Team 7: Quantum Risk

Business_Application.md

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+# Quantum Algorithms for the Sensitivity Analysis of Business Risks
+
+
+# Quantum algorithms
+
+- Problem definition: 
+Quantum algorithms are known for super-polynomial speedup over their classical counterparts.
+They can be applied to various real-world problems such as cryptography, optimization, and simulations. 
+
+- Performance analysis: 
+ Following the methodology by Deutsche Börse Group we present a quantum algorithm (namely Grover's algorithm) to address the sensitivity analysis for a business risk model, which found to be computationally too expensive to perform by classical algorithms. 
+Grover's algorithm (1996) is known for its ability to address unstructured search problems, which are basic problems in computer science.
+Here, we implement and analysis the risk modeling and its representation on quantum circuits using this algorithm and the Quantum Amplitude Estimation (QAE). 
+The fact that quantum algorithms are faster than a classical ones can be tested by the code execution runtime, measured by the number of elementary operations used by an algorithm, and can be done using the quantum circuit model.
+The quantum program employed here is fast (as compared to classical ones, e.g. quasi-Monte Carlo methods), requires low number of qubits (< 200, achievable), and has an interesting nested structure: Grover's algorithm\QAE\Quantum risk model.
+
+
+# Business risk model and analysis: 
+The impact of external adverse developments on future revenues in any business can be addressed within a risk model.
+A risk model estimates the overall likelihood of impacts that would threaten the business.
+Thus we are dealing with a probability problem.
+To define the problem, a threshold $A$ for a financial impact is defined.
+The probability $P(A)$ shows probability that the financial impact bigger than $A$, and $P_{max}$ is the maximal acceptable value of $P(A)$.
+In the business risk model, to avoid loss, an action needs to be taken when $P(A)$ reaches it maximum value $P_{max}$.
+The estimated $P(A)$ is based on some estimated parameters (inputs) and we are interested in finding the parameter(s) that when changed
+slightly, influence the output of the model such that $P(A)>P_{max}$.
+
+The business risks is implemented as follows:
+An intrinsic probability $P_i$ is defined for each relevant event (risk item, e.g. a change in stock market). 
+An item ($i$ th) is also assigned a probability to trigger another item ($j$ th) with the transition probability $P_{ij}$.
+Each triggered risk item (e.g. by other items) generates a specific loss. 
+The sum of the losses of the triggered items gives the total loss for a specific scenario. 
+Finally, the model is evaluated by brute force (cf. ref [1] for the details).
+
+
+- Compilation (circuit representation):
+The implementation of the quantum program and simulations are done using Qiskit, as illustrated in the Risk_Analysis_Hackathon.ipynb file.
+As shown the oracles we consider lower the success probability by a constant factor compared to standard oracles. 
+A success probability of at least 81% (rather than nearly 100% in conventional Grover) is achieved, which is inherited from the QAE. 
+
+
+# Quantum implementation
+The sensitivity analysis of the risk model is considered as a quantum program that analyzes the impact of varying each input parameter in three steps:
+1. Implementing the risk model as a quantum algorithm,
+2. Implementing QAE on the outputs of the risk model,
+3. Search sensitive parameters with Grover's algorithm.
+
+For the first step, the structure of the model is translated into a quantum circuit.
+In the quantum formalism the risk items are represented by qubits.
+A risk item can be put into a superposition of being triggered with probability $P$ and not being triggered with probability $1-P$
+by appying a rotation operator
+$U_3(\theta,\phi,\lambda)$
+on a qubit, with a $\theta$ that fulfil the relation $\sin(\theta/2)=P$, and the phases $\phi$ and $\lambda$ can be set to zero.
+Such quantum implementation of the model turns out to be remarkably efficient as compared to the implementation on a classical computer.
+
+
+# Potential customers
+
+Potential customers for the Sensitivity Analysis of Business Risks includes banks, trading companies, and any financial institute dealing with risk analysis
+and any company dealing with portfolio.
+
+
+# References
+
+[1] M.C. Braun et al., A Quantum Algorithm for the Sensitivity Analysis of Business Risks (2021), https://arxiv.org/abs/2103.05475
+
+[2] A. Montanaro, Quantum algorithms: an overview, npj Quantum Inf 2, 15023 (2016).
+
+[3] S. Woerner and D.J. Egger, Quantum risk analysis, npj Quantum Inf 5, 15 (2019).