Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted we also find localised states at the edges between two bulks belonging to different topological phases. In order to analyse the impact of disorder we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.

Quantum anomalies lead to finite expectation values that defy the apparent symmetries of a system. These anomalies are at the heart of topological effects in electronic, photonic and atomic systems, where they result in a unique response to external fields but generally escape a more direct observation. Here, we implement an optical-network realization of a discrete-time quantum walk, where such an anomaly can be observed directly in the unique circular polarization of a topological midgap state. We base the system on a single-step protocol overcoming the experimental infeasibility of earlier multi-step protocols. The evolution combines a chiral symmetry with a previously unexplored unitary version of supersymmetry. Having experimental access to the position and the coin state of the walker, we perform a full polarization tomography and provide evidence for the predicted anomaly of the midgap states. This approach opens the prospect to dynamically distil topological states for quantum information applications. PACS numbers: 03.67.Ac, 42.50.-p, 03.65.VfIntroduction.-Quantum anomalies take a privileged position amongst fundamental physics as they equip quantum systems with robust topological effects. The historic backdrop for quantum anomalies is provided by the Atiyah-Singer index theorem for the Dirac operator [1], which states that the difference of zero modes with positive and negative chirality is a topological invariant. These zero modes are of fundamental significance not only because of their robustness against smooth deformations, but also since their definite chirality defies an apparent symmetry of the system, which results in an anomalous response to symmetry-breaking external fields. An early practical realization is the Su-Schrieffer-Heeger model for polyacetylene [2], where the anomalous properties of a midgap state result in charge fractionalization and spin-charge separation [3]. Interest in this phenomenon therefore quickly transcended the original setting of continuum and lattice field theories [4], and presently provides a major motivation for research particularly in electronic [5][6][7][8], superconducting [8-11], photonic [12-25] and ultracold atomic [26][27][28][29][30] systems. In all these settings, zero-modes represent symmetryprotected midgap states with unique finite expectation values of a relevant symmetry operator, resulting in a distinct response when probed by suitable external fields. This includes the formation of anomalous currents, as recently observed in Dirac and Weyl semimetals [31,32]. An equally early development was the relation of such anomalous behaviour to supersymmetry. In this case systems appear with partners that differ in the number of zero modes, with the prime example being a Dirac particle exposed to a magnetic field [33,34]. This feature is central to field-theoretic descriptions, but has been much less inquired in practical systems.

We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. The experimental platform is implemented with the well-established time-multiplexing architecture based on fibre-loops and coherent input states. The symmetry protected edge states are approximated with high similarities and we read-out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absence of sign flips between steps. Furthermore, the results show that investigating a bulk-boundary with a single bulk is experimentally feasible when decoupling the walk beforehand.

The dimensionality of the internal coin space of discrete-time quantum walks has a strong impact on the complexity and richness of the dynamics of quantum walkers. While two-dimensional coin operators are sufficient to define a certain range of dynamics on complex graphs, higher-dimensional coins are necessary to unleash the full potential of discrete-time quantum walks. In this work, we present an experimental realization of a discrete-time quantum walk on a line graph that, instead of two-dimensional, exhibits a four-dimensional coin space. Making use of the extra degree of freedom we observe multiple ballistic propagation speeds specific to higher-dimensional coin operators. By implementing a scalable technique, we demonstrate quantum walks on circles of various sizes, as well as on an example of a Husimi cactus graph. The quantum walks are realized via time-multiplexing in a Michelson interferometer loop architecture, employing as the coin degrees of freedom the polarization and the traveling direction of the pulses in the loop. Our theoretical analysis shows that the platform supports implementations of quantum walks with arbitrary 4 × 4 unitary coin operations, and usual quantum walks on a line with various periodic and twisted boundary conditions.

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