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| a | b/Code/All PennyLane QML Demos/31 Quantum Graph RNN 6 of 8 kkawchak.ipynb | ||
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| 1 | { |
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| 2 | "cells": [ |
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| 3 | { |
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| 4 | "cell_type": "code", |
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| 5 | "execution_count": 2, |
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| 6 | "metadata": { |
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| 7 | "id": "e-qZmPKaLbJ7" |
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| 8 | }, |
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| 9 | "outputs": [], |
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| 10 | "source": [ |
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| 11 | "# This cell is added by sphinx-gallery\n", |
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| 12 | "# It can be customized to whatever you like\n", |
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| 13 | "%matplotlib inline\n", |
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| 14 | "# !pip install pennylane" |
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| 15 | ] |
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| 16 | }, |
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| 17 | { |
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| 18 | "cell_type": "markdown", |
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| 19 | "metadata": { |
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| 20 | "id": "ZPrrGxMTLbJ7" |
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| 21 | }, |
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| 22 | "source": [ |
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| 23 | "The Quantum Graph Recurrent Neural Network\n", |
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| 24 | "==========================================\n", |
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| 25 | "\n", |
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| 26 | "::: {.meta}\n", |
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| 27 | ":property=\\\"og:description\\\": Using a quantum graph recurrent neural\n", |
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| 28 | "network to learn quantum dynamics. :property=\\\"og:image\\\":\n", |
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| 29 | "<https://pennylane.ai/qml/_images/qgrnn_thumbnail.png>\n", |
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| 30 | ":::\n", |
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| 31 | "\n", |
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| 32 | "*Author: Jack Ceroni --- Posted: 27 July 2020. Last updated: 25 March\n", |
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| 33 | "2021.*\n" |
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| 34 | ] |
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| 35 | }, |
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| 36 | { |
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| 37 | "cell_type": "markdown", |
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| 38 | "metadata": { |
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| 39 | "id": "CU3x_5JALbJ8" |
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| 40 | }, |
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| 41 | "source": [ |
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| 42 | "This demonstration investigates quantum graph recurrent neural networks\n", |
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| 43 | "(QGRNN), which are the quantum analogue of a classical graph recurrent\n", |
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| 44 | "neural network, and a subclass of the more general quantum graph neural\n", |
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| 45 | "network ansatz. Both the QGNN and QGRNN were introduced in [this paper\n", |
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| 46 | "(2019)](https://arxiv.org/abs/1909.12264).\n" |
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| 47 | ] |
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| 48 | }, |
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| 49 | { |
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| 50 | "cell_type": "markdown", |
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| 51 | "metadata": { |
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| 52 | "id": "3M64CbeaLbJ8" |
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| 53 | }, |
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| 54 | "source": [ |
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| 55 | "The Idea\n", |
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| 56 | "========\n" |
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| 57 | ] |
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| 58 | }, |
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| 59 | { |
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| 60 | "cell_type": "markdown", |
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| 61 | "metadata": { |
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| 62 | "id": "H3vGd3-BLbJ8" |
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| 63 | }, |
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| 64 | "source": [ |
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| 65 | "A graph is defined as a set of *nodes* along with a set of **edges**,\n", |
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| 66 | "which represent connections between nodes. Information can be encoded\n", |
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| 67 | "into graphs by assigning numbers to nodes and edges, which we call\n", |
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| 68 | "**weights**. It is usually convenient to think of a graph visually:\n", |
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| 69 | "\n", |
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| 70 | "{.align-center width=\"70.0%\"}\n", |
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| 71 | "\n", |
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| 72 | "In recent years, the concept of a [graph neural\n", |
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| 73 | "network](https://arxiv.org/abs/1812.08434) (GNN) has been receiving a\n", |
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| 74 | "lot of attention from the machine learning community. A GNN seeks to\n", |
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| 75 | "learn a representation (a mapping of data into a low-dimensional vector\n", |
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| 76 | "space) of a given graph with feature vectors assigned to nodes and\n", |
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| 77 | "edges. Each of the vectors in the learned representation preserves not\n", |
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| 78 | "only the features, but also the overall topology of the graph, i.e.,\n", |
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| 79 | "which nodes are connected by edges. The quantum graph neural network\n", |
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| 80 | "attempts to do something similar, but for features that are\n", |
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| 81 | "quantum-mechanical; for instance, a collection of quantum states.\n" |
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| 82 | ] |
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| 83 | }, |
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| 84 | { |
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| 85 | "cell_type": "markdown", |
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| 86 | "metadata": { |
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| 87 | "id": "T3MTH5BWLbJ8" |
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| 88 | }, |
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| 89 | "source": [ |
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| 90 | "Consider the class of qubit Hamiltonians that are *quadratic*, meaning\n", |
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| 91 | "that the terms of the Hamiltonian represent either interactions between\n", |
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| 92 | "two qubits, or the energy of individual qubits. This class of\n", |
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| 93 | "Hamiltonians is naturally described by graphs, with second-order terms\n", |
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| 94 | "between qubits corresponding to weighted edges between nodes, and\n", |
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| 95 | "first-order terms corresponding to node weights.\n", |
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| 96 | "\n", |
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| 97 | "A well known example of a quadratic Hamiltonian is the transverse-field\n", |
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| 98 | "Ising model, which is defined as\n", |
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| 99 | "\n", |
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| 100 | "$$\\hat{H}_{\\text{Ising}}(\\boldsymbol\\theta) \\ = \\ \\displaystyle\\sum_{(i, j) \\in E}\n", |
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| 101 | "\\theta_{ij}^{(1)} Z_{i} Z_{j} \\ + \\ \\displaystyle\\sum_{i} \\theta_{i}^{(2)} Z_{i} \\ + \\\n", |
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| 102 | "\\displaystyle\\sum_{i} X_{i},$$\n", |
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| 103 | "\n", |
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| 104 | "where $\\boldsymbol\\theta \\ = \\ \\{\\theta^{(1)}, \\ \\theta^{(2)}\\}$. In\n", |
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| 105 | "this Hamiltonian, the set $E$ that determines which pairs of qubits have\n", |
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| 106 | "$ZZ$ interactions can be represented by the set of edges for some graph.\n", |
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| 107 | "With the qubits as nodes, this graph is called the *interaction graph*.\n", |
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| 108 | "The $\\theta^{(1)}$ parameters correspond to the edge weights and the\n", |
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| 109 | "$\\theta^{(2)}$ parameters correspond to weights on the nodes.\n" |
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| 110 | ] |
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| 111 | }, |
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| 112 | { |
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| 113 | "cell_type": "markdown", |
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| 114 | "metadata": { |
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| 115 | "id": "4fIzrPwnLbJ9" |
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| 116 | }, |
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| 117 | "source": [ |
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| 118 | "This result implies that we can think about *quantum circuits* with\n", |
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| 119 | "graph-theoretic properties. Recall that the time-evolution operator with\n", |
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| 120 | "respect to some Hamiltonian $H$ is defined as:\n", |
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| 121 | "\n", |
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| 122 | "$$U \\ = \\ e^{-it H}.$$\n", |
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| 123 | "\n", |
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| 124 | "Thus, we have a clean way of taking quadratic Hamiltonians and turning\n", |
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| 125 | "them into unitaries (quantum circuits) that preserve the same\n", |
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| 126 | "correspondance to a graph. In the case of the Ising Hamiltonian, we\n", |
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| 127 | "have:\n", |
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| 128 | "\n", |
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| 129 | "$$U_{\\text{Ising}} \\ = \\ e^{-it \\hat{H}_{\\text{Ising}} (\\boldsymbol\\theta)} \\ = \\ \\exp \\Big[ -it\n", |
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| 130 | "\\Big( \\displaystyle\\sum_{(i, j) \\in E} \\theta_{ij}^{(1)} Z_{i} Z_{j} \\ + \\\n", |
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| 131 | "\\displaystyle\\sum_{i} \\theta_{i}^{(2)} Z_{i} \\ + \\ \\displaystyle\\sum_{i} X_{i} \\Big) \\Big]$$\n", |
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| 132 | "\n", |
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| 133 | "In general, this kind of unitary is very difficult to implement on a\n", |
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| 134 | "quantum computer. However, we can approximate it using the\n", |
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| 135 | "[Trotter-Suzuki\n", |
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| 136 | "decomposition](https://en.wikipedia.org/wiki/Time-evolving_block_decimation#The_Suzuki-Trotter_expansion):\n", |
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| 137 | "\n", |
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| 138 | "$$\\exp \\Big[ -it \\Big( \\displaystyle\\sum_{(i, j) \\in E} \\theta_{ij}^{(1)} Z_{i} Z_{j} \\ + \\\n", |
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| 139 | "\\displaystyle\\sum_{i} \\theta_{i}^{(2)} Z_{i} \\ + \\ \\displaystyle\\sum_{i} X_{i} \\Big) \\Big]\n", |
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| 140 | "\\ \\approx \\ \\displaystyle\\prod_{k \\ = \\ 1}^{t / \\Delta} \\Bigg[ \\displaystyle\\prod_{j \\ = \\\n", |
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| 141 | "1}^{Q} e^{-i \\Delta \\hat{H}_{\\text{Ising}}^{j}(\\boldsymbol\\theta)} \\Bigg]$$\n", |
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| 142 | "\n", |
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| 143 | "where $\\hat{H}_{\\text{Ising}}^{j}(\\boldsymbol\\theta)$ is the $j$-th term\n", |
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| 144 | "of the Ising Hamiltonian and $\\Delta$ is some small number.\n", |
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| 145 | "\n", |
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| 146 | "This circuit is a specific instance of the **Quantum Graph Recurrent\n", |
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| 147 | "Neural Network**, which in general is defined as a variational ansatz of\n", |
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| 148 | "the form\n", |
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| 149 | "\n", |
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| 150 | "$$U_{H}(\\boldsymbol\\mu, \\ \\boldsymbol\\gamma) \\ = \\ \\displaystyle\\prod_{i \\ = \\ 1}^{P} \\Bigg[\n", |
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| 151 | "\\displaystyle\\prod_{j \\ = \\ 1}^{Q} e^{-i \\gamma_j H^{j}(\\boldsymbol\\mu)} \\Bigg],$$\n", |
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| 152 | "\n", |
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| 153 | "for some parametrized quadratic Hamiltonian, $H(\\boldsymbol\\mu)$.\n" |
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| 154 | ] |
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| 155 | }, |
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| 156 | { |
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| 157 | "cell_type": "markdown", |
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| 158 | "metadata": { |
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| 159 | "id": "uKXgaaF1LbJ9" |
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| 160 | }, |
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| 161 | "source": [ |
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| 162 | "Using the QGRNN\n", |
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| 163 | "===============\n" |
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| 164 | ] |
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| 165 | }, |
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| 166 | { |
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| 167 | "cell_type": "markdown", |
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| 168 | "metadata": { |
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| 169 | "id": "GDA7lU2tLbJ9" |
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| 170 | }, |
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| 171 | "source": [ |
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| 172 | "Since the QGRNN ansatz is equivalent to the approximate time evolution\n", |
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| 173 | "of some quadratic Hamiltonian, we can use it to learn the dynamics of a\n", |
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| 174 | "quantum system.\n", |
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| 175 | "\n", |
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| 176 | "Continuing with the Ising model example, let\\'s imagine we have some\n", |
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| 177 | "system governed by $\\hat{H}_{\\text{Ising}}(\\boldsymbol\\alpha)$ for an\n", |
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| 178 | "unknown set of target parameters, $\\boldsymbol\\alpha$ and an unknown\n", |
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| 179 | "interaction graph $G$. Let\\'s also suppose we have access to copies of\n", |
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| 180 | "some low-energy, non-ground state of the target Hamiltonian,\n", |
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| 181 | "$|\\psi_0\\rangle$. In addition, we have access to a collection of\n", |
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| 182 | "time-evolved states,\n", |
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| 183 | "$\\{ |\\psi(t_1)\\rangle, \\ |\\psi(t_2)\\rangle, \\ ..., \\ |\\psi(t_N)\\rangle \\}$,\n", |
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| 184 | "defined by:\n", |
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| 185 | "\n", |
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| 186 | "$$|\\psi(t_k)\\rangle \\ = \\ e^{-i t_k \\hat{H}_{\\text{Ising}}(\\boldsymbol\\alpha)} |\\psi_0\\rangle.$$\n", |
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| 187 | "\n", |
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| 188 | "We call the low-energy states and the collection of time-evolved states\n", |
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| 189 | "*quantum data*. From here, we randomly pick a number of time-evolved\n", |
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| 190 | "states from our collection. For any state that we choose, which is\n", |
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| 191 | "evolved to some time $t_k$, we compare it to\n", |
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| 192 | "\n", |
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| 193 | "$$U_{\\hat{H}_{\\text{Ising}}}(\\boldsymbol\\mu, \\ \\Delta) |\\psi_0\\rangle \\ \\approx \\ e^{-i t_k\n", |
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| 194 | "\\hat{H}_{\\text{Ising}}(\\boldsymbol\\mu)} |\\psi_0\\rangle.$$\n", |
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| 195 | "\n", |
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| 196 | "This is done by feeding one of the copies of $|\\psi_0\\rangle$ into a\n", |
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| 197 | "quantum circuit with the QGRNN ansatz, with some guessed set of\n", |
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| 198 | "parameters $\\boldsymbol\\mu$ and a guessed interaction graph, $G'$. We\n", |
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| 199 | "then use a classical optimizer to maximize the average \\\"similarity\\\"\n", |
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| 200 | "between the time-evolved states and the states prepared with the QGRNN.\n", |
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| 201 | "\n", |
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| 202 | "As the QGRNN states becomes more similar to each time-evolved state for\n", |
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| 203 | "each sampled time, it follows that\n", |
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| 204 | "$\\boldsymbol\\mu \\ \\rightarrow \\ \\boldsymbol\\alpha$ and we are able to\n", |
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| 205 | "learn the unknown parameters of the Hamiltonian.\n", |
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| 206 | "\n", |
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| 207 | "{.align-center\n", |
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| 209 | "width=\"90.0%\"}\n" |
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| 210 | ] |
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| 211 | }, |
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| 212 | { |
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| 213 | "cell_type": "markdown", |
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| 214 | "metadata": { |
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| 215 | "id": "hYWy2SBeLbJ9" |
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| 216 | }, |
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| 217 | "source": [ |
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| 218 | "Learning an Ising Model with the QGRNN\n", |
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| 219 | "======================================\n" |
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| 220 | ] |
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| 221 | }, |
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| 222 | { |
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| 223 | "cell_type": "markdown", |
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| 224 | "metadata": { |
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| 225 | "id": "HlWNWZZULbJ9" |
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| 226 | }, |
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| 227 | "source": [ |
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| 228 | "We now attempt to use the QGRNN to learn the parameters corresponding to\n", |
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| 229 | "an arbitrary transverse-field Ising model Hamiltonian.\n" |
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| 230 | ] |
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| 231 | }, |
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| 232 | { |
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| 233 | "cell_type": "markdown", |
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| 234 | "metadata": { |
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| 235 | "id": "GpdeYnFwLbJ-" |
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| 236 | }, |
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| 237 | "source": [ |
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| 238 | "Getting Started\n", |
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| 239 | "===============\n" |
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| 240 | ] |
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| 241 | }, |
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| 242 | { |
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| 243 | "cell_type": "markdown", |
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| 244 | "metadata": { |
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| 245 | "id": "-5FbshFpLbJ-" |
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| 246 | }, |
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| 247 | "source": [ |
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| 248 | "We begin by importing the necessary dependencies:\n" |
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| 249 | ] |
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| 250 | }, |
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| 251 | { |
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| 252 | "cell_type": "code", |
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| 253 | "execution_count": 3, |
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| 254 | "metadata": { |
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| 255 | "id": "WoSjhgrvLbJ_" |
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| 256 | }, |
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| 257 | "outputs": [], |
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| 258 | "source": [ |
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| 259 | "import pennylane as qml\n", |
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| 260 | "from matplotlib import pyplot as plt\n", |
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| 261 | "from pennylane import numpy as np\n", |
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| 262 | "import scipy\n", |
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| 263 | "import networkx as nx\n", |
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| 264 | "import copy" |
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| 265 | ] |
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| 266 | }, |
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| 267 | { |
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| 268 | "cell_type": "markdown", |
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| 269 | "metadata": { |
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| 270 | "id": "SgJGdX-wLbJ_" |
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| 271 | }, |
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| 272 | "source": [ |
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| 273 | "We also define some fixed values that are used throughout the\n", |
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| 274 | "simulation.\n" |
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| 275 | ] |
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| 276 | }, |
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| 277 | { |
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| 278 | "cell_type": "code", |
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| 279 | "execution_count": 4, |
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| 280 | "metadata": { |
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| 281 | "id": "FCZr2sEhLbJ_" |
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| 282 | }, |
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| 283 | "outputs": [], |
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| 284 | "source": [ |
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| 285 | "qubit_number = 4\n", |
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| 286 | "qubits = range(qubit_number)" |
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| 287 | ] |
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| 288 | }, |
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| 289 | { |
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| 290 | "cell_type": "markdown", |
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| 291 | "metadata": { |
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| 292 | "id": "LLRw3rbhLbJ_" |
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| 293 | }, |
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| 294 | "source": [ |
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| 295 | "In this simulation, we don\\'t have quantum data readily available to\n", |
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| 296 | "pass into the QGRNN, so we have to generate it ourselves. To do this, we\n", |
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| 297 | "must have knowledge of the target interaction graph and the target\n", |
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| 298 | "Hamiltonian.\n", |
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| 299 | "\n", |
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| 300 | "Let us use the following cyclic graph as the target interaction graph of\n", |
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| 301 | "the Ising Hamiltonian:\n" |
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| 302 | ] |
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| 303 | }, |
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| 304 | { |
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| 305 | "cell_type": "code", |
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| 306 | "execution_count": 5, |
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| 307 | "metadata": { |
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| 308 | "colab": { |
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| 309 | "base_uri": "https://localhost:8080/", |
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| 310 | "height": 534 |
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| 311 | }, |
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| 312 | "id": "BBidusUTLbJ_", |
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| 313 | "outputId": "0c46fd84-ae0e-4151-fa1c-e895585b1e2a" |
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| 314 | }, |
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| 315 | "outputs": [ |
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| 316 | { |
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| 317 | "output_type": "stream", |
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| 318 | "name": "stdout", |
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| 319 | "text": [ |
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| 320 | "Edges: [(0, 1), (0, 3), (1, 2), (2, 3)]\n" |
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| 321 | ] |
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| 322 | }, |
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| 323 | { |
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| 324 | "output_type": "display_data", |
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| 325 | "data": { |
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| 326 | "text/plain": [ |
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| 327 | "<Figure size 640x480 with 1 Axes>" |
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| 328 | ], |
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| 329 | "image/png": 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\n" |
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| 330 | }, |
||
| 331 | "metadata": {} |
||
| 332 | } |
||
| 333 | ], |
||
| 334 | "source": [ |
||
| 335 | "ising_graph = nx.cycle_graph(qubit_number)\n", |
||
| 336 | "\n", |
||
| 337 | "print(f\"Edges: {ising_graph.edges}\")\n", |
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| 338 | "nx.draw(ising_graph)" |
||
| 339 | ] |
||
| 340 | }, |
||
| 341 | { |
||
| 342 | "cell_type": "markdown", |
||
| 343 | "metadata": { |
||
| 344 | "id": "U1Dum11GLbJ_" |
||
| 345 | }, |
||
| 346 | "source": [ |
||
| 347 | "We can then initialize the \"unknown\" target parameters that describe the\n", |
||
| 348 | "target Hamiltonian,\n", |
||
| 349 | "$\\boldsymbol\\alpha \\ = \\ \\{\\alpha^{(1)}, \\ \\alpha^{(2)}\\}$. Recall from\n", |
||
| 350 | "the introduction that we have defined our parametrized Ising Hamiltonian\n", |
||
| 351 | "to be of the form:\n", |
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| 352 | "\n", |
||
| 353 | "$$\\hat{H}_{\\text{Ising}}(\\boldsymbol\\theta) \\ = \\ \\displaystyle\\sum_{(i, j) \\in E}\n", |
||
| 354 | "\\theta_{ij}^{(1)} Z_{i} Z_{j} \\ + \\ \\displaystyle\\sum_{i} \\theta_{i}^{(2)} Z_{i} \\ + \\\n", |
||
| 355 | "\\displaystyle\\sum_{i} X_{i},$$\n", |
||
| 356 | "\n", |
||
| 357 | "where $E$ is the set of edges in the interaction graph, and $X_i$ and\n", |
||
| 358 | "$Z_i$ are the Pauli-X and Pauli-Z on the $i$-th qubit.\n", |
||
| 359 | "\n", |
||
| 360 | "For this tutorial, we choose the target parameters by sampling from a\n", |
||
| 361 | "uniform probability distribution ranging from $-2$ to $2$, with\n", |
||
| 362 | "two-decimal precision.\n" |
||
| 363 | ] |
||
| 364 | }, |
||
| 365 | { |
||
| 366 | "cell_type": "code", |
||
| 367 | "execution_count": 6, |
||
| 368 | "metadata": { |
||
| 369 | "id": "dguKYJmcLbJ_" |
||
| 370 | }, |
||
| 371 | "outputs": [], |
||
| 372 | "source": [ |
||
| 373 | "target_weights = [0.56, 1.24, 1.67, -0.79]\n", |
||
| 374 | "target_bias = [-1.44, -1.43, 1.18, -0.93]" |
||
| 375 | ] |
||
| 376 | }, |
||
| 377 | { |
||
| 378 | "cell_type": "markdown", |
||
| 379 | "metadata": { |
||
| 380 | "id": "AiyO6svHLbJ_" |
||
| 381 | }, |
||
| 382 | "source": [ |
||
| 383 | "In theory, these parameters can be any value we want, provided they are\n", |
||
| 384 | "reasonably small enough that the QGRNN can reach them in a tractable\n", |
||
| 385 | "number of optimization steps. In `matrix_params`, the first list\n", |
||
| 386 | "represents the $ZZ$ interaction parameters and the second list\n", |
||
| 387 | "represents the single-qubit $Z$ parameters.\n", |
||
| 388 | "\n", |
||
| 389 | "Finally, we use this information to generate the matrix form of the\n", |
||
| 390 | "Ising model Hamiltonian in the computational basis:\n" |
||
| 391 | ] |
||
| 392 | }, |
||
| 393 | { |
||
| 394 | "cell_type": "code", |
||
| 395 | "execution_count": 7, |
||
| 396 | "metadata": { |
||
| 397 | "colab": { |
||
| 398 | "base_uri": "https://localhost:8080/", |
||
| 399 | "height": 437 |
||
| 400 | }, |
||
| 401 | "id": "xDSq-w2_LbJ_", |
||
| 402 | "outputId": "b1fd5ee7-6895-4566-ee8a-309d7242394a" |
||
| 403 | }, |
||
| 404 | "outputs": [ |
||
| 405 | { |
||
| 406 | "output_type": "display_data", |
||
| 407 | "data": { |
||
| 408 | "text/plain": [ |
||
| 409 | "<Figure size 480x480 with 1 Axes>" |
||
| 410 | ], |
||
| 411 | "image/png": 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\n" |
||
| 412 | }, |
||
| 413 | "metadata": {} |
||
| 414 | } |
||
| 415 | ], |
||
| 416 | "source": [ |
||
| 417 | "def create_hamiltonian_matrix(n_qubits, graph, weights, bias):\n", |
||
| 418 | "\n", |
||
| 419 | " full_matrix = np.zeros((2 ** n_qubits, 2 ** n_qubits))\n", |
||
| 420 | "\n", |
||
| 421 | " # Creates the interaction component of the Hamiltonian\n", |
||
| 422 | " for i, edge in enumerate(graph.edges):\n", |
||
| 423 | " interaction_term = 1\n", |
||
| 424 | " for qubit in range(0, n_qubits):\n", |
||
| 425 | " if qubit in edge:\n", |
||
| 426 | " interaction_term = np.kron(interaction_term, qml.matrix(qml.PauliZ)(0))\n", |
||
| 427 | " else:\n", |
||
| 428 | " interaction_term = np.kron(interaction_term, np.identity(2))\n", |
||
| 429 | " full_matrix += weights[i] * interaction_term\n", |
||
| 430 | "\n", |
||
| 431 | " # Creates the bias components of the matrix\n", |
||
| 432 | " for i in range(0, n_qubits):\n", |
||
| 433 | " z_term = x_term = 1\n", |
||
| 434 | " for j in range(0, n_qubits):\n", |
||
| 435 | " if j == i:\n", |
||
| 436 | " z_term = np.kron(z_term, qml.matrix(qml.PauliZ)(0))\n", |
||
| 437 | " x_term = np.kron(x_term, qml.matrix(qml.PauliX)(0))\n", |
||
| 438 | " else:\n", |
||
| 439 | " z_term = np.kron(z_term, np.identity(2))\n", |
||
| 440 | " x_term = np.kron(x_term, np.identity(2))\n", |
||
| 441 | " full_matrix += bias[i] * z_term + x_term\n", |
||
| 442 | "\n", |
||
| 443 | " return full_matrix\n", |
||
| 444 | "\n", |
||
| 445 | "\n", |
||
| 446 | "# Prints a visual representation of the Hamiltonian matrix\n", |
||
| 447 | "ham_matrix = create_hamiltonian_matrix(qubit_number, ising_graph, target_weights, target_bias)\n", |
||
| 448 | "plt.matshow(ham_matrix, cmap=\"hot\")\n", |
||
| 449 | "plt.show()" |
||
| 450 | ] |
||
| 451 | }, |
||
| 452 | { |
||
| 453 | "cell_type": "markdown", |
||
| 454 | "metadata": { |
||
| 455 | "id": "EZSqkuz8LbJ_" |
||
| 456 | }, |
||
| 457 | "source": [ |
||
| 458 | "Preparing Quantum Data\n", |
||
| 459 | "======================\n" |
||
| 460 | ] |
||
| 461 | }, |
||
| 462 | { |
||
| 463 | "cell_type": "markdown", |
||
| 464 | "metadata": { |
||
| 465 | "id": "PmSFvIn5LbJ_" |
||
| 466 | }, |
||
| 467 | "source": [ |
||
| 468 | "The collection of quantum data needed to run the QGRNN has two\n", |
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| 469 | "components: (i) copies of a low-energy state, and (ii) a collection of\n", |
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| 470 | "time-evolved states, each of which are simply the low-energy state\n", |
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| 471 | "evolved to different times. The following is a low-energy state of the\n", |
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| 472 | "target Hamiltonian:\n" |
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| 473 | ] |
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| 474 | }, |
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| 475 | { |
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| 476 | "cell_type": "code", |
||
| 477 | "execution_count": 8, |
||
| 478 | "metadata": { |
||
| 479 | "id": "TvhFkhYOLbJ_" |
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| 480 | }, |
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| 481 | "outputs": [], |
||
| 482 | "source": [ |
||
| 483 | "low_energy_state = [\n", |
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| 484 | " (-0.054661080280306085 + 0.016713907320174026j),\n", |
||
| 485 | " (0.12290003656489545 - 0.03758500591109822j),\n", |
||
| 486 | " (0.3649337966440005 - 0.11158863596657455j),\n", |
||
| 487 | " (-0.8205175732627094 + 0.25093231967092877j),\n", |
||
| 488 | " (0.010369790825776609 - 0.0031706387262686003j),\n", |
||
| 489 | " (-0.02331544978544721 + 0.007129899300113728j),\n", |
||
| 490 | " (-0.06923183949694546 + 0.0211684344103713j),\n", |
||
| 491 | " (0.15566094863283836 - 0.04760201916285508j),\n", |
||
| 492 | " (0.014520590919500158 - 0.004441887836078486j),\n", |
||
| 493 | " (-0.032648113364535575 + 0.009988590222879195j),\n", |
||
| 494 | " (-0.09694382811137187 + 0.02965579457620536j),\n", |
||
| 495 | " (0.21796861485652747 - 0.06668776658411019j),\n", |
||
| 496 | " (-0.0027547112135013247 + 0.0008426289322652901j),\n", |
||
| 497 | " (0.006193695872468649 - 0.0018948418969390599j),\n", |
||
| 498 | " (0.018391279795405405 - 0.005625722994009138j),\n", |
||
| 499 | " (-0.041350974715649635 + 0.012650711602265649j),\n", |
||
| 500 | "]" |
||
| 501 | ] |
||
| 502 | }, |
||
| 503 | { |
||
| 504 | "cell_type": "markdown", |
||
| 505 | "metadata": { |
||
| 506 | "id": "nZ5F5vM7LbJ_" |
||
| 507 | }, |
||
| 508 | "source": [ |
||
| 509 | "This state can be obtained by using a decoupled version of the\n", |
||
| 510 | "`Variational Quantum Eigensolver </demos/tutorial_vqe>`{.interpreted-text\n", |
||
| 511 | "role=\"doc\"} algorithm (VQE). Essentially, we choose a VQE ansatz such\n", |
||
| 512 | "that the circuit cannot learn the exact ground state, but it can get\n", |
||
| 513 | "fairly close. Another way to arrive at the same result is to perform VQE\n", |
||
| 514 | "with a reasonable ansatz, but to terminate the algorithm before it\n", |
||
| 515 | "converges to the ground state. If we used the exact ground state\n", |
||
| 516 | "$|\\psi_0\\rangle$, the time-dependence would be trivial and the data\n", |
||
| 517 | "would not provide enough information about the Hamiltonian parameters.\n", |
||
| 518 | "\n", |
||
| 519 | "We can verify that this is a low-energy state by numerically finding the\n", |
||
| 520 | "lowest eigenvalue of the Hamiltonian and comparing it to the energy\n", |
||
| 521 | "expectation of this low-energy state:\n" |
||
| 522 | ] |
||
| 523 | }, |
||
| 524 | { |
||
| 525 | "cell_type": "code", |
||
| 526 | "execution_count": 9, |
||
| 527 | "metadata": { |
||
| 528 | "colab": { |
||
| 529 | "base_uri": "https://localhost:8080/", |
||
| 530 | "height": 0 |
||
| 531 | }, |
||
| 532 | "id": "ZBpXVqTtLbJ_", |
||
| 533 | "outputId": "4d94ef57-a6d0-4476-c92d-b08195c47129" |
||
| 534 | }, |
||
| 535 | "outputs": [ |
||
| 536 | { |
||
| 537 | "output_type": "stream", |
||
| 538 | "name": "stdout", |
||
| 539 | "text": [ |
||
| 540 | "Energy Expectation: -7.244508985189116\n", |
||
| 541 | "Ground State Energy: -7.330689661291244\n" |
||
| 542 | ] |
||
| 543 | } |
||
| 544 | ], |
||
| 545 | "source": [ |
||
| 546 | "res = np.vdot(low_energy_state, (ham_matrix @ low_energy_state))\n", |
||
| 547 | "energy_exp = np.real_if_close(res)\n", |
||
| 548 | "print(f\"Energy Expectation: {energy_exp}\")\n", |
||
| 549 | "\n", |
||
| 550 | "\n", |
||
| 551 | "ground_state_energy = np.real_if_close(min(np.linalg.eig(ham_matrix)[0]))\n", |
||
| 552 | "print(f\"Ground State Energy: {ground_state_energy}\")" |
||
| 553 | ] |
||
| 554 | }, |
||
| 555 | { |
||
| 556 | "cell_type": "markdown", |
||
| 557 | "metadata": { |
||
| 558 | "id": "dQRr5pFSLbKA" |
||
| 559 | }, |
||
| 560 | "source": [ |
||
| 561 | "We have in fact found a low-energy, non-ground state, as the energy\n", |
||
| 562 | "expectation is slightly greater than the energy of the true ground\n", |
||
| 563 | "state. This, however, is only half of the information we need. We also\n", |
||
| 564 | "require a collection of time-evolved, low-energy states. Evolving the\n", |
||
| 565 | "low-energy state forward in time is fairly straightforward: all we have\n", |
||
| 566 | "to do is multiply the initial state by a time-evolution unitary. This\n", |
||
| 567 | "operation can be defined as a custom gate in PennyLane:\n" |
||
| 568 | ] |
||
| 569 | }, |
||
| 570 | { |
||
| 571 | "cell_type": "code", |
||
| 572 | "execution_count": 10, |
||
| 573 | "metadata": { |
||
| 574 | "id": "zxIkTaPiLbKA" |
||
| 575 | }, |
||
| 576 | "outputs": [], |
||
| 577 | "source": [ |
||
| 578 | "def state_evolve(hamiltonian, qubits, time):\n", |
||
| 579 | "\n", |
||
| 580 | " U = scipy.linalg.expm(-1j * hamiltonian * time)\n", |
||
| 581 | " qml.QubitUnitary(U, wires=qubits)" |
||
| 582 | ] |
||
| 583 | }, |
||
| 584 | { |
||
| 585 | "cell_type": "markdown", |
||
| 586 | "metadata": { |
||
| 587 | "id": "auVMu4Q4LbKA" |
||
| 588 | }, |
||
| 589 | "source": [ |
||
| 590 | "We don\\'t actually generate time-evolved quantum data quite yet, but we\n", |
||
| 591 | "now have all the pieces required for its preparation.\n" |
||
| 592 | ] |
||
| 593 | }, |
||
| 594 | { |
||
| 595 | "cell_type": "markdown", |
||
| 596 | "metadata": { |
||
| 597 | "id": "XUMWybNQLbKA" |
||
| 598 | }, |
||
| 599 | "source": [ |
||
| 600 | "Learning the Hamiltonian\n", |
||
| 601 | "========================\n" |
||
| 602 | ] |
||
| 603 | }, |
||
| 604 | { |
||
| 605 | "cell_type": "markdown", |
||
| 606 | "metadata": { |
||
| 607 | "id": "TrYpMK0NLbKA" |
||
| 608 | }, |
||
| 609 | "source": [ |
||
| 610 | "With the quantum data defined, we are able to construct the QGRNN and\n", |
||
| 611 | "learn the target Hamiltonian. Each of the exponentiated Hamiltonians in\n", |
||
| 612 | "the QGRNN ansatz, $\\hat{H}^{j}_{\\text{Ising}}(\\boldsymbol\\mu)$, are the\n", |
||
| 613 | "$ZZ$, $Z$, and $X$ terms from the Ising Hamiltonian. This gives:\n" |
||
| 614 | ] |
||
| 615 | }, |
||
| 616 | { |
||
| 617 | "cell_type": "code", |
||
| 618 | "execution_count": 11, |
||
| 619 | "metadata": { |
||
| 620 | "id": "RewvANn3LbKA" |
||
| 621 | }, |
||
| 622 | "outputs": [], |
||
| 623 | "source": [ |
||
| 624 | "def qgrnn_layer(weights, bias, qubits, graph, trotter_step):\n", |
||
| 625 | "\n", |
||
| 626 | " # Applies a layer of RZZ gates (based on a graph)\n", |
||
| 627 | " for i, edge in enumerate(graph.edges):\n", |
||
| 628 | " qml.MultiRZ(2 * weights[i] * trotter_step, wires=(edge[0], edge[1]))\n", |
||
| 629 | "\n", |
||
| 630 | " # Applies a layer of RZ gates\n", |
||
| 631 | " for i, qubit in enumerate(qubits):\n", |
||
| 632 | " qml.RZ(2 * bias[i] * trotter_step, wires=qubit)\n", |
||
| 633 | "\n", |
||
| 634 | " # Applies a layer of RX gates\n", |
||
| 635 | " for qubit in qubits:\n", |
||
| 636 | " qml.RX(2 * trotter_step, wires=qubit)" |
||
| 637 | ] |
||
| 638 | }, |
||
| 639 | { |
||
| 640 | "cell_type": "markdown", |
||
| 641 | "metadata": { |
||
| 642 | "id": "ZcYEXhauLbKA" |
||
| 643 | }, |
||
| 644 | "source": [ |
||
| 645 | "As was mentioned in the first section, the QGRNN has two registers. In\n", |
||
| 646 | "one register, some piece of quantum data $|\\psi(t)\\rangle$ is prepared\n", |
||
| 647 | "and in the other we have\n", |
||
| 648 | "$U_{H}(\\boldsymbol\\mu, \\ \\Delta) |\\psi_0\\rangle$. We need a way to\n", |
||
| 649 | "measure the similarity between these states. This can be done by using\n", |
||
| 650 | "the fidelity, which is simply the modulus squared of the inner product\n", |
||
| 651 | "between the states,\n", |
||
| 652 | "$| \\langle \\psi(t) | U_{H}(\\Delta, \\ \\boldsymbol\\mu) |\\psi_0\\rangle |^2$.\n", |
||
| 653 | "To calculate this value, we use a [SWAP\n", |
||
| 654 | "test](https://en.wikipedia.org/wiki/Swap_test) between the registers:\n" |
||
| 655 | ] |
||
| 656 | }, |
||
| 657 | { |
||
| 658 | "cell_type": "code", |
||
| 659 | "execution_count": 12, |
||
| 660 | "metadata": { |
||
| 661 | "id": "-5RqUqBKLbKA" |
||
| 662 | }, |
||
| 663 | "outputs": [], |
||
| 664 | "source": [ |
||
| 665 | "def swap_test(control, register1, register2):\n", |
||
| 666 | "\n", |
||
| 667 | " qml.Hadamard(wires=control)\n", |
||
| 668 | " for reg1_qubit, reg2_qubit in zip(register1, register2):\n", |
||
| 669 | " qml.CSWAP(wires=(control, reg1_qubit, reg2_qubit))\n", |
||
| 670 | " qml.Hadamard(wires=control)" |
||
| 671 | ] |
||
| 672 | }, |
||
| 673 | { |
||
| 674 | "cell_type": "markdown", |
||
| 675 | "metadata": { |
||
| 676 | "id": "To4GLL8VLbKA" |
||
| 677 | }, |
||
| 678 | "source": [ |
||
| 679 | "After performing this procedure, the value returned from a measurement\n", |
||
| 680 | "of the circuit is $\\langle Z \\rangle$, with respect to the `control`\n", |
||
| 681 | "qubit. The probability of measuring the $|0\\rangle$ state in this\n", |
||
| 682 | "control qubit is related to both the fidelity between registers and\n", |
||
| 683 | "$\\langle Z \\rangle$. Thus, with a bit of algebra, we find that\n", |
||
| 684 | "$\\langle Z \\rangle$ is equal to the fidelity.\n", |
||
| 685 | "\n", |
||
| 686 | "Before creating the full QGRNN and the cost function, we define a few\n", |
||
| 687 | "more fixed values. Among these is a \\\"guessed\\\" interaction graph, which\n", |
||
| 688 | "we set to be a [complete\n", |
||
| 689 | "graph](https://en.wikipedia.org/wiki/Complete_graph). This choice is\n", |
||
| 690 | "motivated by the fact that any target interaction graph will be a\n", |
||
| 691 | "subgraph of this initial guess. Part of the idea behind the QGRNN is\n", |
||
| 692 | "that we don't know the interaction graph, and it has to be learned. In\n", |
||
| 693 | "this case, the graph is learned *automatically* as the target parameters\n", |
||
| 694 | "are optimized. The $\\boldsymbol\\mu$ parameters that correspond to edges\n", |
||
| 695 | "that don\\'t exist in the target graph will simply approach $0$.\n" |
||
| 696 | ] |
||
| 697 | }, |
||
| 698 | { |
||
| 699 | "cell_type": "code", |
||
| 700 | "execution_count": 13, |
||
| 701 | "metadata": { |
||
| 702 | "colab": { |
||
| 703 | "base_uri": "https://localhost:8080/", |
||
| 704 | "height": 534 |
||
| 705 | }, |
||
| 706 | "id": "C5JNAOMSLbKA", |
||
| 707 | "outputId": "7d2ccda3-2603-4d20-a849-074bb12547ee" |
||
| 708 | }, |
||
| 709 | "outputs": [ |
||
| 710 | { |
||
| 711 | "output_type": "stream", |
||
| 712 | "name": "stdout", |
||
| 713 | "text": [ |
||
| 714 | "Edges: [(4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)]\n" |
||
| 715 | ] |
||
| 716 | }, |
||
| 717 | { |
||
| 718 | "output_type": "display_data", |
||
| 719 | "data": { |
||
| 720 | "text/plain": [ |
||
| 721 | "<Figure size 640x480 with 1 Axes>" |
||
| 722 | ], |
||
| 723 | "image/png": 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\n" |
||
| 724 | }, |
||
| 725 | "metadata": {} |
||
| 726 | } |
||
| 727 | ], |
||
| 728 | "source": [ |
||
| 729 | "# Defines some fixed values\n", |
||
| 730 | "\n", |
||
| 731 | "reg1 = tuple(range(qubit_number)) # First qubit register\n", |
||
| 732 | "reg2 = tuple(range(qubit_number, 2 * qubit_number)) # Second qubit register\n", |
||
| 733 | "\n", |
||
| 734 | "control = 2 * qubit_number # Index of control qubit\n", |
||
| 735 | "trotter_step = 0.01 # Trotter step size\n", |
||
| 736 | "\n", |
||
| 737 | "# Defines the interaction graph for the new qubit system\n", |
||
| 738 | "\n", |
||
| 739 | "new_ising_graph = nx.complete_graph(reg2)\n", |
||
| 740 | "\n", |
||
| 741 | "print(f\"Edges: {new_ising_graph.edges}\")\n", |
||
| 742 | "nx.draw(new_ising_graph)" |
||
| 743 | ] |
||
| 744 | }, |
||
| 745 | { |
||
| 746 | "cell_type": "markdown", |
||
| 747 | "metadata": { |
||
| 748 | "id": "4fc6XyLdLbKA" |
||
| 749 | }, |
||
| 750 | "source": [ |
||
| 751 | "With this done, we implement the QGRNN circuit for some given time\n", |
||
| 752 | "value:\n" |
||
| 753 | ] |
||
| 754 | }, |
||
| 755 | { |
||
| 756 | "cell_type": "code", |
||
| 757 | "execution_count": 14, |
||
| 758 | "metadata": { |
||
| 759 | "id": "iu4hcd2oLbKA" |
||
| 760 | }, |
||
| 761 | "outputs": [], |
||
| 762 | "source": [ |
||
| 763 | "def qgrnn(weights, bias, time=None):\n", |
||
| 764 | "\n", |
||
| 765 | " # Prepares the low energy state in the two registers\n", |
||
| 766 | " qml.QubitStateVector(np.kron(low_energy_state, low_energy_state), wires=reg1 + reg2)\n", |
||
| 767 | "\n", |
||
| 768 | " # Evolves the first qubit register with the time-evolution circuit to\n", |
||
| 769 | " # prepare a piece of quantum data\n", |
||
| 770 | " state_evolve(ham_matrix, reg1, time)\n", |
||
| 771 | "\n", |
||
| 772 | " # Applies the QGRNN layers to the second qubit register\n", |
||
| 773 | " depth = time / trotter_step # P = t/Delta\n", |
||
| 774 | " for _ in range(0, int(depth)):\n", |
||
| 775 | " qgrnn_layer(weights, bias, reg2, new_ising_graph, trotter_step)\n", |
||
| 776 | "\n", |
||
| 777 | " # Applies the SWAP test between the registers\n", |
||
| 778 | " swap_test(control, reg1, reg2)\n", |
||
| 779 | "\n", |
||
| 780 | " # Returns the results of the SWAP test\n", |
||
| 781 | " return qml.expval(qml.PauliZ(control))" |
||
| 782 | ] |
||
| 783 | }, |
||
| 784 | { |
||
| 785 | "cell_type": "markdown", |
||
| 786 | "metadata": { |
||
| 787 | "id": "zUGsNfnsLbKA" |
||
| 788 | }, |
||
| 789 | "source": [ |
||
| 790 | "We have the full QGRNN circuit, but we still need to define a cost\n", |
||
| 791 | "function. We know that\n", |
||
| 792 | "$| \\langle \\psi(t) | U_{H}(\\boldsymbol\\mu, \\ \\Delta) |\\psi_0\\rangle |^2$\n", |
||
| 793 | "approaches $1$ as the states become more similar and approaches $0$ as\n", |
||
| 794 | "the states become orthogonal. Thus, we choose to minimize the quantity\n", |
||
| 795 | "$-| \\langle \\psi(t) | U_{H}(\\boldsymbol\\mu, \\ \\Delta) |\\psi_0\\rangle |^2$.\n", |
||
| 796 | "Since we are interested in calculating this value for many different\n", |
||
| 797 | "pieces of quantum data, the final cost function is the average negative\n", |
||
| 798 | "fidelity\\* between registers:\n", |
||
| 799 | "\n", |
||
| 800 | "$$\\mathcal{L}(\\boldsymbol\\mu, \\ \\Delta) \\ = \\ - \\frac{1}{N} \\displaystyle\\sum_{i \\ = \\ 1}^{N} |\n", |
||
| 801 | "\\langle \\psi(t_i) | \\ U_{H}(\\boldsymbol\\mu, \\ \\Delta) \\ |\\psi_0\\rangle |^2,$$\n", |
||
| 802 | "\n", |
||
| 803 | "where we use $N$ pieces of quantum data.\n", |
||
| 804 | "\n", |
||
| 805 | "Before creating the cost function, we must define a few more fixed\n", |
||
| 806 | "variables:\n" |
||
| 807 | ] |
||
| 808 | }, |
||
| 809 | { |
||
| 810 | "cell_type": "code", |
||
| 811 | "execution_count": 15, |
||
| 812 | "metadata": { |
||
| 813 | "id": "R7jK3kMqLbKB" |
||
| 814 | }, |
||
| 815 | "outputs": [], |
||
| 816 | "source": [ |
||
| 817 | "N = 15 # The number of pieces of quantum data that are used for each step\n", |
||
| 818 | "max_time = 0.1 # The maximum value of time that can be used for quantum data" |
||
| 819 | ] |
||
| 820 | }, |
||
| 821 | { |
||
| 822 | "cell_type": "markdown", |
||
| 823 | "metadata": { |
||
| 824 | "id": "X_e9a1jVLbKB" |
||
| 825 | }, |
||
| 826 | "source": [ |
||
| 827 | "We then define the negative fidelity cost function:\n" |
||
| 828 | ] |
||
| 829 | }, |
||
| 830 | { |
||
| 831 | "cell_type": "code", |
||
| 832 | "execution_count": 16, |
||
| 833 | "metadata": { |
||
| 834 | "id": "qizc8MsqLbKB" |
||
| 835 | }, |
||
| 836 | "outputs": [], |
||
| 837 | "source": [ |
||
| 838 | "rng = np.random.default_rng(seed=42)\n", |
||
| 839 | "\n", |
||
| 840 | "def cost_function(weight_params, bias_params):\n", |
||
| 841 | "\n", |
||
| 842 | " # Randomly samples times at which the QGRNN runs\n", |
||
| 843 | " times_sampled = rng.random(size=N) * max_time\n", |
||
| 844 | "\n", |
||
| 845 | " # Cycles through each of the sampled times and calculates the cost\n", |
||
| 846 | " total_cost = 0\n", |
||
| 847 | " for dt in times_sampled:\n", |
||
| 848 | " result = qgrnn_qnode(weight_params, bias_params, time=dt)\n", |
||
| 849 | " total_cost += -1 * result\n", |
||
| 850 | "\n", |
||
| 851 | " return total_cost / N" |
||
| 852 | ] |
||
| 853 | }, |
||
| 854 | { |
||
| 855 | "cell_type": "markdown", |
||
| 856 | "metadata": { |
||
| 857 | "id": "m5WwkvDHLbKB" |
||
| 858 | }, |
||
| 859 | "source": [ |
||
| 860 | "Next we set up for optimization.\n" |
||
| 861 | ] |
||
| 862 | }, |
||
| 863 | { |
||
| 864 | "cell_type": "code", |
||
| 865 | "execution_count": 17, |
||
| 866 | "metadata": { |
||
| 867 | "id": "3bzbEsNJLbKB" |
||
| 868 | }, |
||
| 869 | "outputs": [], |
||
| 870 | "source": [ |
||
| 871 | "# Defines the new device\n", |
||
| 872 | "qgrnn_dev = qml.device(\"default.qubit\", wires=2 * qubit_number + 1)\n", |
||
| 873 | "\n", |
||
| 874 | "# Defines the new QNode\n", |
||
| 875 | "qgrnn_qnode = qml.QNode(qgrnn, qgrnn_dev, interface=\"autograd\")\n", |
||
| 876 | "\n", |
||
| 877 | "steps = 300\n", |
||
| 878 | "\n", |
||
| 879 | "optimizer = qml.AdamOptimizer(stepsize=0.3)\n", |
||
| 880 | "\n", |
||
| 881 | "weights = rng.random(size=len(new_ising_graph.edges), requires_grad=True) - 0.5\n", |
||
| 882 | "bias = rng.random(size=qubit_number, requires_grad=True) - 0.5\n", |
||
| 883 | "\n", |
||
| 884 | "initial_weights = copy.copy(weights)\n", |
||
| 885 | "initial_bias = copy.copy(bias)" |
||
| 886 | ] |
||
| 887 | }, |
||
| 888 | { |
||
| 889 | "cell_type": "markdown", |
||
| 890 | "metadata": { |
||
| 891 | "id": "Hik8L3erLbKB" |
||
| 892 | }, |
||
| 893 | "source": [ |
||
| 894 | "All that remains is executing the optimization loop.\n" |
||
| 895 | ] |
||
| 896 | }, |
||
| 897 | { |
||
| 898 | "cell_type": "code", |
||
| 899 | "execution_count": 18, |
||
| 900 | "metadata": { |
||
| 901 | "colab": { |
||
| 902 | "base_uri": "https://localhost:8080/", |
||
| 903 | "height": 0 |
||
| 904 | }, |
||
| 905 | "id": "gPvXgB7MLbKB", |
||
| 906 | "outputId": "fedfd42d-6a1c-42aa-de6d-a0786615a80f" |
||
| 907 | }, |
||
| 908 | "outputs": [ |
||
| 909 | { |
||
| 910 | "output_type": "stream", |
||
| 911 | "name": "stdout", |
||
| 912 | "text": [ |
||
| 913 | "Cost at Step 0: -0.9803638573791906\n", |
||
| 914 | "Weights at Step 0: [-0.02603926 0.23887338 0.65859458 0.4973626 -0.10582631 0.17562606]\n", |
||
| 915 | "Bias at Step 0: [-0.03885261 -0.01392638 -0.07189834 0.25038114]\n", |
||
| 916 | "---------------------------------------------\n", |
||
| 917 | "Cost at Step 5: -0.996941435016429\n", |
||
| 918 | "Weights at Step 5: [-0.81324145 1.07476846 1.1061694 1.58556358 0.32683696 -0.30451739]\n", |
||
| 919 | "Bias at Step 5: [-0.69258221 -1.04745285 1.06920295 0.4729811 ]\n", |
||
| 920 | "---------------------------------------------\n", |
||
| 921 | "Cost at Step 10: -0.998631791121192\n", |
||
| 922 | "Weights at Step 10: [-0.42326896 0.79838566 0.61651522 1.83801512 -0.18661762 0.09585344]\n", |
||
| 923 | "Bias at Step 10: [-0.14865635 -1.13290709 1.50067486 -0.17013801]\n", |
||
| 924 | "---------------------------------------------\n", |
||
| 925 | "Cost at Step 15: -0.9989288704590784\n", |
||
| 926 | "Weights at Step 15: [-0.01780188 0.51935145 0.84042409 1.67694816 -0.08570868 -0.17039931]\n", |
||
| 927 | "Bias at Step 15: [ 0.14049396 -0.85137402 1.46620821 0.01876606]\n", |
||
| 928 | "---------------------------------------------\n", |
||
| 929 | "Cost at Step 20: -0.9989877966855808\n", |
||
| 930 | "Weights at Step 20: [-0.04606544 0.61564422 1.20996012 1.53724787 0.03102305 -0.38463701]\n", |
||
| 931 | "Bias at Step 20: [-0.15660669 -0.72249342 1.28027729 0.17742167]\n", |
||
| 932 | "---------------------------------------------\n", |
||
| 933 | "Cost at Step 25: -0.9996824290201453\n", |
||
| 934 | "Weights at Step 25: [-0.15679743 0.6818271 1.14305623 1.52203978 -0.28441025 -0.08613302]\n", |
||
| 935 | "Bias at Step 25: [-0.44370877 -0.83552897 1.05067071 -0.21716541]\n", |
||
| 936 | "---------------------------------------------\n", |
||
| 937 | "Cost at Step 30: -0.9997260854147976\n", |
||
| 938 | "Weights at Step 30: [-0.10965584 0.55377334 1.23770269 1.66310423 -0.22452586 -0.16214067]\n", |
||
| 939 | "Bias at Step 30: [-0.42105765 -1.09901301 0.97692017 -0.25131789]\n", |
||
| 940 | "---------------------------------------------\n", |
||
| 941 | "Cost at Step 35: -0.999720785496688\n", |
||
| 942 | "Weights at Step 35: [ 0.05371611 0.40094669 1.34935989 1.75342972 -0.10426342 -0.3430209 ]\n", |
||
| 943 | "Bias at Step 35: [-0.31829145 -1.20422776 1.00699464 -0.19185141]\n", |
||
| 944 | "---------------------------------------------\n", |
||
| 945 | "Cost at Step 40: -0.9998268061103935\n", |
||
| 946 | "Weights at Step 40: [ 0.14696981 0.42062742 1.30031302 1.74223276 -0.23815364 -0.30014355]\n", |
||
| 947 | "Bias at Step 40: [-0.42731939 -1.12652083 1.05747061 -0.37945587]\n", |
||
| 948 | "---------------------------------------------\n", |
||
| 949 | "Cost at Step 45: -0.999854979792434\n", |
||
| 950 | "Weights at Step 45: [ 0.13980781 0.52546791 1.3589372 1.7401287 -0.24171811 -0.37131231]\n", |
||
| 951 | "Bias at Step 45: [-0.66159346 -1.08685666 1.08351978 -0.44685154]\n", |
||
| 952 | "---------------------------------------------\n", |
||
| 953 | "Cost at Step 50: -0.9999300663096155\n", |
||
| 954 | "Weights at Step 50: [ 0.24206772 0.42578458 1.38259253 1.73759016 -0.17042668 -0.47614044]\n", |
||
| 955 | "Bias at Step 50: [-0.67166662 -1.11916085 1.01155301 -0.45307945]\n", |
||
| 956 | "---------------------------------------------\n", |
||
| 957 | "Cost at Step 55: -0.9999158295165241\n", |
||
| 958 | "Weights at Step 55: [ 0.3107698 0.32643162 1.30955235 1.7661931 -0.18233325 -0.47326321]\n", |
||
| 959 | "Bias at Step 55: [-0.67871142 -1.20848815 0.93740127 -0.56458235]\n", |
||
| 960 | "---------------------------------------------\n", |
||
| 961 | "Cost at Step 60: -0.9999308699951126\n", |
||
| 962 | "Weights at Step 60: [ 0.28372115 0.36451651 1.33221677 1.80954685 -0.13589698 -0.54109403]\n", |
||
| 963 | "Bias at Step 60: [-0.83208754 -1.2754807 0.93311571 -0.60834148]\n", |
||
| 964 | "---------------------------------------------\n", |
||
| 965 | "Cost at Step 65: -0.999959884466806\n", |
||
| 966 | "Weights at Step 65: [ 0.36974336 0.3193621 1.31162222 1.76724553 -0.13338673 -0.583552 ]\n", |
||
| 967 | "Bias at Step 65: [-0.88515714 -1.21379111 0.91018067 -0.66955516]\n", |
||
| 968 | "---------------------------------------------\n", |
||
| 969 | "Cost at Step 70: -0.9999592904739002\n", |
||
| 970 | "Weights at Step 70: [ 0.41857481 0.28942942 1.30436789 1.7750691 -0.10866857 -0.63517285]\n", |
||
| 971 | "Bias at Step 70: [-0.94301792 -1.21574698 0.92016109 -0.71763258]\n", |
||
| 972 | "---------------------------------------------\n", |
||
| 973 | "Cost at Step 75: -0.9999713292245083\n", |
||
| 974 | "Weights at Step 75: [ 0.42211115 0.27516391 1.30028462 1.8124991 -0.08626634 -0.66806052]\n", |
||
| 975 | "Bias at Step 75: [-1.02418273 -1.27352011 0.92542777 -0.77781084]\n", |
||
| 976 | "---------------------------------------------\n", |
||
| 977 | "Cost at Step 80: -0.9999807570513847\n", |
||
| 978 | "Weights at Step 80: [ 0.46823298 0.22196986 1.29819846 1.79885961 -0.07375557 -0.69509066]\n", |
||
| 979 | "Bias at Step 80: [-1.06297822 -1.27152207 0.89615772 -0.8267624 ]\n", |
||
| 980 | "---------------------------------------------\n", |
||
| 981 | "Cost at Step 85: -0.9999834576872019\n", |
||
| 982 | "Weights at Step 85: [ 0.48867472 0.20430434 1.31813101 1.79817224 -0.05572355 -0.73012137]\n", |
||
| 983 | "Bias at Step 85: [-1.12351456 -1.26935382 0.8986494 -0.85965751]\n", |
||
| 984 | "---------------------------------------------\n", |
||
| 985 | "Cost at Step 90: -0.9999848348340454\n", |
||
| 986 | "Weights at Step 90: [ 0.50938179 0.1815515 1.31872181 1.81333952 -0.057935 -0.74241479]\n", |
||
| 987 | "Bias at Step 90: [-1.1648444 -1.28126866 0.91927043 -0.91218166]\n", |
||
| 988 | "---------------------------------------------\n", |
||
| 989 | "Cost at Step 95: -0.9999858404580841\n", |
||
| 990 | "Weights at Step 95: [ 0.53198063 0.14682073 1.32951939 1.81872826 -0.03723321 -0.7724539 ]\n", |
||
| 991 | "Bias at Step 95: [-1.19340463 -1.29189249 0.92624482 -0.93540509]\n", |
||
| 992 | "---------------------------------------------\n", |
||
| 993 | "Cost at Step 100: -0.999985474261986\n", |
||
| 994 | "Weights at Step 100: [ 0.54088845 0.12642269 1.32832267 1.81359343 -0.03539024 -0.78147771]\n", |
||
| 995 | "Bias at Step 100: [-1.23731699 -1.29310153 0.92559765 -0.97196362]\n", |
||
| 996 | "---------------------------------------------\n", |
||
| 997 | "Cost at Step 105: -0.9999849674159557\n", |
||
| 998 | "Weights at Step 105: [ 0.5519262 0.1056115 1.33563262 1.81489961 -0.02321254 -0.80109894]\n", |
||
| 999 | "Bias at Step 105: [-1.26753973 -1.29708393 0.93540955 -0.99148978]\n", |
||
| 1000 | "---------------------------------------------\n", |
||
| 1001 | "Cost at Step 110: -0.9999893366309286\n", |
||
| 1002 | "Weights at Step 110: [ 0.56454314 0.08311651 1.3315906 1.81716268 -0.01884894 -0.81132014]\n", |
||
| 1003 | "Bias at Step 110: [-1.28690663 -1.30170285 0.9490766 -1.01449621]\n", |
||
| 1004 | "---------------------------------------------\n", |
||
| 1005 | "Cost at Step 115: -0.9999910021365572\n", |
||
| 1006 | "Weights at Step 115: [ 0.56547562 0.07231051 1.33386244 1.81426223 -0.010147 -0.82398566]\n", |
||
| 1007 | "Bias at Step 115: [-1.31862939 -1.30416747 0.95796895 -1.02789104]\n", |
||
| 1008 | "---------------------------------------------\n", |
||
| 1009 | "Cost at Step 120: -0.9999892236888762\n", |
||
| 1010 | "Weights at Step 120: [ 0.57542644 0.05174682 1.33008488 1.80845608 -0.00677923 -0.83031403]\n", |
||
| 1011 | "Bias at Step 120: [-1.33280139 -1.30462293 0.96278424 -1.04167386]\n", |
||
| 1012 | "---------------------------------------------\n", |
||
| 1013 | "Cost at Step 125: -0.9999921564251922\n", |
||
| 1014 | "Weights at Step 125: [ 5.74855656e-01 4.41140455e-02 1.33148721e+00 1.81141630e+00\n", |
||
| 1015 | " -1.45922147e-03 -8.37792448e-01]\n", |
||
| 1016 | "Bias at Step 125: [-1.35272154 -1.31313306 0.9769263 -1.05024412]\n", |
||
| 1017 | "---------------------------------------------\n", |
||
| 1018 | "Cost at Step 130: -0.9999896546781485\n", |
||
| 1019 | "Weights at Step 130: [ 5.81148179e-01 3.12881718e-02 1.32648858e+00 1.80476572e+00\n", |
||
| 1020 | " -1.34382543e-04 -8.40486390e-01]\n", |
||
| 1021 | "Bias at Step 130: [-1.36266437 -1.3122935 0.98373353 -1.05698837]\n", |
||
| 1022 | "---------------------------------------------\n", |
||
| 1023 | "Cost at Step 135: -0.999989389789069\n", |
||
| 1024 | "Weights at Step 135: [ 0.58113254 0.02553698 1.32817847 1.8011998 0.00461508 -0.84568556]\n", |
||
| 1025 | "Bias at Step 135: [-1.37617879 -1.31580751 0.99147423 -1.05740398]\n", |
||
| 1026 | "---------------------------------------------\n", |
||
| 1027 | "Cost at Step 140: -0.9999909647373425\n", |
||
| 1028 | "Weights at Step 140: [ 5.84031155e-01 1.80486810e-02 1.32738886e+00 1.80061135e+00\n", |
||
| 1029 | " 1.65406266e-03 -8.43424714e-01]\n", |
||
| 1030 | "Bias at Step 140: [-1.38227067 -1.32143299 0.9995722 -1.06391802]\n", |
||
| 1031 | "---------------------------------------------\n", |
||
| 1032 | "Cost at Step 145: -0.9999882409907013\n", |
||
| 1033 | "Weights at Step 145: [ 0.58696427 0.01254767 1.33270229 1.79866053 0.00341283 -0.84622868]\n", |
||
| 1034 | "Bias at Step 145: [-1.38755681 -1.3249532 1.00609436 -1.06294709]\n", |
||
| 1035 | "---------------------------------------------\n", |
||
| 1036 | "Cost at Step 150: -0.9999882142877635\n", |
||
| 1037 | "Weights at Step 150: [ 5.88782557e-01 9.26749787e-03 1.33607830e+00 1.79737635e+00\n", |
||
| 1038 | " 8.61705971e-04 -8.44657239e-01]\n", |
||
| 1039 | "Bias at Step 150: [-1.39299146 -1.32891592 1.01108686 -1.06590915]\n", |
||
| 1040 | "---------------------------------------------\n", |
||
| 1041 | "Cost at Step 155: -0.9999874507805563\n", |
||
| 1042 | "Weights at Step 155: [ 5.92771490e-01 4.43280303e-03 1.34076089e+00 1.79756493e+00\n", |
||
| 1043 | " -5.09973154e-04 -8.44577799e-01]\n", |
||
| 1044 | "Bias at Step 155: [-1.39410552 -1.33352314 1.01473829 -1.06769553]\n", |
||
| 1045 | "---------------------------------------------\n", |
||
| 1046 | "Cost at Step 160: -0.9999873005360721\n", |
||
| 1047 | "Weights at Step 160: [ 0.59576847 0.00268011 1.34849234 1.79906537 -0.00323521 -0.84434153]\n", |
||
| 1048 | "Bias at Step 160: [-1.3972048 -1.33712094 1.01690619 -1.07141221]\n", |
||
| 1049 | "---------------------------------------------\n", |
||
| 1050 | "Cost at Step 165: -0.9999874561658166\n", |
||
| 1051 | "Weights at Step 165: [ 5.98598863e-01 2.45970557e-04 1.34798706e+00 1.79893319e+00\n", |
||
| 1052 | " -3.96831663e-03 -8.44402361e-01]\n", |
||
| 1053 | "Bias at Step 165: [-1.39840678 -1.3400862 1.01678644 -1.07427356]\n", |
||
| 1054 | "---------------------------------------------\n", |
||
| 1055 | "Cost at Step 170: -0.999988664299768\n", |
||
| 1056 | "Weights at Step 170: [ 0.59556051 0.00301936 1.33720573 1.79440069 0.00246633 -0.84842527]\n", |
||
| 1057 | "Bias at Step 170: [-1.40433835 -1.34054744 1.01793737 -1.06932111]\n", |
||
| 1058 | "---------------------------------------------\n", |
||
| 1059 | "Cost at Step 175: -0.9999876765392001\n", |
||
| 1060 | "Weights at Step 175: [ 0.59507328 0.00398439 1.32779833 1.78879332 0.0043839 -0.84887631]\n", |
||
| 1061 | "Bias at Step 175: [-1.40733842 -1.33831837 1.01934095 -1.06713091]\n", |
||
| 1062 | "---------------------------------------------\n", |
||
| 1063 | "Cost at Step 180: -0.999991473708912\n", |
||
| 1064 | "Weights at Step 180: [ 0.59530716 0.0028657 1.32842831 1.78856307 0.00360646 -0.8477063 ]\n", |
||
| 1065 | "Bias at Step 180: [-1.40721243 -1.34142579 1.02195458 -1.06673697]\n", |
||
| 1066 | "---------------------------------------------\n", |
||
| 1067 | "Cost at Step 185: -0.9999877064247253\n", |
||
| 1068 | "Weights at Step 185: [ 5.97140521e-01 1.29792043e-03 1.33864708e+00 1.78999566e+00\n", |
||
| 1069 | " 1.00191981e-03 -8.46805223e-01]\n", |
||
| 1070 | "Bias at Step 185: [-1.40651854 -1.34399438 1.0238182 -1.06735418]\n", |
||
| 1071 | "---------------------------------------------\n", |
||
| 1072 | "Cost at Step 190: -0.9999945812804165\n", |
||
| 1073 | "Weights at Step 190: [ 6.00719014e-01 -1.62402633e-03 1.34764455e+00 1.79338122e+00\n", |
||
| 1074 | " -4.62571494e-03 -8.43587564e-01]\n", |
||
| 1075 | "Bias at Step 190: [-1.40364571 -1.34711672 1.02503773 -1.07251647]\n", |
||
| 1076 | "---------------------------------------------\n", |
||
| 1077 | "Cost at Step 195: -0.999989989786025\n", |
||
| 1078 | "Weights at Step 195: [ 6.00908732e-01 -1.62012049e-03 1.35014363e+00 1.79380038e+00\n", |
||
| 1079 | " -2.98349731e-03 -8.45624230e-01]\n", |
||
| 1080 | "Bias at Step 195: [-1.40422823 -1.34839724 1.024814 -1.07151688]\n", |
||
| 1081 | "---------------------------------------------\n", |
||
| 1082 | "Cost at Step 200: -0.9999881542248072\n", |
||
| 1083 | "Weights at Step 200: [ 5.99981319e-01 -1.88775908e-04 1.34385950e+00 1.79176993e+00\n", |
||
| 1084 | " -1.80551029e-03 -8.46142226e-01]\n", |
||
| 1085 | "Bias at Step 200: [-1.40629448 -1.34706972 1.02376301 -1.07210325]\n", |
||
| 1086 | "---------------------------------------------\n", |
||
| 1087 | "Cost at Step 205: -0.9999881740456698\n", |
||
| 1088 | "Weights at Step 205: [ 5.99261166e-01 2.51688730e-04 1.33482422e+00 1.78924276e+00\n", |
||
| 1089 | " 2.54997378e-04 -8.46778052e-01]\n", |
||
| 1090 | "Bias at Step 205: [-1.40688336 -1.34566627 1.02294251 -1.07166319]\n", |
||
| 1091 | "---------------------------------------------\n", |
||
| 1092 | "Cost at Step 210: -0.9999849643591882\n", |
||
| 1093 | "Weights at Step 210: [ 0.59556376 0.00290579 1.32712069 1.78534996 0.00520243 -0.8492702 ]\n", |
||
| 1094 | "Bias at Step 210: [-1.4100142 -1.34414691 1.02376047 -1.06630716]\n", |
||
| 1095 | "---------------------------------------------\n", |
||
| 1096 | "Cost at Step 215: -0.9999886489345433\n", |
||
| 1097 | "Weights at Step 215: [ 0.59355916 0.00370791 1.32450808 1.78322703 0.00640902 -0.84894837]\n", |
||
| 1098 | "Bias at Step 215: [-1.41048592 -1.34401778 1.02684022 -1.06268802]\n", |
||
| 1099 | "---------------------------------------------\n", |
||
| 1100 | "Cost at Step 220: -0.9999878707967713\n", |
||
| 1101 | "Weights at Step 220: [ 5.94865214e-01 1.31396790e-03 1.32851226e+00 1.78368971e+00\n", |
||
| 1102 | " 1.99943131e-03 -8.44829812e-01]\n", |
||
| 1103 | "Bias at Step 220: [-1.40762506 -1.34564162 1.02914859 -1.06395825]\n", |
||
| 1104 | "---------------------------------------------\n", |
||
| 1105 | "Cost at Step 225: -0.9999894983980553\n", |
||
| 1106 | "Weights at Step 225: [ 5.96129030e-01 -2.70446381e-04 1.33726404e+00 1.78530720e+00\n", |
||
| 1107 | " -9.14375384e-04 -8.43087376e-01]\n", |
||
| 1108 | "Bias at Step 225: [-1.40591305 -1.34783606 1.03097914 -1.06389792]\n", |
||
| 1109 | "---------------------------------------------\n", |
||
| 1110 | "Cost at Step 230: -0.9999922381489225\n", |
||
| 1111 | "Weights at Step 230: [ 5.95698562e-01 2.12153829e-05 1.34097628e+00 1.78640344e+00\n", |
||
| 1112 | " -2.66939871e-04 -8.43866656e-01]\n", |
||
| 1113 | "Bias at Step 230: [-1.40598424 -1.35013 1.03301139 -1.06176633]\n", |
||
| 1114 | "---------------------------------------------\n", |
||
| 1115 | "Cost at Step 235: -0.9999915503513058\n", |
||
| 1116 | "Weights at Step 235: [ 5.94412462e-01 1.26937987e-03 1.33568147e+00 1.78429986e+00\n", |
||
| 1117 | " 1.10164011e-03 -8.44057595e-01]\n", |
||
| 1118 | "Bias at Step 235: [-1.40759178 -1.35003317 1.03306412 -1.06029115]\n", |
||
| 1119 | "---------------------------------------------\n", |
||
| 1120 | "Cost at Step 240: -0.9999899139875156\n", |
||
| 1121 | "Weights at Step 240: [ 0.59325925 0.00212116 1.32698164 1.78079875 0.00265113 -0.84377279]\n", |
||
| 1122 | "Bias at Step 240: [-1.40862419 -1.34902462 1.03344422 -1.05847877]\n", |
||
| 1123 | "---------------------------------------------\n", |
||
| 1124 | "Cost at Step 245: -0.9999889983186859\n", |
||
| 1125 | "Weights at Step 245: [ 5.95248184e-01 1.16515643e-03 1.32888367e+00 1.77990336e+00\n", |
||
| 1126 | " 3.18940688e-04 -8.42352373e-01]\n", |
||
| 1127 | "Bias at Step 245: [-1.40780015 -1.34809049 1.03368968 -1.05975704]\n", |
||
| 1128 | "---------------------------------------------\n", |
||
| 1129 | "Cost at Step 250: -0.9999881214395573\n", |
||
| 1130 | "Weights at Step 250: [ 5.94746365e-01 3.52226215e-04 1.33273458e+00 1.78271878e+00\n", |
||
| 1131 | " 2.35450291e-04 -8.42131285e-01]\n", |
||
| 1132 | "Bias at Step 250: [-1.40703427 -1.35285926 1.03469553 -1.05970064]\n", |
||
| 1133 | "---------------------------------------------\n", |
||
| 1134 | "Cost at Step 255: -0.9999893074495982\n", |
||
| 1135 | "Weights at Step 255: [ 5.95128451e-01 1.41833640e-03 1.33495243e+00 1.78118103e+00\n", |
||
| 1136 | " -1.66495422e-04 -8.42564017e-01]\n", |
||
| 1137 | "Bias at Step 255: [-1.40849133 -1.35075613 1.03571014 -1.05899157]\n", |
||
| 1138 | "---------------------------------------------\n", |
||
| 1139 | "Cost at Step 260: -0.9999911831425025\n", |
||
| 1140 | "Weights at Step 260: [ 5.93954624e-01 3.65997215e-04 1.32747775e+00 1.78014389e+00\n", |
||
| 1141 | " 1.86270231e-03 -8.42368868e-01]\n", |
||
| 1142 | "Bias at Step 260: [-1.40745686 -1.35292854 1.03598998 -1.05730236]\n", |
||
| 1143 | "---------------------------------------------\n", |
||
| 1144 | "Cost at Step 265: -0.9999898569345608\n", |
||
| 1145 | "Weights at Step 265: [ 5.94054550e-01 2.22631406e-03 1.32948594e+00 1.77704330e+00\n", |
||
| 1146 | " 1.19766250e-03 -8.42715291e-01]\n", |
||
| 1147 | "Bias at Step 265: [-1.41030685 -1.34876472 1.03609132 -1.05643617]\n", |
||
| 1148 | "---------------------------------------------\n", |
||
| 1149 | "Cost at Step 270: -0.9999877705287914\n", |
||
| 1150 | "Weights at Step 270: [ 5.94781072e-01 -1.11477042e-04 1.33392437e+00 1.78305723e+00\n", |
||
| 1151 | " 3.76174702e-04 -8.42183574e-01]\n", |
||
| 1152 | "Bias at Step 270: [-1.40704689 -1.35586199 1.03817742 -1.058197 ]\n", |
||
| 1153 | "---------------------------------------------\n", |
||
| 1154 | "Cost at Step 275: -0.9999913225494264\n", |
||
| 1155 | "Weights at Step 275: [ 5.94417898e-01 1.55898287e-03 1.33236765e+00 1.77833787e+00\n", |
||
| 1156 | " 8.61473931e-04 -8.42756926e-01]\n", |
||
| 1157 | "Bias at Step 275: [-1.4102245 -1.35092356 1.03638933 -1.05664399]\n", |
||
| 1158 | "---------------------------------------------\n", |
||
| 1159 | "Cost at Step 280: -0.999991247464759\n", |
||
| 1160 | "Weights at Step 280: [ 5.96133973e-01 -6.81094713e-04 1.33226199e+00 1.78221990e+00\n", |
||
| 1161 | " -1.23665053e-03 -8.41018922e-01]\n", |
||
| 1162 | "Bias at Step 280: [-1.40689524 -1.35460515 1.03788072 -1.06013602]\n", |
||
| 1163 | "---------------------------------------------\n", |
||
| 1164 | "Cost at Step 285: -0.9999871732223706\n", |
||
| 1165 | "Weights at Step 285: [ 5.97358644e-01 -9.15319018e-04 1.33478275e+00 1.78040181e+00\n", |
||
| 1166 | " -2.90160465e-03 -8.40389367e-01]\n", |
||
| 1167 | "Bias at Step 285: [-1.40803754 -1.35171341 1.03412961 -1.06239103]\n", |
||
| 1168 | "---------------------------------------------\n", |
||
| 1169 | "Cost at Step 290: -0.9999918619409637\n", |
||
| 1170 | "Weights at Step 290: [ 5.94771896e-01 8.18512334e-04 1.32655817e+00 1.77910435e+00\n", |
||
| 1171 | " 2.12609569e-03 -8.43221244e-01]\n", |
||
| 1172 | "Bias at Step 290: [-1.40897233 -1.35158162 1.03631282 -1.05834328]\n", |
||
| 1173 | "---------------------------------------------\n", |
||
| 1174 | "Cost at Step 295: -0.9999908671570564\n", |
||
| 1175 | "Weights at Step 295: [ 5.95660428e-01 9.09973737e-04 1.33134943e+00 1.77946500e+00\n", |
||
| 1176 | " -2.53296584e-04 -8.42278191e-01]\n", |
||
| 1177 | "Bias at Step 295: [-1.40945183 -1.35032335 1.03559811 -1.06039016]\n", |
||
| 1178 | "---------------------------------------------\n" |
||
| 1179 | ] |
||
| 1180 | } |
||
| 1181 | ], |
||
| 1182 | "source": [ |
||
| 1183 | "for i in range(0, steps):\n", |
||
| 1184 | " (weights, bias), cost = optimizer.step_and_cost(cost_function, weights, bias)\n", |
||
| 1185 | "\n", |
||
| 1186 | " # Prints the value of the cost function\n", |
||
| 1187 | " if i % 5 == 0:\n", |
||
| 1188 | " print(f\"Cost at Step {i}: {cost}\")\n", |
||
| 1189 | " print(f\"Weights at Step {i}: {weights}\")\n", |
||
| 1190 | " print(f\"Bias at Step {i}: {bias}\")\n", |
||
| 1191 | " print(\"---------------------------------------------\")" |
||
| 1192 | ] |
||
| 1193 | }, |
||
| 1194 | { |
||
| 1195 | "cell_type": "markdown", |
||
| 1196 | "metadata": { |
||
| 1197 | "id": "TLYEfeyjLbKB" |
||
| 1198 | }, |
||
| 1199 | "source": [ |
||
| 1200 | "With the learned parameters, we construct a visual representation of the\n", |
||
| 1201 | "Hamiltonian to which they correspond and compare it to the target\n", |
||
| 1202 | "Hamiltonian, and the initial guessed Hamiltonian:\n" |
||
| 1203 | ] |
||
| 1204 | }, |
||
| 1205 | { |
||
| 1206 | "cell_type": "code", |
||
| 1207 | "execution_count": 19, |
||
| 1208 | "metadata": { |
||
| 1209 | "colab": { |
||
| 1210 | "base_uri": "https://localhost:8080/", |
||
| 1211 | "height": 210 |
||
| 1212 | }, |
||
| 1213 | "id": "QnPHQC1PLbKB", |
||
| 1214 | "outputId": "4418f7de-b924-4c2d-ae0b-6ad626551982" |
||
| 1215 | }, |
||
| 1216 | "outputs": [ |
||
| 1217 | { |
||
| 1218 | "output_type": "display_data", |
||
| 1219 | "data": { |
||
| 1220 | "text/plain": [ |
||
| 1221 | "<Figure size 600x600 with 3 Axes>" |
||
| 1222 | ], |
||
| 1223 | "image/png": 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\n" |
||
| 1224 | }, |
||
| 1225 | "metadata": {} |
||
| 1226 | } |
||
| 1227 | ], |
||
| 1228 | "source": [ |
||
| 1229 | "new_ham_matrix = create_hamiltonian_matrix(\n", |
||
| 1230 | " qubit_number, nx.complete_graph(qubit_number), weights, bias\n", |
||
| 1231 | ")\n", |
||
| 1232 | "\n", |
||
| 1233 | "init_ham = create_hamiltonian_matrix(\n", |
||
| 1234 | " qubit_number, nx.complete_graph(qubit_number), initial_weights, initial_bias\n", |
||
| 1235 | ")\n", |
||
| 1236 | "\n", |
||
| 1237 | "fig, axes = plt.subplots(nrows=1, ncols=3, figsize=(6, 6))\n", |
||
| 1238 | "\n", |
||
| 1239 | "axes[0].matshow(ham_matrix, vmin=-7, vmax=7, cmap=\"hot\")\n", |
||
| 1240 | "axes[0].set_title(\"Target\", y=1.13)\n", |
||
| 1241 | "\n", |
||
| 1242 | "axes[1].matshow(init_ham, vmin=-7, vmax=7, cmap=\"hot\")\n", |
||
| 1243 | "axes[1].set_title(\"Initial\", y=1.13)\n", |
||
| 1244 | "\n", |
||
| 1245 | "axes[2].matshow(new_ham_matrix, vmin=-7, vmax=7, cmap=\"hot\")\n", |
||
| 1246 | "axes[2].set_title(\"Learned\", y=1.13)\n", |
||
| 1247 | "\n", |
||
| 1248 | "plt.subplots_adjust(wspace=0.3, hspace=0.3)\n", |
||
| 1249 | "plt.show()" |
||
| 1250 | ] |
||
| 1251 | }, |
||
| 1252 | { |
||
| 1253 | "cell_type": "markdown", |
||
| 1254 | "metadata": { |
||
| 1255 | "id": "zspKar5SLbKB" |
||
| 1256 | }, |
||
| 1257 | "source": [ |
||
| 1258 | "These images look very similar, indicating that the QGRNN has done a\n", |
||
| 1259 | "good job learning the target Hamiltonian.\n", |
||
| 1260 | "\n", |
||
| 1261 | "We can also look at the exact values of the target and learned\n", |
||
| 1262 | "parameters. Recall how the target interaction graph has $4$ edges while\n", |
||
| 1263 | "the complete graph has $6$. Thus, as the QGRNN converges to the optimal\n", |
||
| 1264 | "solution, the weights corresponding to edges $(1, 3)$ and $(2, 0)$ in\n", |
||
| 1265 | "the complete graph should go to $0$, as this indicates that they have no\n", |
||
| 1266 | "effect, and effectively do not exist in the learned Hamiltonian.\n" |
||
| 1267 | ] |
||
| 1268 | }, |
||
| 1269 | { |
||
| 1270 | "cell_type": "code", |
||
| 1271 | "execution_count": 20, |
||
| 1272 | "metadata": { |
||
| 1273 | "id": "-ql-50F_LbKB" |
||
| 1274 | }, |
||
| 1275 | "outputs": [], |
||
| 1276 | "source": [ |
||
| 1277 | "# We first pick out the weights of edges (1, 3) and (2, 0)\n", |
||
| 1278 | "# and then remove them from the list of target parameters\n", |
||
| 1279 | "\n", |
||
| 1280 | "weights_noedge = []\n", |
||
| 1281 | "weights_edge = []\n", |
||
| 1282 | "for ii, edge in enumerate(new_ising_graph.edges):\n", |
||
| 1283 | " if (edge[0] - qubit_number, edge[1] - qubit_number) in ising_graph.edges:\n", |
||
| 1284 | " weights_edge.append(weights[ii])\n", |
||
| 1285 | " else:\n", |
||
| 1286 | " weights_noedge.append(weights[ii])" |
||
| 1287 | ] |
||
| 1288 | }, |
||
| 1289 | { |
||
| 1290 | "cell_type": "markdown", |
||
| 1291 | "metadata": { |
||
| 1292 | "id": "WL7HVML3LbKB" |
||
| 1293 | }, |
||
| 1294 | "source": [ |
||
| 1295 | "Then, we print all of the weights:\n" |
||
| 1296 | ] |
||
| 1297 | }, |
||
| 1298 | { |
||
| 1299 | "cell_type": "code", |
||
| 1300 | "execution_count": 21, |
||
| 1301 | "metadata": { |
||
| 1302 | "colab": { |
||
| 1303 | "base_uri": "https://localhost:8080/", |
||
| 1304 | "height": 0 |
||
| 1305 | }, |
||
| 1306 | "id": "rlB_pAzGLbKB", |
||
| 1307 | "outputId": "ae59834f-dead-437f-ac7c-d030cd9da2f1" |
||
| 1308 | }, |
||
| 1309 | "outputs": [ |
||
| 1310 | { |
||
| 1311 | "output_type": "stream", |
||
| 1312 | "name": "stdout", |
||
| 1313 | "text": [ |
||
| 1314 | "Target parameters Learned parameters\n", |
||
| 1315 | "Weights\n", |
||
| 1316 | "-----------------------------------------\n", |
||
| 1317 | "0.56 | 0.5974609311535144\n", |
||
| 1318 | "1.24 | 1.3411495568824126\n", |
||
| 1319 | "1.67 | 1.784741915545322\n", |
||
| 1320 | "-0.79 | -0.8434429713923873\n", |
||
| 1321 | "\n", |
||
| 1322 | "Bias\n", |
||
| 1323 | "-----------------------------------------\n", |
||
| 1324 | "-1.44 | -1.4067252706628723\n", |
||
| 1325 | "-1.43 | -1.3546960101894798\n", |
||
| 1326 | "1.18 | 1.0354062517348281\n", |
||
| 1327 | "-0.93 | -1.0615531663839644\n", |
||
| 1328 | "\n", |
||
| 1329 | "Non-Existing Edge Parameters: [-0.0013558219901317857, -0.0007373999961358714]\n" |
||
| 1330 | ] |
||
| 1331 | } |
||
| 1332 | ], |
||
| 1333 | "source": [ |
||
| 1334 | "print(\"Target parameters Learned parameters\")\n", |
||
| 1335 | "print(\"Weights\")\n", |
||
| 1336 | "print(\"-\" * 41)\n", |
||
| 1337 | "for ii_target, ii_learned in zip(target_weights, weights_edge):\n", |
||
| 1338 | " print(f\"{ii_target : <20}|{ii_learned : >20}\")\n", |
||
| 1339 | "\n", |
||
| 1340 | "print(\"\\nBias\")\n", |
||
| 1341 | "print(\"-\"*41)\n", |
||
| 1342 | "for ii_target, ii_learned in zip(target_bias, bias):\n", |
||
| 1343 | " print(f\"{ii_target : <20}|{ii_learned : >20}\")\n", |
||
| 1344 | "\n", |
||
| 1345 | "print(f\"\\nNon-Existing Edge Parameters: {[val.unwrap() for val in weights_noedge]}\")" |
||
| 1346 | ] |
||
| 1347 | }, |
||
| 1348 | { |
||
| 1349 | "cell_type": "markdown", |
||
| 1350 | "metadata": { |
||
| 1351 | "id": "K4jKWHsELbKB" |
||
| 1352 | }, |
||
| 1353 | "source": [ |
||
| 1354 | "The weights of edges $(1, 3)$ and $(2, 0)$ are very close to $0$,\n", |
||
| 1355 | "indicating we have learned the cycle graph from the complete graph. In\n", |
||
| 1356 | "addition, the remaining learned weights are fairly close to those of the\n", |
||
| 1357 | "target Hamiltonian. Thus, the QGRNN is functioning properly, and has\n", |
||
| 1358 | "learned the target Ising Hamiltonian to a high degree of accuracy!\n" |
||
| 1359 | ] |
||
| 1360 | }, |
||
| 1361 | { |
||
| 1362 | "cell_type": "markdown", |
||
| 1363 | "metadata": { |
||
| 1364 | "id": "0BWaZBY7LbKB" |
||
| 1365 | }, |
||
| 1366 | "source": [ |
||
| 1367 | "References\n", |
||
| 1368 | "==========\n", |
||
| 1369 | "\n", |
||
| 1370 | "1. Verdon, G., McCourt, T., Luzhnica, E., Singh, V., Leichenauer, S., &\n", |
||
| 1371 | " Hidary, J. (2019). Quantum Graph Neural Networks. arXiv preprint\n", |
||
| 1372 | " [arXiv:1909.12264](https://arxiv.org/abs/1909.12264).\n", |
||
| 1373 | "\n", |
||
| 1374 | "About the author\n", |
||
| 1375 | "================\n" |
||
| 1376 | ] |
||
| 1377 | } |
||
| 1378 | ], |
||
| 1379 | "metadata": { |
||
| 1380 | "kernelspec": { |
||
| 1381 | "display_name": "Python 3", |
||
| 1382 | "language": "python", |
||
| 1383 | "name": "python3" |
||
| 1384 | }, |
||
| 1385 | "language_info": { |
||
| 1386 | "codemirror_mode": { |
||
| 1387 | "name": "ipython", |
||
| 1388 | "version": 3 |
||
| 1389 | }, |
||
| 1390 | "file_extension": ".py", |
||
| 1391 | "mimetype": "text/x-python", |
||
| 1392 | "name": "python", |
||
| 1393 | "nbconvert_exporter": "python", |
||
| 1394 | "pygments_lexer": "ipython3", |
||
| 1395 | "version": "3.9.17" |
||
| 1396 | }, |
||
| 1397 | "colab": { |
||
| 1398 | "provenance": [] |
||
| 1399 | } |
||
| 1400 | }, |
||
| 1401 | "nbformat": 4, |
||
| 1402 | "nbformat_minor": 0 |
||
| 1403 | } |