The subset sum problem is a typical NP-complete problem that is hard to solve efficiently in time due to the intrinsic superpolynomial-scaling property. Increasing the problem size results in a vast amount of time consuming in conventionally available computers. Photons possess the unique features of extremely high propagation speed, weak interaction with environment and low detectable energy level, therefore can be a promising candidate to meet the challenge by constructing an a photonic computer computer. However, most of optical computing schemes, like Fourier transformation, require very high operation precision and are hard to scale up. Here, we present a chip built-in photonic computer to efficiently solve the subset sum problem. We successfully map the problem into a waveguide network in three dimensions by using femtosecond laser direct writing technique. We show that the photons are able to sufficiently dissipate into the networks and search all the possible paths for solutions in parallel. In the case of successive primes the proposed approach exhibits a dominant superiority in time consumption even compared with supercomputers. Our results confirm the ability of light to realize a complicated computational function that is intractable with conventional computers, and suggest the subset sum problem as a good benchmarking platform for the race between photonic and conventional computers on the way towards "photonic supremacy".

Low-decoherence regime plays a key role in constructing multi-particle quantum systems and has therefore been constantly pursued in order to build quantum simulators and quantum computers in a scalable fashion. Quantum error correction and quantum topological computing have been proved being able to protect quantumness but haven't been experimentally realized yet.Recently, topological boundary states are found inherently stable and are capable of protecting physical fields from dissipation and disorder, which inspires the application of such a topological protection on quantum correlation. Here, we present an experimental demonstration of topological protection of two-photon quantum states on a photonic chip. By analyzing the quantum correlation of photons out from the topologically nontrivial boundary state, we obtain a high cross-correlation and a strong violation of Cauchy-Schwarz inequality up to 30 standard deviations. Our results, together with our integrated implementation, provide an alternative way of protecting quantumness, and may inspire many more explorations in 'quantum topological photonics', a crossover between topological photonics and quantum information.Single photons inherently hold the features of single qubit in quantum computing, and has been widely used in various quantum simulation protocols, such as quantum walk [1,2], boson sampling [3-6] and quantum fast hitting [7]. The single-particle quantum walks have a precise mapping to classical wave phenomena. However, the advantage of uniquely quantum mechanical behavior is limited due to the single walker.In contrast, multiple indistinguishable particles can provide distinctly non-classical correlations, and this quantum behavior becomes a computational advantage. For example, two-particle quantum walk can be an algorithmic tool for the graph isomorphism problem [8], and the universal computation can be achieved by multiparticle quantum walk efficiently [9]. Thus, it is crucial to preserve the non-classical features when constructing a multi-particle quantum computers. * xianmin.jin@sjtu.edu.cn Quantum error correction is proposed to preserve logical quantum states in a subspace and rectify errors according to the measurement outcomes of ancillary particles [10,11]. Quantum topological computing, meanwhile, strives to store and manipulate quantum information with topological protection in a nonlocal manner using non-Abelian anyons [12]. Both of them are promising candidates in theoretical predictions but are still in their initial stage for experimental implementations [10][11][12][13].Topological photonics, derived from the discovery of topological phases in condensed-matter physics, aims to topologically protect photons from the inevitable fabrication-induced dissipation and disorder [14,15]. Many types of topological phases have been observed, for instance, Hall effect [16,17], edge states [18][19][20][21], topological insulators [22,23] and Weyl points [24,25], implying the capability of protecting physical fields. Inspired by these, we m...

Topology manifesting in many branches of physics deepens our understanding on state of matters.Topological photonics has recently become a rapidly growing field since artificial photonic structures can be well designed and constructed to support topological states, especially a promising large-scale implementation of these states using photonic chips. Meanwhile, due to the inapplicability of Hall conductance to photons, it is still an elusive problem to directly measure the integer topological invariants and topological phase transitions for photons. Here, we present a direct observation of topological winding numbers by using bulk-state photon dynamics on a chip. Furthermore, we for the first time experimentally observe the topological phase transition points via single-photon dynamics. The integrated topological structures, direct measurement in the single-photon regime and strong robustness against disorder add the key elements into the toolbox of 'quantum topological photonics' and may enable topologically protected quantum information processing in large scale.The introduction of topology into condensed-matter and material sciences originates from the connection of integer quantum Hall conductances with topological Chern invariants [1], which greatly expands our knowledge on state of matters. With the birth of topological insulators, searching topological state of matters in solid state materials [2,3] and photonic systems [4,5] has recently become a leading research field. In contrast to the challenging experimental requirements for realizing topological states in solid state materials, photonic systems provide a convenient and versatile platform to design various topological lattice models and study different topological * meifeng@sxu.edu.cn † xianmin.jin@sjtu.edu.cn states, including topological insulator states [6-10] and topological Weyl points [11,12]. The found topological boundary states potentially can be utilized for developing inherently robust and efficient artificial photonic devices [13-16].In the view of fundamental physics, the topological invariant is a crucial parameter to characterize the topological matter state. In fermion systems, the topological invariant can be revealed by conductance measurements, while the concept of Hall conductance is inapplicable in photonic systems. New methods for directly detecting the topological invariants in topological photonics remain to be developed. The pioneering proposals in theory [17][18][19] and experimental observations have been dedicated in both integrated photonic lattices [20][21][22] and bulk optics [23][24][25][26].Different from the efforts made to detect the topological invariant based on probing Berry curvature [17,20], non-Hermitian photon loss [18,21] or the dynamics of edge states [19,22], we propose a new approach to directly detect the topological invariant via the bulk-state photon dynamics in the real space, which beyonds the physical picture where topological invariant is defined on the equilibrium Bloch state in the momentum spac...