Problem definition

Suppose we are managing a credit risk porfolio with $K$ assets. The default probability of every asset $k$ follows the Gaussian Conditional Independnce model, i.e., given a value $z$ sampled from a latent random variable $Z$ following a standard normal distribution, the default probability of asset $k$ can be described as

$$p_{k}(z) = \Phi \bigg(\frac{\Phi^{-1}(p_{k}^{0} - \sqrt{\rho_k}z)}{\sqrt{1-\rho_{k}}} \bigg)$$

where $\Phi$ denotes the cumulative distribution function of $Z$, $p_{k}^{0}$ is the default probability of asset $k$ for $z = 0$ and $\rho_{k}$ is the sensitivity of the default probability of asset $k$ with respect to $Z$. By giving a concrete realization of $Z$ the individual default events are assumed to be independent from each other.

Then, the topic we are interested in is measuring the total loss

$$L = \sum_{k=1}^{K}\lambda_{k}X_{k}(Z)$$

where $\lambda_k$ denotes the loss given default of asset $k$, and $X_{k}(Z)$ denotes a Bernoulli variable representing the default event of asset $k$.

In reality, we are insterested in the expected value $\mathbb{E}[L]$, the Value at Risk (VaR) of $L$ and the Conditional Value at Risk of $L$. The VaR and CVaR are defined as

$$\text{VaR}_{\alpha}(L) = \text{inf}\{x|\mathbb{P}[L\leq x] \geq 1 - \alpha\}$$

with confidence level $\alpha \in [0,1]$, and

$$\text{CVaR}_{\alpha}(L) = \mathbb{E}[L|L \geq \text{VaR}_{\alpha}(L)].$$

where:

• number of qubits used to represent $Z$, denoted by $n_z$,
• truncation value for $Z$, denoted by $z_\text{max}$, i.e., $Z$ is assumed to take $2^{n_z}$ equidistant values in $\{-z_\text{min}, \cdots, z_\text{max} \}$,
• the base default probabilities for each asset $p_0^{k} \in (0,1), k = 1, \cdots, K$,
• sensitivities of the default probabilities with respect to $Z$, denoted by $\rho_k \in [0,1)$,
• loss given default for asset $k$, denoted by $\lambda_k$
• confidence level for VaR/CVaR $\alpha \in [0,1]$

Uncertainty model

An uncertainty model is a methematical model that is used to quantify and manage unknowns espeically future outcomes that are not determinstic. Here, in our risk analysis, this uncertainty model can be achieved by creating a quantum state in a register of $n_z$ qubits that represents $Z$ following a standard normal distribution. This state is then used to control single qubit Y-rotations on a second qubit register of $K$ qubits, where a $|1\rangle$ state of qubit $k$ represents the default even of asset $k$. We can encode our quantum state $|\psi\rangle$ into

$$|\psi\rangle = \sum_{i=0}^{2^{n_z}-1}\sqrt{p_{z}^{i}}|z_{i}\rangle \bigotimes_{k=1}^{K}\bigg( \sqrt{1-p_{k}(z_{i})}|0\rangle + \sqrt{p_{k}(z_i)}|1\rangle \bigg),$$

where we denote by $z_i$ the $i$-th value of the discretized and truncated $Z$.

See Qiskit GCI for how to construct this model using Qiskit.

Linear Approximation

From Amplitude Estimation, we know that we want to encode a function $p(x)$

$$p_{k}(z) = \Phi \bigg(\frac{\Phi^{-1}(p_{k}^{0} - \sqrt{\rho_k}z)}{\sqrt{1-\rho_{k}}} \bigg)$$

into a rotational angle in a quantum circuit

$$\theta_{k}(z) = 2\text{sin}^{-1}(\sqrt(p_k)(z)).$$

This angle is needed to prepare a qubit state:

$$\sqrt{1-p_{k}(z)}|0\rangle + \sqrt{p_{k}(z)}|1\rangle$$

since the quantum gate $Ry$ work with linear functions of the input register, we approximate around $z = 0$:

$$\theta_{k}(\theta) \approx \text{slope}_k \cdot z + \text{offset}_k$$

From [2], we can approximate using

$$\theta_k(z) = 2 \arcsin\left( \sqrt{p_k(z)} \right)$$

Then you approximate this $\theta$ as:

$$\theta_k(z) \approx \underbrace{\theta_k(0)}_{\text{offset}} + \underbrace{ \left. \frac{d\theta_k}{dz} \right|_{z=0} }_{\text{slope}} \cdot z$$

Step by step

1. Linear Dependency Between Risk and Latent Factor

From paper Regulatory Capital Modelling for Credit Risk, we know that given a firm $k$, the default probabilites and the latent vairable can be linearlized as

$$X_k = a_{k} Z_{k} + b_k$$

where $X_k$ is the individual risk variables, $Z$ is the latent variable, $a_k, b_k$ are constants.

2. Default event Modeling

A firm defaults if its latent variable fall below a threshold:

$$\text{Firm Default if } X_{k} \leq \Phi^{-1}(p_{k}^{0})$$

where $p^{0}_{k}$ is the baseline (unconditional) default probability.

3. Conditional Default Probability as a Function of Z

From above, the conditional probability that firm $k$ defaults given $Z = z$ is:

$$p_{k}(z) = \Phi\bigg(\frac{\Phi^{-1}(p_{k}^{0}) - \sqrt{\rho_k}z}{\sqrt{1-\rho_{k}}} \bigg)$$

where $\Phi(z)$ is a standard normal CDF. This models how worsening economy (negative $z$) increases default probability.

4. Rotation Mapping for Quantum Circuit

In quantum, we want to prepare qubit amplitudes based on probabilities. Thus, we map $p_{k}(z)$ into a rotation angle:

$$\theta_{k}(\theta) = 2 \text{sin}^{-1}(\sqrt{p_{k}(z)})$$

we must let rotation angle build qubit state with correct probability $p_{k}(z)$.

5. Problem with Angle Rotation

It's clear that above formula is nonlinear in $z$ and $\text{sin}^{-1}$ introduces another nonlinearity. This is too complex for us for direct quantum control.

6. Linear Approximation Using Taylor Expansion

We want to approximate $\theta_{k}(z)$ by first-order Taylor expansion around $z = 0$:

$$\theta_k(z) \approx \underbrace{\theta_k(0)}_{\text{offset}} + \underbrace{ \left. \frac{d\theta_k}{dz} \right|_{z=0} }_{\text{slope}} \cdot z$$

where $\theta_{k}(0) = 2 \text{sin}^{-1}(\sqrt{p_{k}(0)})$ is the offset and $\frac{d\theta_k}{dz}$ is the slope computed using chain rule.

Here we go through the details of chain rule. We want to expand:

$$\theta_{k}(z) = 2 \text{sin}^{-1}\sqrt{p_{k}(z)}$$

where

$$p_{k}(z) = \Phi\bigg(\frac{\Phi^{-1}(p_{k}^{0}) - \sqrt{\rho_k}z}{\sqrt{1-\rho_{k}}} \bigg)$$

we compute the chain rule by

$$\frac{d\theta_{k}}{dz} = \frac{d\theta_{k}}{dp_k} \times \frac{dp_{k}}{dz}$$

where

$$\frac{d\theta_{k}}{dp_k}= \frac{d}{dp_k}\bigg( 2 \text{sin}^{-1}(\sqrt{p_{k}(z)}) \bigg) = \frac{1}{\sqrt{p_k(1- p_k)}}$$

then $\frac{dp_{k}}{dz}$ is

$$\frac{dp_{k}}{dz} = \frac{d}{dz}\Phi\bigg(\frac{\Phi^{-1}(p_{k}^{0}) - \sqrt{\rho_k}z}{\sqrt{1-\rho_{k}}} \bigg) = -\sqrt{\rho_k} \times \frac{1}{1-\rho_k} \times \phi \bigg(\frac{\Phi^{-1}(p_{k}^{0}) - \sqrt{\rho_k}z}{\sqrt{1-\rho_{k}}} \bigg)$$

where $\phi(\cdot)$ is the standard normal PDF. Since we expand our function around $z = 0$,

$$\psi = \frac{\Phi^{-1}(p_{k}^{0}) - \sqrt{\rho_k}0}{\sqrt{1-\rho_{k}}} = \frac{\Phi^{-1}(p_{k}^{0})}{\sqrt{1-\rho_{k}}}$$

thus,

$$\left. \frac{d\theta_k}{dz} \right|_{z=0} = -\sqrt{\rho_k} \times \frac{1}{1-\rho_k} \times \phi(\psi)$$

the final form of $\frac{d\theta_{k}}{dz} = \frac{d\theta_{k}}{dp_k} \times \frac{dp_{k}}{dz}$ will be

$$\text{slope} = \bigg(\frac{1}{\sqrt{p_k(0)(1 - p_{k}(0))}} \bigg) \times \bigg( -\sqrt{\rho_k} \times \frac{1}{1-\rho_k} \times \phi(\psi)\bigg)$$

note that $p_{k}(0) = \Phi(\psi)$ and $\phi(psi)$ is the standard normal pdf evaluated at $\psi$.

7. Final Linear Form for Quantum Amplitude Encoding

After approximation:

$$\theta_{k}(z) \approx \text{offset}_{k} + \text{slop}_{k} \cdot z$$

The quantum circuit can easily implement rotation $\text{RY}(\theta_{k})$ using only a simple linear function of the quantum register represenattaing $z$. In short,

$$Z \xrightarrow{ X_k = a_{k} Z_{k} + b_k} X_k \xrightarrow{\Phi(\cdot)} p_{k}(Z) \xrightarrow{2 \text{sin}^{-1}\sqrt{\cdot}} \theta_{k}(z) \xrightarrow{\text{Linear Approx}} \text{Quantum Rotatopm RY}$$

8. Scaling

Motivation: Why Do We Shift and Scale?

In quantum circuits (like Qiskit's GCI model), quantum registers naturally hold integer states:

$$i \in [0, 2^n-1]$$

But the real-world credit risk model operates in a continuous real domain:

$$z \in [-c, c]$$

Our goal is to correctly map quantum integer states $i$ to real systemic risk factors $z$ and preserve the real-world meaning: systemic factor $z$ drives default probabilities and quantum rotations.

Adjusting for integer to normal range ensures that quantum integers correctly simulate real-world systemic risk factors in $[-c,c].$

Step 1: Shifting (Centering)

Problem:

• Quantum integers start from 0.
• Real $z$ domain is centered at 0 ( $-c$ to $c$).

Solution:

• Shift $i$ so that the middle integer maps to $z = 0$.
• Use shift:

$$\text{shift} = \frac{2^n-1}{2}$$

• Adjust offset:

$$\text{offset} += \text{real slope} \times (-\text{normal_max_value})$$

Step 2: Scaling (Stretching/Compressing)

Problem:

• Integer domain $[0, 2^n-1]$ is not the same size as real $[-c, c]$.

Solution:

• Scale slope to match step sizes:

$$\text{scale factor} = \frac{2c}{2^n-1}$$

• Adjust slope:

$$\text{slope} *= \text{scale factor}$$

---

Super Simple Example

Given:

• $c = 3$
• $n = 3$ qubits → $2^3 = 8$ states
• Real slope = $-0.02$
• Real offset = $0.1$

Correct Process:

1. Shift:

$$\text{offset} += (-0.02) \times (-3) = 0.1 + 0.06 = 0.16$$

1. Scale:

$$\text{scale factor} = \frac{6}{7} \approx 0.857$$

$$\text{new slope} = (-0.02) \times 0.857 \approx -0.01714$$

Now $i=0$ maps to $z=-3$ and $i=7$ maps to $z=+3$ correctly. Makesure you always adjust offset using real (unscaled) slope and only scale slope once after shifting. Thus, real-world meaning is preserved while quantum integers are mapped correctly into physical systemic factors $z$. Combininig shift and scale:

$$z(i) = \frac{2c}{2^{n}-1}\bigg(i - \frac{2^{n}-1}{2}\bigg)$$

Now the ==quantum states $|i\rangle$==s behave as if they are sampling real $z \in [-c,c]$.

Normal Distribution Quantum Circuit Modeling

The probability density function of a normal distribution is defined as

$$\mathbb{P}(X = x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{\sigma^2}}$$

Note that $\sigma^{2}$ is the variance. This is fro consistency with multivariate distributions, where the uppercase sigma $\Sigma$ is associated with the covariance.

The circuit considers the discretized version of the normal distribution on $2^{\text{num_qubits}}$ equidistant points, $x_i$, truncated to $\text{bonds}$. For a one-dimensional random variable, which means the num_qubits is a single integer, it applies the operation

$$\mathcal{P}_X |0\rangle^n = \sum_{i=0}^{2^n - 1} \sqrt{\mathbb{P}(x_i)} |i\rangle$$

where $n$ is num_qubits. The circuit loads the square root of the probabilities into the qubit amplitudes such that the sampling probability, which is the square of the amplitude, equals the probability of the distribution.

In the multi-dimensional case, the distribution is defined as

$$\mathbb{P}(X = x) = \frac{1}{\sqrt{2\pi\Sigma^{2}}} e^{-\frac{(x - \mu)^2}{\Sigma}}$$

where $\Sigma$ is the covariance. To specify a multivariate normal distribution num_qubits is a list of integers, each specifying how many qubits are used to discretize the respective dimension. The arguments $\mu$ and $\sigma$ in this case are a vector and square matrix.

If for instance, num_qubits = [2, 3] then $\mu$ is a 2d vector and $\sigma$ is the $2 \times 2$ covariance matrix. The first dimension is discretized using 2 qubits, hence on 4 points, and the second dimension on 3 qubits, hence 8 points. Therefore the random variable is discretized on $4 \times 8 = 32$ points.

Since, in general, it is not yet known how to efficiently prepare the qubit amplitudes to represent a normal distribution, this class computes the expected amplitudes and then uses the QuantumCircuit.initialize method to construct the corresponding circuit.

This circuit is for example used in amplitude estimation applications, such as finance [1, 2], where customer demand or the return of a portfolio could be modeled using a normal distribution.

References

[1]. https://qiskit-community.github.io/qiskit-finance/tutorials/09_credit_risk_analysis.html

[2]. https://arxiv.org/abs/1412.1183