arXiv subjects

Jan W. Delfs

Publications and source records attributed to Jan W. Delfs.

2 recordsLinked to original sources

arXiv2026

A Quantum Algorithm for Solving the Poisson Equation for Free Field Conditions via the Hockney Method

For the often encountered problem of the Poisson equation, this work presents a quantum algorithm solving it based on the quantum Fourier transform (QFT) for periodic boundary conditions as well as free field conditions, where the latter is realized via the Hockney method. Besides the QFT and an initialization procedure for amplitude encoding, the algorithm just uses a procedure for multiplying the state vector by a diagonal matrix w.r.t. amplitude encoding. For the latter, two alternative implementations are considered here. The first variant is a version of the LCU method and the second is a sequence of multi-controlled rotation gates that represents a factoring of the multiplied values into absolute values and complex phase factors. The functionality of the algorithm is verified via comparing the results obtained from state vector simulations for one- and two-dimensional test examples with their analytical solutions. For the considered test examples, it is found that the success probability for obtaining the desired ancilla qubit subspace in the LCU version is a factor of around two higher than that for the sequence of multi-controlled rotation gates. However, the LCU version requires a number of ancilla qubits up to the number of qubits that is set to store the discretized source term of the Poisson equation in amplitude encoding, whereas the sequence of multi-controlled rotation gates demands only one ancilla qubit. Computations of the success probabilities for both variants furthermore indicate that the success probability converges for a specific problem with increasing resolution. Concerning the required computational resources for the quantum algorithm, the conclusion is drawn that while the QFT is a more efficient procedure than its classical counterpart, the current implementations of the other necessary steps in the algorithm diminish the efficiency w.r.t. the runtime.

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arXiv2026

Resource Implications of Different Encodings for Quantum Computational Fluid Dynamics

For quantum algorithms for problems in which the task is to compute an entire field of values, like e.g. computational fluid dynamics (CFD), it is often proposed amplitude encoding w.r.t. multiple qubits; however, the efforts implied by it for initialization and read-out are not addressed. This work is devoted specifically to this issue: It reviews different encoding schemes in quantum computing, discussing their computational costs for initialization and read-out as well as resulting aspects for their usage via minimal examples. The considerations in previous literature on the required computational resources for amplitude encoding w.r.t. multiple qubits are extended in the presented quantification by explicitly deducing the circuit depth that results for the decomposed initialization procedure of V. V. Shende et al. [1, 2] and deriving an upper bound for the necessary number of executions of a quantum algorithm to extract the encoded values with a specific accuracy. For these two results, an empirical verification via the means provided by IBM's quantum computing simulation framework $\textit{Qiskit}$ [3] is given. In the framework of the study on the required number of runs to achieve a desired accuracy, it is however found that the derived upper bound, scaling like $ {\tilde{n}^2} ~ {\ln( {\tilde{n}} )} $ with the number of encoded values $ {\tilde{n}} $, is too conservative to be used for precise estimations. Therefore, a corresponding study of the required runs for the reference distribution of equal probabilities for all basis states is done in particular, which suggests $ {\tilde{n}} ~ { \ln( {\tilde{n}} ) } $ as an empirical scaling law. Since the view regarding CFD applications is taken here, it is presented in particular that the insights from this work lead to a new encoding approach, which is proposed specifically for a quantum algorithm for the lattice Boltzmann method.

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