Figure 1.
CVaR-driven VQA workflow: classical input, variational encoding, threshold oracle, MLAE, and CVaR evaluation. Dashed lines indicate feedback and spintronic mapping. Total qubits N = n + 2.
Table 1.
Quantum approaches to financial risk estimation.
| Authors | Method | Circuit Depth | Hardware | Limitation / Gaps |
|---|---|---|---|---|
| Woerner & Egger [9] | QAE | 500–1000 | Theoretical | Deep circuits unsuitable for NISQ; no hardware results |
| Orús et al. [12] | Quantum annealing | 200–500 | D-Wave | Limited to small portfolios; lacks CVaR support |
| Herman et al. [13] | VQA | 50–100 | IBM Simulator | No scalability analysis on real datasets |
| Chakrabarti et al. [14] | QPE + QAA | 600–1200 | Theoretical | Deep circuits not NISQ compatible; synthetic dataset only |
| Fontana et al. [15] | VQA for VaR | 100–200 | IBM Quantum | Focused only on VaR; no CVaR extension |
| Li & Zhang [16] | Optimized QPE Oracle | 150–300 | Simulated | Assumes ideal distributions; lacks robustness test |
| Pérez-Salinas et al. [17] | VQE | 200–400 | Simulated | High classical cost; limited scalability |
| Miyamoto & Kubo [18] | Quantum walk | 300–600 | N/A | Lacks practical integration with CVaR pipelines |
| Gilles et al. [19] | QAE with EVaR/RVaR | 500–1000 | NISQ Simulator | No hardware runs; circuit depth remains a bottleneck |
| Matsakos & Nield [20] | Quantum-enhanced Monte Carlo | 400–800 | Simulated | Complex mapping for highdimensional risks |
| Wu et al. [21] | End-to-end QAE | 300–600 | Simulated | No extension to American options; limited derivatives covered |
| Ghosh et al. [22] | Dynamic Amplitude Estimation | 200–400 | Simulated | Domain-specific; lacks generalizability |
| Yohichi et al. [23] | Quantum PDE solver | 300–600 | None | No quantum hardware implementation yet |
| Cong & Thi [24] | VQA Survey | 100–500 | N/A | No empirical tests; theoretical-only |
| Thakkar et al. [25] | Quantum ML | 150–300 | Simulated | Not integrated into enterprise-grade risk systems |
Figure 2.
Proposed 4-qubit variational ansatz with Ry(θi) rotations and CNOT entanglement chain for encoding portfolio loss amplitudes.
Figure 3.
Conceptual spintronic view of the proposed 4-qubit Variational ansatz.
Figure 4.
Oracle Uf implemented as a multi-controlled-X (MCX) gate: the ancilla qubit flips if the input register encodes a loss L(x) ≥ τ. For τ = 8, the MSB (x3 = 1) triggers the ancilla flip, marking tail states.
Figure 5.
Conceptual spintronic view of the proposed oracle Uf.
Figure 6.
MLAE circuit for amplifying the quantum states with depth k.
Figure 7.
Conceptual spintronic MLAE architecture.
Figure 8.
Convergence of the variational quantum ansatz (VQA) during CVaR-based optimization. The smooth decay of the loss proxy indicates stable parameter adaptation within 40 iterations.
Table 2.
Circuit and execution metrics.
| Platform | Avg. Depth | Fidelity |
|---|---|---|
| AerSimulator (ideal) | 98 | 1.000 |
| AerSimulator (noisy) | 98 | 0.991 |
| IBM Brisbane (real) | 112 | 0.967 |
Figure 9.
Threshold oracle validation showing ancilla measurement outcomes. The state ancilla = 1 corresponds to flagged events with L(x) > 0.75.
Figure 10.
Effect of noise models on CVaR estimation error. Measurement mitigation (M3) reduces total estimation error by over 12%, confirming enhanced robustness under realistic NISQ conditions.
Figure 11.
Log-likelihood distribution for the MLAE estimator showing a sharp maximum at the true probability p∗ ≈ 0.048.
Table 3.
Estimated energy per logical gate using spintronic analogues.
| Logical Gate | Physical Mechanism | Energy (fJ) |
|---|---|---|
| Hadamard | STO precession | 1.9 |
| Ry(θ) | Rashba SO coupling | 1.7 |
| CNOT | Exchange interaction | 2.3 |
| Measurement | MTJ readout | 2.0 |
Figure 12.
Scalability of the MLAE–QCVaR framework: runtime increases approximately linearly with the number of state qubits for fixed estimation precision.
Figure 13.
CVaR sensitivity to α: consistent monotonic increase indicating proper representation of tail risk.
Table 4.
Algorithmic and resource comparison.
| Metric | Classical MC | Canonical QAE | VQA+MLAE (Ours) |
|---|---|---|---|
| Sampling complexity | O(1/ϵ2) | O(1/ϵ) | O(1/ϵ) |
| Circuit depth (max k) | – | 20 | 6 (–70%) |
| Tail-probability MAE | – | 2.1 × 10−3 | 1.7 × 10−3 |
| CVaR error | 0.4% | 1.3% | <1.1% |
| Total shots (K = 6) | 106 | 20,480 | 6,144 |
| Runtime (ibm_brisbane) | 184 s | 42 s | 13 s |
Figure 14.
Workflow comparison: classical → QAE → hybrid MLAE.
Figure 15.
Fidelity degradation under depolarizing and readout noise.
Figure 16.
Spintronic energy–fidelity trade-off (conceptual).