arXiv subjects

Pani W. Fernando

Publications and source records attributed to Pani W. Fernando.

3 recordsLinked to original sources

arXiv2025

Stochastic quantum adiabatic algorithm with fractional Brownian motion

Adiabatic Quantum Computing relies on the quantum adiabatic theorem, which states that a quantum system evolves along its ground state with time if the governing Hamiltonian varies infinitely slowly. However, practical limitations force computations to be performed within limited times, exposing the system to transitions into excited states, and thereby reducing the success probability. Here we investigate the counterintuitive hypothesis that incorporating stochastic noise, specifically noise driven by fractional Brownian motion, in a non-Markovian setup can enhance the performance of adiabatic quantum computing by improving its success probability at limited evolution times. The study begins by developing the mathematical framework to introduce stochastic noise multiplicatively into the Schrödinger equation, resulting in a stochastic Schrödinger equation. To preserve Itô integrability within the non-Markovian framework, a semimartingale approximation for fractional Brownian motion is employed. We perform numerical simulations to compare the performance of the quantum adiabatic algorithm with and without noise driven by fractional Brownian motion using the NP-complete Exact Cover-3 problem, transformed into the Ising model. Our results exhibit an improvement in success probability in the presence of noise driven by fractional Brownian motion with Hurst parameter $0<H<\frac{1}{2}$ and an increase in speedup as $H$ approaches 0. Although simulations are limited to problems involving a modest number of qubits, evidence suggests that the proposed approach scales favorably with the system size.

quant-ph↗

arXiv2020

On Markovian semigroups of Lévy driven SDEs, symbols and pseudo--differential operators

We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in $\mathbb{R}^d$ driven by pure jump--type Lévy processes. First, we will show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m(\mathbb{R}^d)$ with $1\le p, q<\infty$ and $m\in\mathbb{R}$. Secondly, we present some applications by proving the strong Feller property and give weak error estimates for approximating schemes of the SDEs over the Besov space $B_{\infty,\infty}^ m(\mathbb{R}^d)$.

math.PR↗

arXiv2015

Analytic properties of Markov semigroup generated by Stochastic Differential Equations driven by Lévy processes

We consider the stochastic differential equations of the form \begin{equation*} \begin{cases} dX^ x(t) = σ(X(t-)) dL(t) \\ X^ x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where $σ:\mathbb{R}^ d\to \mathbb{R}^ d$ is Lipschitz continuous and $L=\{L(t):t\ge 0\}$ is a Lévy process. Under this condition on $σ$ it is well known that the above problem has a unique solution $X$. Let $(\mathcal{P}_{t})_{t\ge0}$ be the Markovian semigroup associated to $X$ defined by $( \mathcal{P}_t f) (x) := \mathbb{E} [ f(X^ x(t))]$, $t\ge 0$, $x\in \mathbb{R}^d$, $f\in \mathcal{B}_b(\mathbb{R}^d)$. Let $B$ be a pseudo--differential operator characterized by its symbol $q$. Fix $ρ\in\mathbb{R}$. In this article we investigate under which conditions on $σ$, $L$ and $q$ there exist two constants $γ>0$ and $C>0$ such that $$ \lvert B \mathcal{P}_t u \rvert_{H^ρ_2} \le C \, t^{-γ} \,\lvert u \rvert_{H^ρ_2}, \quad \forall u \in {H^ρ_2}(\mathbb{R}^d ),\, t>0. $$

math.PR↗