We consider the problem of pricing discretely monitored Asian options over $T$ monitoring points where the underlying asset is modeled by a geometric Brownian motion. We provide two quantum algorithms with complexity poly-logarithmic in $T$ and polynomial in $1/ε$ , where $ε$ is the additive approximation error. Our algorithms are obtained respectively by using an $O(\log T)$ -qubit semi-digital quantum encoding of the Brownian motion that allows for exponentiation of the stochastic process and by analyzing classical Monte Carlo algorithms inspired by the semi-digital encodings. The best quantum algorithm obtained using this approach has complexity $\widetilde{O}(1/ε^{3})$ where the $\widetilde{O}$ suppresses factors poly-logarithmic in $T$ and $1/ε$ . The methods proposed in this work generalize to pricing options where the underlying asset price is modeled by a smooth function of a sub-Gaussian process and the payoff is dependent on the weighted time-average of the underlying asset price.We consider the problem of pricing discretely monitored Asian options over $T$ monitoring points where the underlying asset is modeled by a geometric Brownian motion. We provide two quantum algorithms with complexity poly-logarithmic in $T$ and polynomial in $1/ε$, where $ε$ is the additive approximation error. Our algorithms are obtained respectively by using an $O(\log T)$-qubit semi-digital quantum encoding of the Brownian motion that allows for exponentiation of the stochastic process and by analyzing classical Monte Carlo algorithms inspired by the semi-digital encodings. The best quantum algorithm obtained using this approach has complexity $\widetilde{O}(1/ε^{3})$ where the $\widetilde{O}$ suppresses factors poly-logarithmic in $T$ and $1/ε$. The methods proposed in this work generalize to pricing options where the underlying asset price is modeled by a smooth function of a sub-Gaussian process and the payoff is dependent on the weighted time-average of the underlying asset price.

arXiv:2402.10132Feb 15, 20240 citations

Brownian Motion in Orthogonal and Symplectic Groups

Zhiyang Tan, Piet W. Brouwer

Matrix Brownian motion provides a powerful framework for studying crossover ensembles in quantum chaos and quantum transport, as well as thermalization and information scrambling in many-body dynamics. Here, we develop a unified diagrammatic framework to characterize Brownian ensembles for orthogonal and symplectic random matrices, which describe systems with particle-hole symmetry. We compute polynomial averages up to fourth order and construct an orthogonally invariant interpolation for the disconnected $\mathrm{SO}^-(q)$ sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.Matrix Brownian motion provides a powerful framework for studying crossover ensembles in quantum chaos and quantum transport, as well as thermalization and information scrambling in many-body dynamics. Here, we develop a unified diagrammatic framework to characterize Brownian ensembles for orthogonal and symplectic random matrices, which describe systems with particle-hole symmetry. We compute polynomial averages up to fourth order and construct an orthogonally invariant interpolation for the disconnected $\mathrm{SO}^-(q)$ sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.

quant-pharXiv:2607.05094Jul 7, 20260 citations

On the theory of quantum brownian motion

R.F Roose, R. Dagonnier

PhysicaFeb 1, 19720 citations

Quantum Brownian Motion. II

Fabián H. Gaioli, Edgardo T. Garcia Alvarez, Diego G. Arbo

arXiv:quant-ph/9807049International Journal of Theoretical PhysicsJan 1, 19993 citations

Quantum Brownian Motion Revisited

Aniello Lampo, Miguel Ángel García March, Maciej Lewenstein

SpringerBriefs in physicsJan 1, 201918 citations

We consider the problem of pricing discretely monitored Asian options over

T

monitoring points where the underlying asset is modeled by a geometric Brownian motion. We provide two quantum algorithms with complexity poly-logarithmic in

T

and polynomial in

1/ε

, where

ε

is the additive approximation error. Our algorithms are obtained respectively by using an

O(\log T)

-qubit semi-digital quantum encoding of the Brownian motion that allows for exponentiation of the stochastic process and by analyzing classical Monte Carlo algorithms inspired by the semi-digital encodings. The best quantum algorithm obtained using this approach has complexity

\widetilde{O}(1/ε^{3})

where the

\widetilde{O}

suppresses factors poly-logarithmic in

T

and

1/ε

. The methods proposed in this work generalize to pricing options where the underlying asset price is modeled by a smooth function of a sub-Gaussian process and the payoff is dependent on the weighted time-average of the underlying asset price.

\mathrm{SO}^-(q)

sector of the orthogonal group. We consider applications relating to the fields of quantum information, quantum chaos, and quantum transport.