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arXiv:2504.19801

Stochastic quantum adiabatic algorithm with fractional Brownian motion

Osanda Chinthila, Pani W. Fernando, Anuradha Mahasinghe, Kaushika De Silva, Kumara, Sarath

Published Apr 28, 2025• 0 citations
arXivHTMLPDF

Abstract

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

DOI
10.48550/arxiv.2504.19801

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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
We analyze the power of the quantum adiabatic evolution algorithm (QAA) for solving random computationally hard optimization problems within a theoretical framework based on random matrix theory (RMT). We present two types of driven RMT models. In the first model, the driving Hamiltonian is represented by Brownian motion in the matrix space. We use the Brownian motion model to obtain a description of multiple avoided crossing phenomena. We show that nonadiabatic corrections in the QAA are due to the interaction of the ground state with the ``cloud'' formed by most of the excited states, confirming that in driven RMT models, the Landau-Zener scenario of pairwise level repulsions is not relevant for the description of nonadiabatic corrections. We show that the QAA has a finite probability of success in a certain range of parameters, implying a polynomial complexity of the algorithm. The second model corresponds to the standard QAA with the problem Hamiltonian taken from the RMT Gaussian unitary ensemble (GUE). We show that the level dynamics in this model can be mapped onto the dynamics in the Brownian motion model. For this reason, the driven GUE model can also lead to polynomial complexity of the QAA. The main contribution to the failure probability of the QAA comes from the nonadiabatic corrections to the eigenstates, which only depend on the absolute values of the transition amplitudes. Due to the mapping between the two models, these absolute values are the same in both cases. Our results indicate that this ``phase irrelevance'' is the leading effect that can make both the Markovian- and GUE-type QAAs successful.
The Time-Fractional Schrödinger Equation (TFSE) is an effective tool for simulating the dynamics of non-Markovian quantum systems. The Quantum Speed Limit (QSL) time characterizes the minimum time required for the evolution of a non-Markovian quantum system. In this paper, Wei's TFSE is employed to simulate the non-Markovian quantum accelerated evolution process in the Resonant Dissipative Jaynes-Cummings (RDJC) model. By solving the QSL time of a time-fractional single-qubit open system, the enhancement mechanism of the system evolution speed induced by the non-Markovian memory effects of the environment is revealed. Further studies show that the optimized acceleration of the system evolution can be achieved by jointly regulating the fractional order, coupling strength, and photon number. Comparative analyses indicate that Wei's TFSE can accurately capture the non-Markovian accelerated dynamical features of the system over the entire fractional order range, whereas Naber's TFSE is applicable only within a limited fractional order interval. In addition, the comparisons of the average simulation time for calculating the dynamical trajectory of the excited-state probability demonstrate that Wei's TFSE has a significant simulation advantage in computational efficiency. Therefore, Wei's TFSE is more accurate and efficient for simulating the accelerated dynamics of non-Markovian quantum systems.
Adiabatic quantum computation requires that the system remains in its ground state. However, an adiabatic process often loses partly or entirely its quantumness, such as entanglement in its long runtime, due to the system-environment interactions. Here we put forward an effective quantum control technique to realize the adiabatic quantum search algorithm in a nonadiabatic regime and in the presence of environment. Using the non-Markovian quantum state diffusion equation approach, we numerically study the system dynamics characterized by the success probability. The results show that the probability increases with decreasing coupling strength and temperature. In particular, non-Markovianity from the environment can help to enhance the success probability. By choosing a suitable pulse intensity and period, a high success probability can be obtained in a short runtime for weak system-bath coupling, low temperature, and strong non-Markovian heat baths.
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A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
Quantum computing seeks to use the powers of quantum mechanics to accomplish tasks that classical computers cannot easily accomplish. Adiabatic quantum computing is one flavor of quantum computing that slowly changes a system with time in such a way that the state remains close to the ground state for the entire evolution. An easy way to implement and study adiabatic computing is using stoquastic Hamiltonians with no-sign problem. These Hamiltonians can be readily simulated on classical computers using quantum Monte Carlo schemes; though, it is unknown whether such simulation captures the speed and power of the quantum computation.In this work, I study stoquastic Hamiltonians, focusing on quantum mechanical tunneling through classically inaccessible regions and barriers. I study multi-qubit tunneling problems to determine what properties of the barrier make this problem quantum mechanically or classically hard or inefficient. I compare quantum adiabatic computing with a path-integral Quantum Monte Carlo algorithm, which is a classical algorithm designed to simulate the quantum mechanical dynamics of the adiabatic evolution. Limited numerical data shows strong correlation between adiabatic and Monte Carlo runtimes, but due to computational limits, only small system sizes could be sampled.Additionally, I study the properties of the quantum adiabatic algorithm in both the asymptotic limit of a large number of qubits and long runtimes and more realistic finite system sizes and non-adiabatic conditions. I develop a modification of the existing Villain transformation that allows it to find the asymptotic running time of the quantum evolution for different barrier sizes. Furthermore, I compare this to finite system sizes and discover the extremely large numbers of qubits are needed for these asymptotics to be useful.When the quantum dynamics are run faster than adiabatically, new behavior arises that could potentially lead to enhancements in the quantum runtimes. I explore these enhancements, discovering when they do and do not occur. I also develop a broad framework for determining the nature of the near-adiabatic behavior of any quantum evolution, using only information about the energy gap between the ground state and the first excited state.