We propose a simple quantum algorithm for implementing the diffusion step of grid-based Bayesian filters. The method encodes the advected state density and the process noise density into quantum registers and realizes diffusion using a quantum Fourier transform--based adder. This avoids the explicit convolution required in classical implementations and the repeated coin-flip operations used in quantum random walk approaches. Numerical simulations using a gate-based quantum computing simulator confirm that the proposed approach reproduces the desired probability densities while requiring significantly fewer quantum gates and much shallower circuit depth.
We propose a simple quantum algorithm for implementing the diffusion step of grid-based Bayesian filters. The method encodes the advected state density and the process noise density into quantum registers and realizes diffusion using a quantum Fourier transform--based adder. This avoids the explicit convolution required in classical implementations and the repeated coin-flip operations used in quantum random walk approaches. Numerical simulations using a gate-based quantum computing simulator confirm that the proposed approach reproduces the desired probability densities while requiring significantly fewer quantum gates and much shallower circuit depth.
We propose a gate-based quantum algorithm for the prediction step of Bayesian state estimation based on the Fokker-Planck equation on a discretized position-velocity state space. The probability density is encoded in the amplitudes of a quantum state, enabling a compact representation of high-dimensional distributions. Exploiting the circulant structure of finite-difference operators, the evolution is realized in the spectral domain using quantum Fourier transforms and phase rotations.
A key result is that the drift component can be implemented exactly in amplitude space, leading to an accurate reproduction of the classical transport dynamics. In contrast, the diffusion term does not admit a linear representation in amplitude space due to the nonlinear relation between probability density and wave function. To enable a quantum implementation, we introduce a unitary surrogate based on a Wick rotation, transforming diffusion into a dispersive phase evolution. This yields a fully unitary propagation that can be implemented efficiently on a gate-based quantum computer. The proposed method is evaluated numerically for different scenarios and shows strong agreement with the exact solution of the Fokker-Planck equation. The approach demonstrates the potential of quantum computing for Bayesian state estimation, as the representable state space grows exponentially with the number of qubits. This allows the efficient representation and propagation of probability densities that would otherwise require complex tensor decompositions on classical hardware, making the method a promising candidate for high-dimensional filtering problems.
Quantum computing promises a theoretical computational advantage for solving certain problems, thus it holds significant academic and commercial interest. This thesis begins by reviewing established quantum algorithms and their applications. Then I expand this set to demonstrate quantum computational advantage for solving the drift-diffusion equation, a computationally intensive problem with applications in finance, modelling, and manufacturing. First, I introduce a complexity analysis for four classical algorithms designed to solve the multi-dimensional drift-diffusion equation. I then present four quantum algorithms, leveraging a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. By comparing the complexities of these methods to their classical counterparts, I find that diagonalisation via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. I employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer. Finally, I compare the circuit depths for five gate sets to implement a quantum algorithm solving the drift-diffusion equation in two spatial dimensions. This uses the diagonalisation by the quantum Fourier transform algorithm. The gate sets are: an unconstrained gate set, TK1 gate set from Quantinuum, the native gate sets of IBM Heron and IonQ, and Fujitsu’s space-time efficient analog rotation (STAR) gate set. This analysis covers a set of illustrative scenarios using up to 22 qubits. I find that while scaling with spatial resolution aligns with theoretical predictions in one dimension, scaling with spatial dimension is less efficient than theorised due to overhead from block encoding. Finally, using the STAR gate set, I find that even minimal problem instances exceed the operational limits of current quantum hardware. We find that for a test scenario using 22 logical qubits, requires 1 × 106 logical operations, assuming full connectivity.
Simulating differential equations on classical computers becomes an intractable problem if the grid size is extremely large. Quantum computers are believed to achieve a possibly exponential speedup in the matrix operation. In this paper, we propose a quantum algorithm for solving the advection-diffusion-reaction equation by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. First, the effectiveness of the proposed approximate PITE operator is justified by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit to realize the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. Compared to the existing algorithms for the quantum linear system problem, our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound. Moreover, numerical simulations using gate-based quantum emulator for 1D/2D examples are also provided to verify our algorithm. Finally, we extend our algorithm to the coupled system of advection-diffusion equations to show the prospects for practical applications.
The prediction step is a crucial element of the Bayesian recursion for target tracking and state estimation in general. Discrete representations of the probability density function (pdf) can deal with non-linear models and nonGaussian noise, however, the prediction step is challenging to solve on classical computers. In this paper, a novel concept of quantum simulation to solve the application of a continuous noise motion model to a pdf is presented. The pdf is prepared as the squared amplitudes for the basis states spanned by the number of used qubits. The number of required qubits grows linearly in the number of time steps to simulate in a prediction phase. One step performs a single Brownian motion, which is used to generate the diffusion of the Wiener increments. A drift function can be implemented based on a separate register, which holds the pdf on the velocity information for an adequate discretization. The approach will be visualized in terms of quantum circuits and evaluated based on a quantum simulator.
The Quantum Lattice Boltzmann Method (QLBM) is one of the most promising approaches for realizing the potential of quantum computing in simulating computational fluid dynamics. Many recent works mostly focus on classical simulation, and rely on full state tomography. Several key algorithmic issues like observable readout, data encoding, and impractical circuit depth remain unsolved. As a result, these are not directly realizable on any quantum hardware. We present a series of novel algorithmic advances which allow us to implement the QLBM algorithm, for the first time, on a quantum computer. Hardware results for the time evolution of a 2D Gaussian initial density distribution subject to a uniform advection-diffusion field are presented. Furthermore, 3D simulation results are presented for particular non-uniform advection fields, devised so as to avoid the problem of diminishing probability of success due to repeated post-selection operations required for multiple timesteps. We demonstrate the evolution of an initial quantum state governed by the advection-diffusion equation, accounting for the iterative nature of the explicit QLBM algorithm. A tensor network encoding scheme is used to represent the initial condition supplied to the advection-diffusion equation, significantly reducing the two-qubit gate count affording a shorter circuit depth. Further reductions are made in the collision and streaming operators. Collectively, these advances give a path to realizing more practical, 2D and 3D QLBM applications with non-trivial velocity fields on quantum hardware.