Quantum analog-encoding for correlated Gaussian vectors and their exponentiation with application to rough volatility

Paper: arXiv 2604.22463

Authors: Tassa Thaksakronwong, Koichi Miyamoto

Abstract

Quantum computing may speed up numerical problems involving large matrices that are demanding for classical computers, and active research on this possibility is ongoing. In this work, we propose quantum algorithms for the exact simulation of a normalised correlated Gaussian random vector $|x\rangle=\vec{x}/\lVert\vec{x}\rVert$, $\vec{x}\sim\mathcal{N}(0,Σ)$, and its exponentiation $|e^{\vec{x}} \rangle= e^{\vec{x}}/\lVert e^{\vec{x}}\rVert$. When an $O(\mathrm{polylog} N)$-gate-depth quantum data loader for the covariance matrix $Σ\in\mathbb{R}^{N\times N}$ is available, preparing $|x\rangle$ and $|e^{\vec{x}}\rangle$ require $\widetilde{O}\left(\frac{\lVertΣ\rVert_F}{λ_{\max}}κ^{1.5}\right)$ and $\widetilde{O}\left(\lVert\vec{x}\rVert\frac{\lVertΣ\rVert_F}{λ_{\max}}κ^{1.5}\right)$ elementary gate depth respectively, where $\lVertΣ\rVert_F$, $λ_{\max}$, $κ$ denote the Frobenius norm, maximal eigenvalue, and condition number of $Σ$. Motivated by financial applications, we provide an end-to-end resource analysis when $\vec{x}$ represents a sample path of a Riemann-Liouville or standard fractional Brownian motion, or of a stationary fractional Ornstein-Uhlenbeck process. As a concrete example, we construct the quantum state encoding the rough Bergomi variance process and analyse the extraction of the integrated variance via quantum amplitude estimation. Under specific conditions, the dependence of $\lVertΣ\rVert_F/λ_{\max}$ and $κ$ on $N$ is small, and subcubic complexity in $N$ is achieved, indicating a quantum advantage over classical Cholesky-based sampling methods. To our knowledge, this constitutes the first quantum algorithmic framework for the amplitude encoding of exponentiated Gaussian processes, providing foundational primitives for quantum-enhanced financial modelling.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper features highly advanced quantum algorithm construction and rigorous complexity analysis involving spectral properties of covariance matrices. While it provides numerical analysis of complexity exponents, it focuses on theoretical quantum frameworks rather than practical backtesting on empirical market data.

Research Flowchart

  flowchart TD
    A[<b>Research Goal:</b> Speeding up simulation of high-dimensional Gaussian processes] --> B[<b>Inputs:</b> Covariance matrix Σ & Stochastic processes (fBM, fOU)]
    B --> C[<b>Methodology:</b> Quantum analog-encoding & Amplitude estimation]
    C --> D[<b>Computational Process:</b> Preparing states |x⟩ and |e˟⟩ via gate-depth circuits]
    D --> E[<b>Application:</b> Pricing derivative models (e.g., rough Bergomi)]
    E --> F[<b>Outcomes:</b> Subcubic complexity & Quantum advantage over classical Cholesky]

Cited References (82)

1. Grand unification of quantum algorithms (2021)
2. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics (2019) — link
3. Hamiltonian simulation by uniform spectral amplification (2017) — link
4. Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra (2022)
5. Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications (2023) — link
6. Quantum linear system algorithm for dense matrices (2018)
7. A quantum spectral method for simulating stochastic processes, with applications to M onte C arlo (2023) — link
8. Nearest centroid classification on a trapped ion quantum computer (2021)

0 of 82 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.