A Quantum Algorithm for the Sensitivity Analysis of Business Risks

Abstract

We present a novel use case for quantum computation: the sensitivity analysis for a risk model used at Deutsche Börse Group. Such an analysis is computationally too expensive to perform on classical computers. We show in detail how the risk model and its analysis can be implemented as a quantum circuit. We test small scale versions of the model in simulation and find that the expected quadratic speedup compared to the classical implementation used at Deutsche Börse Group can be realized. Full scale production usage would be possible with less than 200 error corrected qubits. Our quantum algorithm introduces unitary but imperfect oracles which use Quantum Amplitude Estimation to detect and mark states. This construction should be of general interest and we present theoretical results regarding the performance of Grover's search algorithm with imperfect oracles.

Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation

“…ref. [3], where the result of a QAE is the input of a Grover search. The same circuit performs a QAE if the operator U is a Grover operator (see 2.1).…”

“…The problem we solve in this section is finding the probability of the worst case scenario in a small scale business risk model described in section 2.1 of ref. [3]. In that example, there are four events with a certain dependency structure, where the first two events are mutually exclusive.…”

“…Many potential commercial applications for quantum computers rely on quantum algorithms that can execute the quantum equivalent of Monte-Carlo simulations quadratically faster than classical Monte-Carlo implementations on classical computers. Well-known examples are Value-at-Risk (VaR) calculations [1], the calculation of option prices [2] or the calculation of sensitivities of risk models [3], all of which are based on the Quantum Amplitude Estimation algorithm (QAE, see [4]), which itself is a variant of the Quantum Phase Estimation algorithm (QPE, see [5]). Unfortunately, quantum computers, which satisfy the fidelity requirements for such applications 1 , are not available today and are not expected to be available in the foreseeable future (see [7][8][9][10]).…”

“…This cannot be improved by an increase of the number of times the circuit is measured. Consider an example: If a Grover operator has a 20% probability of picking up an error, then the precision of any form of serial QAE will be limited to a single decimal digit 3 , regardless of how many times we measure or how we post-processes the results of measurements. This precision is several orders of magnitude away from what is required for typical practical applications.…”

Hybrid Quantum-Classical Algorithms for Loan Collection Optimization with Loan Loss Provisions

“…Applications of quantum computing on finance have been recently explored both theoretically and experimentally on small quantum devices. Examples include portfolio optimization [4][5][6][7], option pricing [8][9][10], risk analysis [11], transaction settlement [12], and credit valuation adjustment [13].…”