arXiv subjects

Martin Štefaňák

Publications and source records attributed to Martin Štefaňák.

6 recordsLinked to original sources

arXiv2026

Quantum Walk on a Line with Absorbing Boundaries

Absorption of two-state coined quantum walks on a finite line with two sinks located at $N$ and $-N$ is investigated. Elaborating on the results of Konno et al., J. Phys. A: Math. Gen. {\bf 36} 241 (2003), we derive closed formulas for the absorption probabilities at the boundaries in the limit of large system size $N$. Two limiting cases are considered, with the starting position $k$ being independent of $N$, or kept at a constant distance $δ$ from one of absorbers. In the first scenario, the absorption probability is determined only by the coin parameter and polar angle of the initial coin state decomposed into the eigenbasis of the coin operator. In the second case, a correction depending exponentially on $δ$ is introduced. Finally, we perform an extensive numerical investigation for small system size $N$, showing excellent agreement between numerical and analytical results.

quant-ph↗

arXiv2022

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping -- there can be non-zero probability of the quantum walker being localised in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned non-zero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.

quant-ph↗

arXiv2021

Relation between quantum walks with tails and quantum walks with sinks on finite graphs

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in the graph theory.

math-ph↗

arXiv2020

Complete classification of trapping coins for quantum walks on the 2D square lattice

One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, albeit the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four dimensional coin Hilbert space. The well known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the 2D square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or non-existence of the escaping state and the area covered by the spreading wave-packet.

quant-ph↗

arXiv2019

A counterintuitive role of geometry in transport by quantum walks

Quantum walks are accepted as a generic model for quantum transport. The character of the transport crucially depends on the properties of the walk like its geometry and the driving coin. We demonstrate that increasing transport distance between source and target or adding redundant branches to the actual graph may surprisingly result in a significant enhancement of transport efficiency. We explain analytically the observed non-classical effects using the concept of trapped states for several intriguing geometries including the ladder graph, the Cayley tree and its modifications.

quant-ph↗

arXiv2018

Probing Measurement Induced Effects in Quantum Walks via Recurrence

Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a non-unitary projection operation. In order to probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks due to their high control over all relevant parameters. However, measurement-induced dynamics in such a platform have not yet been realized. Here we implement controlled measurements in a discrete-time quantum walk based on time multiplexing. This is achieved by adding a deterministic out-coupling of the optical signal to include measurements constrained to specific positions resulting in the projection of the walker's state on the remaining ones. With this platform and coherent input light we experimentally simulate measurement-induced single particle quantum dynamics. We demonstrate the difference between dynamics with only a single measurement at the final step and those including measurements during the evolution. To this aim we study recurrence as a figure of merit, i.e. the return probability to the walker's starting position, which is measured in the two cases. We track the development of the return probability over 36 time steps and observe the onset of both recurrent and transient evolution as an effect of the different measurement schemes, a signature which only emerges for quantum systems. Our simulation of the observed one particle conditional quantum dynamics does not require a genuine quantum particle but is demonstrated with coherent light.

quant-ph↗