Fig. 1. (a) Direct coupling of two independent waveguides; (b) Schematic of one-dimensional coinless quantum walks in optical waveguides

Fig. 2. Topological phase diagram

Fig. 3. Spectra and topological edge states in the open boundary condition when the number of the lattice sites is even.(a) Spectrum varying with parameter c1, another parameter c2=c1-0.5π; (b) Spectrum when the parameters are chosen as c1=1.9π and c2=1.4π; (c) Spectrum when the parameters are chosen as c1=1.1π and c2=0.6π; (d) Spectrum when the parameters are chosen as c1=1.4π and c2=0.9π

Fig. 4. The probability distributions of the one-dimensional multi-step coinless quantum walks when the number of the lattice sites is even. (a1)-(i1) The probability distributions of 30-step coinless quantum walks; (a2)-(i2) The probabilities localized at the edge of the lattice during 100-step coinless quantum walks [(a1)(a2) c1=0.6π; (b1)(c1)(b2)(c2) c1=1.9π; (d1)(e1)(d2)(e2) c1=1.1π; (f1)-(i1)(f2)-(i2) c1=1.4π, c2=c1-0.5π]

Fig. 5. Two different schematic of the unit cells when the number of the lattice sites is odd.(a) c' is the intracell coupling strength, c″ is the intercell coupling strength; (b) c″ is the intracell coupling strength, c' is the intercell coupling strength

Fig. 6. Spectra and topological edge states in the open boundary condition when the number of the lattice sites is odd. (a) Spectrum varying with parameter c', another parameter c''=c'-0.5π; (b) Spectrum when the parameters are chosen as c'=0.6π and c''=0.1π; (c) Spectrum when the parameters are chosen as c'=0.1π and c''=0.6π; (d) Spectrum when the parameters are chosen as c'=1.1π and c''=0.6π; (e) Spectrum when the parameters are chosen as c'=0.6π and c''=1.1π

Fig. 7. The probability distributions of the one-dimensional multi-step coinless quantum walk when the number of the lattice sites is odd. (a1)-(h1) The probability distributions of 30-step coinless quantum walks; (a2)-(h2) The probabilities localized at the edge of the lattice during 100-step coinless quantum walks; [ (a1)(b1)(a2)(b2) c'=0.6π, c''=0.1π; (c1)-(e1)(c2)-(e2) c'=1.1π, c''=0.6π; (f1)-(h1)(f2)-(h2) c'=0.6π, c''=1.1π; c2=c1-0.5π ]