arXiv subjects

Lukas Hantzko

Publications and source records attributed to Lukas Hantzko.

5 recordsLinked to original sources

arXiv2025

Sub-Riemannian geometry of measurement based quantum computation

The computational power of quantum phases of matter with symmetry can be accessed through local measurements, but what is the most efficient way of doing so? In this work, we show that minimizing operational resources in measurement-based quantum computation on subsystem symmetric resource states amounts to solving a sub-Riemannian geodesic problem between the identity and the target logical unitary. This reveals a geometric structure underlying MBQC and offers a principled route to optimize quantum processing in computational phases.

quant-ph↗

arXiv2025

Classification of coined quantum walks on the line and comparison to correlated classical random walks

We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins, and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation and a bijective parametrisation modulo equivalence of the walks' limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with the ones in the classical case.

quant-ph↗

arXiv2024

Fast generation of Pauli transfer matrices utilizing tensor product structure

Analysis of quantum processes, especially in the context of noise, errors, and decoherence is essential for the improvement of quantum devices. An intuitive representation of those processes modeled by quantum channels are Pauli transfer matrices. They display the action of a linear map in the $n$-qubit Pauli basis in a way, that is more intuitive, since Pauli strings are more tangible objects than the standard basis matrices. We set out to investigate classical algorithms that convert the various representations into Pauli transfer matrices. We propose new algorithms that make explicit use of the tensor product structure of the Pauli basis. They convert a quantum channel in a given representation (Chi or process matrix, Choi matrix, superoperator, or Kraus operators) to the corresponding Pauli transfer matrix. Moreover, the underlying principle can also be used to calculate the Pauli transfer matrix of other linear operations over $n$-qubit matrices such as left-, right-, and sandwich multiplication as well as forming the (anti-)commutator with a given operator. Finally, we investigate the runtime of these algorithms, derive their asymptotic scaling and demonstrate improved performance using instances with up to seven qubits.

quant-ph↗

arXiv2024

Tensorized Pauli decomposition algorithm

This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a detailed complexity analysis, we show that the algorithm admits the best known worst-case scaling and more favorable runtimes for many practical examples. Numerical experiments are provided to validate the asymptotic speed-up already for small instance sizes, underscoring the algorithm's potential significance in the realm of quantum computing and quantum chemistry simulations.

quant-ph↗

arXiv2022

Conserved charges for rational electromagnetic knots

We revisit a newfound construction of rational electromagnetic knots based on the conformal correspondence between Minkowski space and a finite $S^3$-cylinder. We present here a more direct approach for this conformal correspondence based on Carter-Penrose transformation that avoids a detour to de Sitter space. The Maxwell equations can be analytically solved on the cylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed into Minkowski coordinates using the conformal map. The resultant "knot basis" electromagnetic field configurations have non-trivial topology in that their field lines form closed knots. We consider finite, complex linear combinations of these knot-basis solutions for a fixed spin $j$ and compute all the $15$ conserved Noether charges associated with the conformal group. We find that the scalar charges either vanish or are proportional to the energy. For the non-vanishing vector charges, we find a nice geometric structure that facilitates computation of their spherical components as well. We present analytic results for all charges for up to $j{=}1$. We demonstrate possible applications of our findings through some known previous results.

math-ph↗