# A More General Quantum Credit Risk Analysis Framework

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## Abstract

**:**

## 1. Introduction

#### Quantum Finance and Credit Risk Analysis

## 2. Methods

#### 2.1. SOTA Quantum Credit Risk Analysis

- $\mathcal{U}$, which loads the domain-dependent uncertainty model.
- $\mathcal{S}$, which computes the total loss over ${n}_{S}$ qubits.
- $\mathcal{C}$, which flips a target qubit if the total loss is equal to or lower than a certain threshold x.

#### 2.2. Multiple Risk Factors

#### 2.3. Arbitrary LGD

## 3. Results

#### 3.1. Noiseless Simulation

#### 3.2. Real Hardware and Noisy Simulations

**Ibm_perth**and**ibm_lagos**, each with 7 qubits and a quantum volume of 32.**Ibm_canberra**and**ibm_algiers**, each with 27 qubits and quantum volumes of 32 and 128, respectively.

## 4. Discussion

**ibm_algiers**: this machine with a higher quantum volume (128) shows a greater variance around the central value, a potential indication of how the continuous improvement of this particular dimension suggests a future successful application of this and other algorithms on quantum machines.

#### 4.1. Scalability and Complexity

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QPU | Quantum processing unit |

CRA | Credit risk analysis |

VaR | Value at risk |

QAE | Quantum amplitude estimation |

PD | Probability of default |

LGD | Loss given default |

## Appendix A. Quantum Processor Topologies

#### Appendix A.1. ibm_lagos and ibm_perth

#### Appendix A.2. ibm_algiers and ibm_canberra

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**Figure 2.**An instance of the multi-factor version of the quantum circuit that encodes the canonical uncertainty model, using multiple rotations. The example involves $K=2$ assets and ${n}_{z}=2$, which means that two qubits are used to encode each normal standard distribution. The example also takes into account two risk factors ($R=2$).

**Figure 3.**An instance of the multi-factor version of the quantum circuit that encodes the canonical uncertainty model. It has identical parameters to the circuit illustrated in Figure 2 but uses only one rotation per asset.

**Figure 4.**Noiseless simulation: probability distribution function of total loss. The green dashed line shows the expected loss while the orange dashed line shows the value at risk.

**Figure 5.**Noiseless simulation: CDF of total loss $\mathcal{L}$ in green and target level of 95 percent in orange.

**Figure 6.**Ratio frequency distribution for the experiments conducted on classical machines, simulating the effects of noise thanks to noise models from the quantum experiments.

Asset Number | Loss Given Default | Default Prob. | Sensitivity | Risk Factor Weights |
---|---|---|---|---|

$\mathit{k}$ | ${\mathit{LGD}}_{\mathit{k}}$ | ${\mathit{p}}_{\mathit{k}}^{\mathbf{0}}$ | ${\mathsf{\rho}}_{\mathit{k}}$ | ${({\mathsf{\alpha}}_{\mathbf{1}},{\mathsf{\alpha}}_{\mathbf{2}})}_{\mathit{k}}$ |

1 | 1000.5 | $0.15$ | $0.1$ | $0.35,0.2$ |

2 | 2000.5 | $0.25$ | $0.05$ | $0.1,0.25$ |

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**MDPI and ACS Style**

Dri, E.; Aita, A.; Giusto, E.; Ricossa, D.; Corbelletto, D.; Montrucchio, B.; Ugoccioni, R. A More General Quantum Credit Risk Analysis Framework. *Entropy* **2023**, *25*, 593.
https://doi.org/10.3390/e25040593

**AMA Style**

Dri E, Aita A, Giusto E, Ricossa D, Corbelletto D, Montrucchio B, Ugoccioni R. A More General Quantum Credit Risk Analysis Framework. *Entropy*. 2023; 25(4):593.
https://doi.org/10.3390/e25040593

**Chicago/Turabian Style**

Dri, Emanuele, Antonello Aita, Edoardo Giusto, Davide Ricossa, Davide Corbelletto, Bartolomeo Montrucchio, and Roberto Ugoccioni. 2023. "A More General Quantum Credit Risk Analysis Framework" *Entropy* 25, no. 4: 593.
https://doi.org/10.3390/e25040593