Fig. 1. (a) Experimental setup of one step of a 1D quantum walk. (b) Schematic of N steps of a quantum walk, where module G denotes the setup shown in (a)^[38].

Fig. 2. Schematic of the specially devised OAM beam splitter (BS′), consisting of a BS (symmetric BS) and two mirrors^[39].

Fig. 3. Experimental scheme of a 1D two-state quantum walk with the specially devised OAM BS′. A normal symmetric BS1 is still used in this scheme, while BS2′ and BS3′ are both OAM BS′s. Single-mode fibers (SMFs), computer generated holograms with proper ±l (Holol), pinholes (P1−P6), and power meters (Du and Dd) are used for the measurement^[39].

Fig. 4. Conceptual schematic of quantum walks with the spin-orbital system^[41].

Fig. 5. Schematic of the experimental setup to implement 1D quantum walks by using photon SAM as the quantum coin and OAM space as the walk space. A set of three wave-plates [two quarter-wave-plates (QWPs) and one half-wave-plate (HWP)] are used for initial state preparation. The gray box implements one step of quantum walks, consisting of three wave-plates, one QWP and two HWPs, and one q-plate (QP). The OAM analysis module is made up of a computer generated hologram, an SMF, and a single-photon counting module (SPCM)^[42].

Fig. 6. Detailed sketch of the setup for demonstration of a two-particle quantum walk^[41].

Fig. 7. Experimental setup of 2D quantum walks (see text for more details)^[7].

Fig. 8. Schematic of using bulk optics as the basic elements of 1D quantum walks. The PBS¯ consists of a PBS and HWPs (R90), where a PBS splits the input light into the two output ports denoted by “left” and “right” for |V〉 and |H〉, respectively^[44].

Fig. 9. Optical layout of the experimental setup for implementing N-step quantum walks on a line^[44].

Fig. 10. Scheme of implementing a quantum quincunx with optical elements^[52].

Fig. 11. Schematic of the experimental demonstration six steps of the quantum walks with single photons. (a) Ci and Si indicate six pairs of coin and shift operators, separately. (b) The first two steps are illustrated in detail. (c) Adjusting the relative angle between adjacent BDs, resulting in a temporal lag Δt and a transversal mode mismatch Δx between interfering wave-packets, thereby introducing decoherence^[15].

Fig. 12. Experimental results. Probability distributions of the (a) quantum walks and (b) classical walks when introducing decoherence. (c) Normalized standard deviation of the probability distribution, where the lines indicate the theoretical values^[15].

Fig. 13. Experimental setup for realization of quantum walks on cycles with N nodes^[56].

Fig. 14. Detailed schematic of the experimental setup for implementation of arbitrary coined 2D quantum walks (see text for details)^[60].

Fig. 15. (a) Encoding method. The walker’s positions are encoded with the polarization states of single photons. (b) Implementation of the conditional shift operator, where two HWPs at 0° and another two at Δθ/2 and −Δθ/2 are used^[63].

Fig. 16. Experimental setup. The photon pairs are generated by the BBO crystal through the SPDC technique, one of which is detected by the single-photon detector (D0) as the trigger, while the other one is initialized in the state initialization part and then incident into the optical loops to demonstrate the quantum walks, of which the arrival times can be detected by D1 and D2^[63].

Fig. 17. Schematic setup of the detector. 50:50, symmetric fiber couplers^[64].

Fig. 18. Schematic diagram of the principle of the time-multiplexed framework. The HWP is used to implement the coin flipping operator. The first PBS can separate photons by their polarization to different paths, while the second PBS recombines these photons from different paths to the same one. The red wave indicates the evolution of the first step, and the yellow wave indicates the second step. The arrows represent the direction of polarization.

Fig. 19. Experimental scheme of the time-multiplexed framework^[23].

Fig. 20. (a) Experimental setup. (b) Probability distribution of a Hadamard walk after 28 steps^[24].

Fig. 21. (a) Experimental setup of the 2D quantum walk with one walker. (b) Diagram of a 2D quantum walk separated by two different-direction 1D quantum walks^[25].

Fig. 22. Initial state, produced by a type-II PPKTP crystal, is prepared as a Bell state, and one of the photon pairs is sent to Alice, while the other is sent to Bob. At Alice’s side, projection measurement is implemented to choose the initial state of the system, while Bob that realizes a 2D quantum walk is the same as above. The initial state of Bob can be chosen rather than the beginning of Bob, but not the projection measurement of Alice’s side^[66].

Fig. 23. (a) Experimental setup of 2D time-bin quantum walk. The x direction quantum walk was realized in the fiber loop, while the y direction quantum walk was in the free space loop. The AOM acts as a photo switch to guarantee the effective detection of photons. (b) Reconstruction of 2D quantum walk, where the walker walks along the x direction in step 1, while it walks along the y direction in step 2^[71].

Fig. 24. (a), (b) Probability distribution of 2D quantum walk after 25 steps. (c) n−P(B;n) in different phase parameters. (d) |d|−〈z〉 for the parameter of θ in different phases^[71].

Fig. 25. (a) Experimental setup of time-bin quantum walk in the birefringence crystal. (b) Details and the time interval for different steps^[65].

Fig. 26. Normalized probability distribution of Hadamard quantum walk after 50 steps^[65].