The preparation of high-dimensional quan- tum states is of great significance in quantum information sci- ence and technology. Compared to qubits, qudit states – de- scribing quantum systems in d-dimensional spaces – enable stronger foundational tests of quantum mechanics [1–3] and better-performing applications in secure quantum communi- cations [4–9], quantum emulation [10, 11], quantum error cor- rection [12–14], fault-tolerant quantum computation [15–19], and quantum machine learning [20–22].
Needless to say, protocols performed on systems living in Hilbert spaces of large dimension require an increasing degree of control, in light of the large number of parameters required to describe states and operations. Nonetheless, the endeav- our of preparing arbitrary qudit states has been successfully achieved in various physical platforms [11, 23–32]. However, most of these works rely on ad hoc strategies, whose specific dependence on the underpinning dynamics makes their trans- lation across different physical platforms very difficult.
A very promising way to achieve the desired full inde- pendence of the physical platform, and thus a higher degree of universality, is the use of the rich dynamics offered by Quantum Walks (QWs) [33–35]. QWs, which can be thought of as the quantum counterparts of classical random walks, describe in their discrete version a high-dimensional qudit, named walker, embedded with an internal two-dimensional degree of freedom, conventionally dubbed coin. At every time step, the walker’s state moves coherently to the neighbouring sites in the lattice, conditionally to its coin state . QWs have been successfully implemented  in systems as di- verse as trapped atoms  and ions [39, 40], photonic cir- cuits [41–50], and optical lattices . Hence, an approach for state engineering based on their dynamics offers hope of being applicable effectively in a variety of different systems, independently of the details of the physical implementation.
While the QW dynamics was previously shown to allow the
engineering of specific walker’s states [52, 53], in Ref.  a scheme was proposed to use discrete-time QWs on a line to prepare arbitrary qudit states. This is achieved by enhanc- ing the degree of control over the walk’s dynamics through the arrangement of suitable step-dependent coin operations, which affect the coin-walker quantum correlations by de facto steering the state of the walker towards the desired final state, and finally projecting in the coin space. This last operation removes the correlations between walker and coin, thus pro- ducing a pure walker state with the desired features. In light of the large parameter space that characterizes the problem at hand, a systematic approach to the identification of the right set of coin operations and final projection is necessary. Such an analysis was presented in Ref. , in which a set of an- alytic conditions, together with suitable numerical optimiza- tions, was shown to guarantee the reaching of arbitrary target states with high probability.
In this paper, we make use of the scheme of Ref.  to give the first demonstration of a state-engineering protocol based on the controlled dynamics generated by QWs. We use the orbital angular momentum (OAM) degree of freedom of single-photon states as a convenient embodiment of the walker [48, 55, 56]. OAM-based experiments offer the pos- sibility to cover Hilbert spaces of large dimensions in light of the favourable (linear) scaling of the number of optical ele- ments with the size of the walk. Moreover, the scheme al- lows for the full control of the coin operation that is key to the implementation of the walk. In order to demonstrate the ver- satility of our scheme, we focus on the interesting classes of cat-like states and spin-coherent states [57, 58]. Those classes play a critical role in the exploration of the boundaries be- tween quantum and classical physics and whose implementa- tion is, in general, still a challenging task. Furthermore, we show experimentally the capability of engineering arbitrary states. The quality of the states synthesized in our endeavours, and the relative simplicity of the experimental protocol that we have put in place, demonstrate the effectiveness of a hybrid platform for quantum state engineering. Such platform holds together a programmable quantum system, the photonic QW in the angular momentum, and classical optimization al- gorithms for finding the best evolution to reach a certain quantum target.
 T. Ve ́rtesi, S. Pironio, and N. Brunner, Physical Review Letters 104, 060401 (2010).
 N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Reviews of Modern Physics 86, 419 (2014).
 R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wies ́niak, and A. Zeilinger, Nature 474, 490 (2011).
 H. Bechmann-Pasquinucci and A. Peres, Physical Review Let- ters 85, 3313 (2000).
 M. Fitzi, N. Gisin, and U. Maurer, Physical Review Letters 87, 217901 (2001).
 N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Physical Review Letters 88, 127902 (2002).
 D. Bruß and C. Macchiavello, Physical Review Letters 88, 127901 (2002).
 A. Acin, N. Gisin, and V. Scarani, (2003), quant-ph/0303009.
 N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Physical Review Letters 93, 053601 (2004).
 I. Buluta and F. Nori, Science 326, 108 (2009).
 M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz,
E. Lucero, A. D. O’Connell, D. Sank, H. Wang, J. Wenner, A. N. Cleland, M. R. Geller, and J. M. Martinis, Science 325, 722 (2009).
 I. L. Chuang, D. W. Leung, and Y. Yamamoto, Physical Review A 56, 1114 (1997).
 G. Duclos-Cianci and D. Poulin, Physical Review A 87, 062338 (2013).
 M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, Physical Review X 6, 031006 (2016).
 S. D. Bartlett, H. d. Guise, and B. C. Sanders, Physical Review A 65, 052316 (2002).
 T. C. Ralph, K. J. Resch, and A. Gilchrist, Physical Review A 75, 022313 (2007).
 B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, Nature Physics 5, 134 (2009).
 E. T. Campbell, H. Anwar, and D. E. Browne, Physical Review X 2, 041021 (2012).
 E. T. Campbell, Physical Review Letters 113, 230501 (2014).
 M. Schuld, I. Sinayskiy, and F. Petruccione, Contemporary
Physics 56, 172 (2015).
 V. Dunjko and H. J. Briegel, arXiv:1709.02779.
 J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe,
and S. Lloyd, Nature 549, 195 (2017).
 D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, Physical Review Letters 77, 4281 (1996).
 M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner,
J. M. Martinis, and A. N. Cleland, Nature 459, 546 (2009).
 B. Anderson, H. Sosa-Martinez, C. Riofr ́ıo, I. H. Deutsch, and P. S. Jessen, Physical Review Letters 114, 240401 (2015).
 S. P. Walborn, D. S. Lemelle, M. P. Almeida, and P. H. S. Ribeiro, Physical Review Letters 96, 090501 (2006).
 G. Lima, L. Neves, R. Guzma ́n, E. S. Go ́mez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, Optics Express 19, 3542 (2011).
 A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, Physical Review Letters 102, 153902 (2009).
 A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, Nature Physics 7, 677 (2011).
 R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Nature Communications 8, 94 (2017).
 S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Nature Comm. 9, 652 (2018).
 Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, (2018), arXiv:1804.07426.
 Y. Aharonov, L. Davidovich, and N. Zagury, Physical Review A 48, 1687 (1993).
 J. Kempe, Contemporary Physics 44, 307 (2003).
 S. E. Venegas-Andraca, Quantum Information Processing 11, 1015 (2012).
 A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in STOC ’01 Proceedings of the thirty-third annual ACM symposium on Theory of computing (2001) pp. 37–49.
 K. Manouchehri and J. Wang, Physical Implementation of Quantum Walks (Springer Berlin Heidelberg, Berlin, Heidelberg, 2014).
 R. Coˆte ́, A. Russell, E. E. Eyler, and P. L. Gould, New Journal of Physics 8, 156 (2006).
 H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Physical Review Letters 103, 090504 (2009).
 F. Za ̈hringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt,
and C. F. Roos, Physical Review Letters 104, 100503 (2010).  H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, Physical Review Letters 100, 170506 (2008).
 A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wo ̈rhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and
J. L. O’Brien, Science 329, 1500 (2010).
 M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-
Guzik, and A. G. White, Phys. Rev. Lett. 104, 153602 (2010).  A. Schreiber, K. N. Cassemiro, V. Potocˇek, A. Ga ́bris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, Physical Re-
view Letters 104, 050502 (2010).
 P. P. Rohde, A. Schreiber, M. Sˇtefanˇa ́k, I. Jex, and C. Silber-
horn, New Journal of Physics 13, 013001 (2011).
 L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, Physical Review Letters 108,
 J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S.
Kolthammer, and I. A. Walmsley, Journal of Optics 18, 094007
 F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko,
D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd,
and L. Marrucci, Science Advances 1, e1500087 (2015).
 F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. D. Filippis, V. Cataudella, E. Santamato, and L. Marrucci,
Nature Communications 7, 11439 (2016).
 F. Caruso, A. Crespi, A. G. Ciriolo, F. Sciarrino, and R. Osel-
lame, Nature Communications 7, 11682 (2016).
 F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Grobner, A. J. Daley, H.-C. Nagerl, N. Ismail, K. Wo ̈rhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L.
OBrien, Science 344, 1259 (2014).
 C. M. Chandrashekar, R. Srikanth, and R. Laflamme, Physical
Review A 77, 032326 (2008).
 H. Majury, J. Boutari, E. O’Sullivan, A. Ferraro, and M. Pater-
nostro, EPJ Quant. Tech. 5, 1 (2018).
 L. Innocenti, H. Majury, T. Giordani, N. Spagnolo, F. Sciar-
rino, M. Paternostro, and A. Ferraro, Phys. Rev. A 96, 062326
 P.Zhang,B.H.Liu,R.F.Liu,H.R.Li,F.L.Li, andG.C.
Guo, Physical Review Letters 81 (2010).
 S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, Physical
Review Letters 110, 263602 (2013).
 M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and
N. Zagury, Phys. Rev. A 45, 5193 (1992).
 C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland,
Science 272, 1131 (24 May 1996).