# Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer

Dynamically inﬂuencing the walker’s evolution gives a high degree of ﬂexibility for studying various applications. Additionally to the described applications for Anderson localization and percolation we realised time-multiplexed finite quantum walks of variable size, the preparation of non-localised input states and their dynamical evolution. As a further application, we implement a state transfer scheme for an arbitrary input state to two different output modes.

First, we vary the size of the underlying graph by introducing corresponding reflection operations with the EOM as illustrated in fig. 1.

The timing of the reflection switching consequently determines how large the graph is and thus how far the walker’s wavefunction is spread out. Three examples for finite graphs in contrast to an unlimited walk are presented in figure 2.

In figure 2a, we observe an additional interesting feature of the walk on a finite graph: The walker, initially spread out, is not only reflected at the boundaries, but even exhibits revival (within experimental errors) at the initial position 0 in step 10.

Most quantum walk experiments including those presented above are performed with localised input states. However, there are theory proposals, e.g. for boson sampling in a time-multiplexing setup or for quantum search, which rely on initial states occupying several positions. The ''in-situ'' state preparation, i.e. the preparation of different initial states within the quantum walk setup, guarantees phase-stable pulse trains with controlled polarization inherently exhibiting a coherent evolution. Two examples of transformations accessible by the EOM used in the present setup are illustrated in Figure 3. We use these schemes to prepare two states with four positions, but different polarizations.

For these given input states, we observe the evolution in a balanced walk on an infinite graph (see Fig. 4). The comparison of the two cases reveals the relevance of the input polarisation: For the | VVHH> state, the walker evolves along two branches which are more distinctively defined than for the | VHVH>state. Here in comparison, especially the right branch is much more smeared out leading to a very different intensity distribution in position space. By reprogramming the EOM switching patterns, the preparation of well-defined input states distributed over a higher number of positions is possible as well.

Zhan et al. in [1] proposed a state transfer scheme via discrete time quantum walks. The possibility of dynamical coin control allows for the easy implementation of this protocol in our setup. We are able to observe a revival of the walker's initial polarisation state at a chosen position, although it is beforehand spread out in space and undergoes switchings in polarization. Different protocols transferring the state to different positions can be realised simply by reprogramming the EOM. In figure 5, we show the implementation of 2 exemplary non-trivial schemes.

All the presented experiments rely on the full dynamical control of the time-multiplexed quantum walk, which includes adjustable coin operation as well as the possibility to ﬂexibly conﬁgure the underlying graph structures. Details about the presented applications as well as a thorough explanation of all the experimental challenges and pitfalls including a full analysis of the EOM features, timings and loss handling are published in [2].

**References:**

- Zhan, X., Qin, H., Bian, Z., Li, J., Xue, P., 2014. Perfect state transfer and efficient quantum routing: A discrete-time quantum-walk approach. Phys. Rev. A 90, 012331. doi:10.1103/PhysRevA.90.012331
- Nitsche, T., Elster, F., Novotný, J., Gábris, A., Jex, I., Barkhofen, S., Christine Silberhorn, 2016. Quantum walks with dynamical control: graph engineering, initial state preparation and state transfer. New J. Phys. 18, 063017. doi:10.1088/1367-2630/18/6/063017