",
"$$U = e^{i A \\Delta t } = e^{i \\Delta t} \\left| 0 \\right\\rangle \\left\\langle 0 \\right| + e^{-i \\Delta t} \\left| 1 \\right\\rangle \\left\\langle 1 \\right|$$\n",
"\n",
"两个相位分别是 $\\varphi_{1} = \\Delta t / 2 \\pi$ 和 $\\varphi_{2} = -\\Delta t / 2 \\pi$ 。\n",
"\n",
"如果我们使用 $t = 4$ 个辅助比特，并且取 $\\Delta t = \\pi / 4$，那么 $\\varphi_{1} = 1 / 8$ 和 $\\varphi_{2} = - 1 / 8$ 。\n",
"\n",
"两个相位的估计值分别是 $\\widetilde{\\varphi_1} = 0.0010 \\times 2^4 = 2$ 和 $\\widetilde{\\varphi_2} = 0.1110 \\times 2^4 = 14$ 。\n",
"因为相位是 $[0, 1)$ 之间的小数 $0.q_{t_1}q_{t_2}\\ldots q_{t_t}$，\n",
"乘上 $2^t$ 得到其对应的量子态 $|q_{t_1} \\ldots q_{t_t} \\rangle$ 。\n",
"需要指出的是，因为相位是模 $1$ 的小数，$-0.0010$ 被映射到了 $0.1110$，我们想要还原相位（为了还原 $\\lambda$），首先通过选取足够小的 $\\Delta t$ 使得 $|\\varphi| < 1 / 2$，这样如果 $|q_{t_1} \\ldots q_{t_t}\\rangle < 2^{t-1}$ 对应正数，$|q_{t_1}\\ldots q_{t_t}\\rangle > 2^{t-1}$ 对应负数。\n",
"\n",
"通过简单的定量计算，我们得到如下的映射关系：\n",
"\n",
"$$\n",
"\\frac{\\lambda \\Delta t}{2 \\pi} = \\varphi = \\begin{cases}\n",
"\\tilde{\\varphi} / 2^{t} & , \\varphi > 0 & , \\tilde{\\varphi} < 2^{t-1} \\\\\n",
"\\tilde{\\varphi} / 2^{t} - 1 & , \\varphi < 0 & , \\tilde{\\varphi} > 2^{t-1}\\\\\n",
"\\end{cases}\n",
"$$\n",
"\n",
"其中 $\\lambda$ 是 $A$ 的特征值，$\\varphi$ 是 $U = e^{i A \\Delta t}$ 的相位，$\\tilde{\\varphi}$ 是 $\\varphi$ 的估计。"
]
},
{
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"source": [
"### 条件旋转\n",
"\n",
"经过 QPE 后，整个量子态如下\n",
"\n",
"$$\n",
"\\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| \\widetilde{\\varphi_j} \\right\\rangle \\left| 0 \\right\\rangle\n",
"$$\n",
"\n",
"想要将 $\\lambda_j$ 的信息从量子态 $\\left| \\widetilde{\\varphi_j} \\right\\rangle$ 中提取出来，我们需要使用条件旋转门 $CR(k)$，其作用效果如下\n",
"\n",
"$$\n",
"CR(k) \\left| \\tilde{\\varphi} \\right\\rangle \\left| b \\right\\rangle = \\begin{cases}\n",
"\\left| \\tilde{\\varphi} \\right\\rangle \\left| b \\right\\rangle & , k \\neq \\tilde{\\varphi} \\\\\n",
"\\left| \\tilde{\\varphi} \\right\\rangle R_y \\left( 2 \\arcsin \\frac{C}{\\lambda} \\right) \\left| b \\right\\rangle & , k = \\tilde{\\varphi} \\\\\n",
"\\end{cases}\n",
"$$\n",
"\n",
"简单来说，只有当 $k$ 选择了正确的 $\\tilde{\\varphi}$，才会对后面的量子比特作用旋转操作。\n",
"\n",
"因为我们不知道正确的 $\\tilde{\\varphi}$ 是什么，所以暴力的枚举所有的可能 $CR(k)$， $k = 1, \\ldots , 2^{t}-1$ 。\n",
"\n",
"旋转的效果是很简单的\n",
"\n",
"$$\n",
"\\prod_{k=1}^{2^{t} - 1} I\\otimes CR(k) \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| \\widetilde{\\varphi_j} \\right\\rangle \\left| 0 \\right\\rangle\n",
"= \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| \\widetilde{\\varphi_j} \\right\\rangle \\left( \\sqrt{1 - \\left( \\frac{C}{\\lambda_{j}} \\right)^{2}} \\left| 0 \\right\\rangle + \\frac{C}{\\lambda_{j}} \\left| 1 \\right\\rangle \\right)\n",
"$$\n",
"\n",
"再作用一次逆 QPE，整体的量子态如下\n",
"\n",
"$$\n",
"\\left| \\psi \\right\\rangle = \\sum_{j=1}^{N} \\beta_{j} \\left| u_{j} \\right\\rangle \\left| 0 \\right\\rangle^{\\otimes t} \\left( \\sqrt{1 - \\left( \\frac{C}{\\lambda_{j}} \\right)^{2}} \\left| 0 \\right\\rangle + \\frac{C}{\\lambda_{j}} \\left| 1 \\right\\rangle \\right)\n",
"$$"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"### 测量\n",
"\n",
"对最后一个量子比特进行测量，当测量结果是 $\\left| 1 \\right\\rangle$ 时，其概率是\n",
"\n",
"$$\n",
"p_1 = C^{2} \\sum_{j=1}^{N} ( \\beta_{j} / \\lambda_{j} )^{2}\n",
"$$\n",
"\n",
"测量后的量子态变为\n",
"\n",
"$$\n",
"\\left| \\psi_{1} \\right\\rangle = \\frac{1}{\\left( \\sum_{j=1}^{N} \\left( \\beta_{j} / \\lambda_{j} \\right)^{2} \\right)^{-1}} \\sum_{j=1}^{N} \\frac{\\beta_{j}}{\\lambda_{j}} \\left| u_{j} \\right\\rangle \\left| 0 \\right\\rangle^{\\otimes t} \\left| 1 \\right\\rangle\n",
"$$\n",
"\n",
"显然此时三个量子寄存器之间不存在纠缠，如果只看第一个量子寄存器，它已经是 $|x\\rangle$ 的状态了（忽略掉一个归一化系数）。\n",
"\n",
"需要指出的是，旋转操作里面有一个参数 $C$，我们看到 $C$ 不会影响最后结果的正确性，但是却实实在在的影响了得到结果的概率 $p_{1}$ 。\n",
"我们肯定希望 $C$ 能够尽可能大，从而使 $p_1$ 尽可能大。但是 $C \\leq \\left\\vert \\lambda_{j} \\right\\vert$ ，它要比所有特征值的绝对值还要小，如果没有先验信息，不知道绝对值最小的 $\\lambda$ 有多大，那么就只能保守的取一个很小的 $C$，然后可以通过振幅放大技术来增大得到结果的概率。"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## MindQuantum 实现\n",
"\n",
"这里使用 MindQuantum 实现一个简单的例子，仅说明 HHL 算法的过程和正确性。\n",
"\n",
"为了计算方便，我们选取一个简单的\n",
"$A = Z = \\begin{bmatrix}\n",
"1 & 0 \\\\\n",
"0 & -1\n",
"\\end{bmatrix}$，因为其时间演化算子 $e^{i Z \\Delta t} = R_z(- 2 \\Delta t)$ ，比较容易实现。"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"eigenvalues of A:\n",
" [ 1. -1.]\n",
"eigenvectors of A:\n",
" [[1. 0.]\n",
" [0. 1.]]\n",
"b: [0.6 0.8]\n",
"Solution of Ax=b is: [ 0.6 -0.8]\n"
]
}
],
"source": [
"import numpy as np\n",
"\n",
"A = np.array([[1, 0], [0, -1]])\n",
"es, vs = np.linalg.eig(A)\n",
"\n",
"print(f\"eigenvalues of A:\\n {es}\")\n",
"print(f\"eigenvectors of A:\\n {vs}\")\n",
"\n",
"b = np.array([0.6, 0.8])\n",
"print(f\"b: {b}\")\n",
"print(f\"Solution of Ax=b is: {np.linalg.solve(A, b)}\")"
]
},
{
"attachments": {},
"cell_type": "markdown",
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"source": [
"这里导入所需要的各种函数："
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from mindquantum.core.gates import H, X, RY, RZ, Measure, Power, BARRIER\n",
"from mindquantum.core.circuit import Circuit\n",
"from mindquantum.simulator import Simulator"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"对于量子态 $|b\\rangle = \\cos\\theta |0\\rangle + \\sin\\theta |1\\rangle$ 的制备，可以通过一个 $R_y(2 \\theta)$ 实现。\n",
"\n",
"下面的代码制备了 $|b\\rangle = 0.6 |0\\rangle + 0.8 |1\\rangle$ 并进行了测量。"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
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],
"source": [
"circ = Circuit()\n",
"circ += RY(2 * np.arcsin(0.8)).on(0)\n",
"circ += Measure().on(0)\n",
"\n",
"sim = Simulator(backend=\"mqvector\", n_qubits=1)\n",
"res = sim.sampling(circ, shots=10000)\n",
"res.svg()"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"下面一步步构建出完整的量子线路，取 $t = 4$，$\\Delta t = \\pi / 4$， $C = 0.5$。\n",
"\n",
"- 首先是 QPE\n",
"- 然后是 $CR(k)$，$k = 1, \\ldots, 15$\n",
"- 接着是 逆QPE\n",
"- 最后测量"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
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],
"text/plain": [
"<mindquantum.io.display.circuit_svg_drawer.SVGCircuit at 0x7f13de288fd0>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from mindquantum.algorithm.library import qft\n",
"\n",
"# q1 ~ q4 : for QPE\n",
"t = 4\n",
"qc = [4, 3, 2, 1]\n",
"\n",
"# store |b> , and store the result |x>\n",
"qb = 5\n",
"\n",
"# use for conditional rotation\n",
"qs = 0\n",
"\n",
"# choose a time evolution\n",
"dt = np.pi / 4\n",
"\n",
"# choose a small C\n",
"C = 0.5\n",
"\n",
"circ = Circuit()\n",
"\n",
"# prepare b\n",
"circ += RY(2 * np.arcsin(0.8)).on(qb)\n",
"\n",
"# QPE\n",
"for i in qc:\n",
" circ += H.on(i)\n",
"\n",
"for (i, q) in enumerate(qc):\n",
" circ += Power(RZ(-2 * dt), 2**i).on(qb, q)\n",
"\n",
"# apply inverse QFT\n",
"circ += qft(list(reversed(qc))).hermitian()\n",
"\n",
"# conditional rotate\n",
"circ += BARRIER\n",
"for k in range(1, 2**t):\n",
" for i in range(t):\n",
" if (k & (1 << i)) == 0:\n",
" circ += X.on(qc[i])\n",
" phi = k / (2**t)\n",
" if k > 2**(t-1):\n",
" phi -= 1.0\n",
" l = 2 * np.pi / dt * phi\n",
" circ += RY(2 * np.arcsin(C / l)).on(qs, qc)\n",
"\n",
" for i in range(t):\n",
" if (k & (1 << i)) == 0:\n",
" circ += X.on(qc[i])\n",
" circ += BARRIER\n",
"\n",
"# apply inverse QPE\n",
"circ += qft(list(reversed(qc)))\n",
"\n",
"for (i, q) in enumerate(qc):\n",
" circ += Power(RZ(2 * dt), 2**i).on(qb, q)\n",
"\n",
"for i in qc:\n",
" circ += H.on(i)\n",
"\n",
"circ.svg()"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"如何验证我们的结果的正确性？测量。因为结果是量子态 $|x\\rangle$，目前 MindQuantum 没有相关的办法能够单独取出某一个量子比特的内部值，我们只能通过测量。"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
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],
"text/plain": [
"<mindquantum.io.display.measure_res_svg_drawer.SVGMeasure at 0x7f13de2c7490>"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sim = Simulator(backend=\"mqvector\", n_qubits=2 + t)\n",
"sim.apply_circuit(circ)\n",
"\n",
"circ_m = Circuit()\n",
"circ_m += Measure().on(qs)\n",
"circ_m += Measure().on(qb)\n",
"\n",
"res = sim.sampling(circ_m, shots=100000)\n",
"res.svg()"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3602052179446389"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res.data.get(\"01\", 0) / (res.data.get(\"01\", 0) + res.data.get(\"11\", 0))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"通过统计当 `q0` 是 1 时，`q5` 的测量结果，我们看到 `q5 = 0` 的占比是 `0.36`，和答案的预期一致。\n",
"\n",
"当然这种方法只能得到振幅大小，想要得到更多信息，例如相位，需要对测量进行调整，这里不去讨论。"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"<table border=\"1\">\n",
" <tr>\n",
" <th>Software</th>\n",
" <th>Version</th>\n",
" </tr>\n",
"<tr><td>mindquantum</td><td>0.9.11</td></tr>\n",
"<tr><td>scipy</td><td>1.10.1</td></tr>\n",
"<tr><td>numpy</td><td>1.23.5</td></tr>\n",
"<tr>\n",
" <th>System</th>\n",
" <th>Info</th>\n",
"</tr>\n",
"<tr><td>Python</td><td>3.9.16</td></tr><tr><td>OS</td><td>Linux x86_64</td></tr><tr><td>Memory</td><td>8.3 GB</td></tr><tr><td>CPU Max Thread</td><td>8</td></tr><tr><td>Date</td><td>Fri Dec 29 23:43:57 2023</td></tr>\n",
"</table>\n"
],
"text/plain": [
"<mindquantum.utils.show_info.InfoTable at 0x7f13de238af0>"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from mindquantum.utils.show_info import InfoTable\n",
"\n",
"InfoTable('mindquantum', 'scipy', 'numpy')"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "base",
"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": "3.9.16"
},
"orig_nbformat": 4
},
"nbformat": 4,
"nbformat_minor": 2
}