Quantum walks are powerful tools for building quantum algorithms, modeling transport phenomena, and designing topological systems. Here we present a photonic implementation of a quantum walk in two spatial dimensions, where the lattice of walker positions is encoded in the transversewavevector components of a paraxial light beam. The desired quantum dynamics is obtained by means of a sequence of liquid-crystal devices ("g-plates"), which apply polarization-dependent transverse kicks to the photons in the beam. We first characterize our setup, and then benchmark it by implementing a periodically-driven Chern insulator and probing its topological features. Our platform is compact, versatile and cost-effective: most evolution parameters are controlled dynamically, the walker distribution is detected in a single shot, and the input state can be tailored at will. These features offer exciting prospects for the photonic simulation of two-dimensional quantum systems. * AD'E and FC contributed equally to this work †

Boson Sampling is a computational paradigm representing one of the most viable and pursued approaches to demonstrate the regime of quantum advantage. Recent results have shown significant technological leaps in single-photon generation and detection, leading to progressively larger instances of Boson Sampling experiments in different photonic systems. However, a crucial requirement for a fully-fledged platform solving this problem is the capability of implementing large-scale interferometers, that must simultaneously exhibit low losses, high degree of reconfigurability and the realization of arbitrary transformations. In this work, we move a step forward in this direction by demonstrating the adoption of a compact and reconfigurable 3D-integrated platform for photonic Boson Sampling. We perform 3- and 4-photon experiments by using such platform, showing the possibility of programming the circuit to implement a large number of unitary transformations. These results show that such compact and highly-reconfigurable layout can be scaled up to experiments with larger number of photons and modes, and can provide a viable direction for hybrid computing with photonic processors.

Photon indistinguishability plays a fundamental role in information processing, with applications such as linear-optical quantum computation and metrology. It is then necessary to develop appropriate tools to quantify the amount of this resource in a multiparticle scenario. Here we report a four-photon experiment in a linear-optical interferometer designed to simultaneously estimate the degree of indistinguishability between three pairs of photons. The interferometer design dispenses with the need of heralding for parametric down-conversion sources, resulting in an efficient and reliable optical scheme. We then use a recently proposed theoretical framework to quantify fourphoton indistinguishability, as well as to obtain bounds on three unmeasured two-photon overlaps. Our findings are in high agreement with the theory, and represent a new resource-effective technique for the characterization of multiphoton interference.

Quantum walks represent paradigmatic quantum evolutions, enabling powerful applications in the context of topological physics and quantum computation. They have been implemented in diverse photonic architectures, but the realization of two-particle dynamics on a multidimensional lattice has hitherto been limited to continuous-time evolutions. To fully exploit the computational capabilities of quantum interference it is crucial to develop platforms handling multiple photons that propagate across multidimensional lattices. Here, we report a discrete-time quantum walk of two correlated photons in a two-dimensional lattice, synthetically engineered by manipulating a set of optical modes carrying quantized amounts of transverse momentum. Mode-couplings are introduced via the polarization-controlled diffractive action of thin geometric-phase optical elements. The entire platform is compact, efficient, scalable, and represents a versatile tool to simulate quantum evolutions on complex lattices. We expect that it will have a strong impact on diverse fields such as quantum state engineering, topological quantum photonics, and Boson Sampling.

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