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"cells": [
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"source": [
"# Quantum Finance Application on Arbitrage Opportunity Optimization\n",
"\n",
" Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Overview\n",
"\n",
"Current finance problems can be mainly tackled by three areas of quantum algorithms: quantum simulation, quantum optimization, and quantum machine learning [1,2]. Many financial problems are essentially combinatorial optimization problems, and corresponding algorithms usually have high time complexity and are difficult to implement. Due to the power of quantum computing, these complex problems are expected to be solved by quantum algorithms in the future.\n",
"\n",
"The Quantum Finance module of Paddle Quantum focuses on quantum optimization: how to apply quantum algorithms in real finance optimization problems. This tutorial focuses on how to use quantum algorithms to solve the arbitrage opportunity optimization problem [3].\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Arbitrage Opportunity Optimization \n",
"\n",
"Arbitrage describes the fact that the same asset has different prices in different markets and can be traded between multiple markets to generate a positive return. That is, given a set of assets and transaction costs, it is possible to have a cycle among different markets that can generate a positive return.\n",
"\n",
"This problem can be formulated in the language of graph theory: given a weighted directed graph $G$ with vertex $i$ denoting $i$ th market currency, the weight of the edge from vertex $i$ to vertex $j$ denoting the exchange rate $c_{ij} $ that converts currency $i$ to currency $j$. That is if we have a currency $i$ of quantity $x_i$, then by the exchange rate, we will get a currency $j$ of quantity $c_{ij}x_i$. In general, the exchange rate is not symmetric, i.e. $c_{ij} \\neq c_{ji}$. We also assume that transaction costs (service fees etc.) are already included in the exchange rate.\n",
"\n",
"Given the number of tradable currency types $n$, the optimization problem is to find a cycle of the maximal profit on the given directed graph with $K (K \\leq n)$ vertices. In this finance module, users can define the number of vertices contained in the arbitrage cycle $K$ according to their requirements.\n"
]
},
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"cell_type": "markdown",
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"source": [
"### Encoding Arbitrage Opportunity Optimization Problem\n",
"\n",
"To transform the arbitrage opportunity optimization problem into a problem applicable for parameterized quantum circuits, we need to encode the arbitrage opportunity optimization problem into a Hamiltonian. We realize the encoding by first constructing an integer programming problem. Suppose there are $|V|=n$ vertices in graph $G$. Then for each vertex $i \\in V$, we define $n$ binary variables $x_{i,k}$, where $k \\in [0,K-1]$, such that\n",
"\n",
"$$\n",
"x_{i, k}=\n",
"\\begin{cases}\n",
"1, & \\text{the order of traded to vertex } i \\text { in the arbitrage cycle is $k$}\\\\\n",
"0, & \\text{otherwise}\n",
"\\end{cases}.\n",
"\\tag{1}\n",
"$$\n",
"\n",
"Refering to the Travelling Salesman Problem ([TSP tutorial](./TSP_CN.ipynb)), since $G$ has $n$ vertices, we have $n^2$ variables in total, whose value we denote by a bit string $x=x_{0,0}x_{0,1}\\dots x_{n-1,K-1}$. Assume for now that the bit string $x$ represents a arbitrage cycle. Then for each edge $(i,j,w_{i,j}) \\in E$, we will have $x_{i,k} = x_{j,k+1}=1$, i.e., $x_{i,k}\\cdot x_{j,k+1}=1$, if and only if the arbitrage cycle visits vertex $i$ at time $k$ and vertex $j$ at time $k+1$. Otherwise, $x_{i,k}\\cdot x_{j,k+1}$ will be $0$. Therefore the logarithm of the profit of the cycle is:\n",
"\n",
"$$\n",
"P(x) = - \\sum_{i,j\\in V} \\log(c_{ij}) \\sum_{k=0}^{K-1} x_{i,k}x_{j,k+1}, \\tag{2}\n",
"$$\n",
"\n",
"For $x$ to represent a valid arbitrage cycle, the following constraint needs to be met:\n",
"\n",
"$$\n",
"\\sum_{i=0}^{n-1} x_{i,k} = 1 \\quad \\text{and} \\quad \\sum_{k=0}^{K-1}\\sum_{(i,j)\\notin E}x_{i,k}x_{j, k+1}, \\tag{3}\n",
"$$\n",
"\n",
"where the first equation guarantees that only one vertex is visited at each time $k$. The second term constrains that a nonexistent edge does not appear in the found arbitrage cycle. These two equations ensure that the parameterized quantum circuits find $𝑥$ as a simple loop. Then the cost function under the constraint can be formulated below:\n",
"$$\n",
"C_x = - P(x) + A\\sum_{k=0}^{K-1} \\left(1 - \\sum_{i=0}^{n-1} x_{i,k}\\right)^2 + A\\sum_{k=0}^{K-1}\\sum_{(i,j)\\notin E}\n",
"x_{i,k}x_{j,k+1}.\n",
"\\tag{4}\n",
"$$\n",
"In this equation, $V$ is the number of vertices of the graph, $E$ is the edge set of the graph and $K$ is the number of vertices of the most profitable cycle. Note that as we would like to maximize the $P(x)$ while ensuring $x$ represents a valid arbitrage cycle, we had better set $A$ large, at least larger than the largest weight of edges.\n",
"\n",
"We now need to transform the cost function $C_x$ into a Hamiltonian to realize the encoding of the arbitrage opportunity optimization problem. Each variable $x_{i,k}$ has two possible values, $0$ and $1$, corresponding to quantum states $|0\\rangle$ and $|1\\rangle$. Note that every variable corresponds to a qubit and so $n^2$ qubits are needed for solving the arbitrage opportunity optimization problem. The Pauli $Z$ operator has two eigenstates, $|0\\rangle$ and $|1\\rangle$ . Their corresponding eigenvalues are 1 and -1, respectively. So we consider encoding the cost function as a Hamiltonian using the Pauli $Z$ matrix.\n",
"\n",
"Now we would like to consider the mapping\n",
"$$\n",
"x_{i,k} \\mapsto \\frac{I-Z_{i,k}}{2}, \\tag{3}\n",
"$$\n",
"\n",
"where $Z_{i,k} = I \\otimes I \\otimes \\ldots \\otimes Z \\otimes \\ldots \\otimes I$ with $Z$ operates on the qubit at position $(i,k)$. Under this mapping, the value of $x_{i,k}$ can be illustrated in a different way. If the qubit $(i,k)$ is in state $|1\\rangle$, then $x_{i,k} |1\\rangle = \\frac{I-Z_{i,k}}{2} |1\\rangle = 1|1\\rangle $, which means vertex $i$ is visited at time $k$. Also, for the qubit $(i,k)$ in state $|0\\rangle$, $x_{i,k}|0\\rangle = \\frac{I-Z_{i,k}}{2} |0\\rangle = 0 |0\\rangle $.\n",
"\n",
"Thus using the above mapping, we can transform the cost function $C_x$ into a Hamiltonian $H_C$ for the system of $n^2$ qubits and realize the quantumization of the arbitrage opportunity optimization problem. Then the ground state of $H_C$ is the optimal solution to the arbitrage opportunity optimization problem. In the following section, we will show how to use a parameterized quantum circuit to find the ground state, i.e., the eigenvector with the smallest eigenvalue."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Paddle Quantum Implementation\n",
"\n",
"To investigate the arbitrage opportunity optimization problem using Paddle Quantum, there are some required packages to import, which are shown below. The ``networkx`` package is the tool to handle graphs."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"ExecuteTime": {
"end_time": "2021-05-17T08:00:15.901429Z",
"start_time": "2021-05-17T08:00:12.708945Z"
}
},
"outputs": [],
"source": [
"# Import packages needed\n",
"import numpy as np\n",
"import networkx as nx\n",
"import matplotlib.pyplot as plt\n",
"\n",
"# Import related modules from Paddle Quantum and PaddlePaddle\n",
"import paddle\n",
"import paddle_quantum\n",
"from paddle_quantum.ansatz import Circuit\n",
"from paddle_quantum.finance import arbitrage_opportunities_hamiltonian"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next, we generate a weighted directed graph $G$ with $3$ vertices. For the convenience of computation, the vertices here are labeled starting from $0$.\n",
"\n",
"At the same time, in order to verify the solution more easily, special values are assigned for the weights in the directed graph. In practice, users can construct their own desired graphs and set the real exchange rates."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"ExecuteTime": {
"end_time": "2021-05-17T08:00:16.212260Z",
"start_time": "2021-05-17T08:00:15.918792Z"
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"outputs": [
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",
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