Credit Risk Analysis Using Quantum Computers

Abstract

We present and analyze a quantum algorithm to estimate credit risk more efficiently than Monte Carlo simulations can do on classical computers. More precisely, we estimate the economic capital requirement, i.e. the difference between the Value at Risk and the expected value of a given loss distribution. The economic capital requirement is an important risk metric because it summarizes the amount of capital required to remain solvent at a given confidence level. We implement this problem for a realistic loss distribution and analyze its scaling to a realistic problem size. In particular, we provide estimates of the total number of required qubits, the expected circuit depth, and how this translates into an expected runtime under reasonable assumptions on future faulttolerant quantum hardware.

Quantum Algorithms for Mixed Binary Optimization Applied to Transaction Settlement

“…This allows us to define trial wavefunctions |ψ H(y) (θ) that now depend on y. As in (5) we can translate the MBO to a continuous optimization problem, where the measurements x j (θ, y) depend on θ as well as y. In summary, A(y), b(y), and c(y) define the Hamiltonian H(y), which in turn defines the θand y-dependent trial wavefunction |ψ H(y) (θ) .…”

“…Next, we introduce a heuristic that is designed to handle slack variables resulting from modelling inequality constraints explicitly. This can help the classical optimizer used to solve (5) to move out of local minima. Suppose, for simplicity, that we have a problem with only binary variables and a single slack variable, coming from an inequality constraint, formally given by…”

“…They are well suited for a number of tasks such as simulating quantum mechanical systems [1], [2] and factoring [3]. Additionally, quantum computers can provide a quadratic speed-up over classical Monte-Carlo simulations which may be used to evaluate risk [4], [5] and price financial derivatives [6], [7]. Another possible application area for quantum computers is optimization, particularly combinatorial optimization.…”

Iterative quantum amplitude estimation

“…Quantum Amplitude Estimation (QAE) 1 is a fundamental quantum algorithm with the potential to achieve a quadratic speedup for many applications that are classically solved through Monte Carlo (MC) simulation. It has been shown that we can leverage QAE in the financial service sector, e.g., for risk analysis 2,3 or option pricing [4][5][6] , and also for generic tasks such as numerical integration 7 . While the estimation error bound of classical MC simulation scales as Oð1= ffiffiffiffi M p Þ, where M denotes the number of (classical) samples, QAE achieves a scaling of Oð1=MÞ for M (quantum) samples, indicating the aforementioned quadratic speedup.…”

Grover Adaptive Search for Constrained Polynomial Binary Optimization

“…Quantum computers are theoretically proven to solve certain problems faster than a classical device [1][2][3] and are well-equipped to handle tasks such as factoring [2], linear systems of equations [4,5], Monte-Carlo simulations [6][7][8][9], as well as combinatorial optimization problems [10][11][12][13][14].…”