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**In principle, it is possible to model any physical system exactly using quantum mechanics; in practice, it quickly becomes infeasible. Recognising this, Feynman suggested that quantum systems be used to model quantum problems ^{1}.**

There are two approaches: *simulation*, where digital outcomes yield the desired physical quanties—e.g. via a universal quantum computer—and *emulation*, where physical measurements yield the physical quantities, e.g. spatial probabilities of a quantum walk. In recent years both approaches have been explored, for example the first quantum simulation was of the smallest quantum chemistry problem: obtaining the energies of H2, the hydrogen molecule, in a minimal basis^{2}.

Here we report on our efforts using quantum emulation to explore systems in condensed matter physics ^{3} , biology, and complexity theory. Engineered photonic systems, with their precise controllability, provide a versatile platform for creating and probing a wide variety of quantum phenomena.

One of the most striking features of quantum mechanics is the appearance of phases of matter with topological origins. First recognised in the integer and fractional quantum Hall effects in the 1980s, topological phases have been identified in physical systems ranging from condensed-matter^{4} and highenergy physics ^{5} to quantum optics ^{6} and atomic physics ^{7} . Topological phases are parametrised by integer topological invariants: as integers cannot change continuously, a consequence is exotic phenomena at the interface between systems with different values of topological invariants. For example, a topological insulator supports conducting states at the surface, precisely because its bulk topology is different to that of its surroundings ^{4} . Creating and studying new topological phases remains a difficult task in a solid-state setting because the properties of electronic systems are often hard to control.

We use photonic quantum walks to investigate topological phenomena: the photon evolution simulates the dynamics of topological phases which have been predicted to arise in, for example, polyacetylene. We experimentally confirm the long-standing prediction of topologically protected localised states associated with these phases by directly imaging their wavefunctions ^{3} . Moreover, we reveal an entirely new topological phenomenon: the existence of a topologically protected pair of bound states which is unique to periodically driven systems—demonstrating a powerful new approach for controlling topological properties of quantum systems through periodic driving^{3}.

In light-harvesting molecules in photosynthesis the energy is localised much faster than can be explained by Naíve tunelling model: it has instead been suggested that it is in fact due to a partially-decohered quantum walk. Subsequently, coherence in light-harvesting complexes has been measured so many times ^{8};^{9} that its existence is now unquestioned in the field. The initial surprise that long-lived quantum coherences occur in biology was in large part a consequence of using old theoretical models from other fields outside of their range of validity--with an appropriate treatment, long-lived coher- ences are quite natural ^{10} . But many questions remain open: how robust is the coherence? Does it assist transport? Is it optimised by natural selection? These are difficult to address experimentally because it is very difficult to modify the struc- ture of a biological complex. Here we report our efforts to understand quantum transport by engineering a photonic emulator for biological systems ^{11} , with the goal of being able to easily turn handles to vary the structure, the degree of coherence, or even the environment.

A landmark recent paper shows that even simple quantum computers—built entirely from linear photonic elements with nonadaptive measurements—cannot be efficiently simulated by classical computers ^{12} . Such devices are able to solve sampling problems and search problems that are classically intractable under plausible assumptions; alternatively if such devices can be efficiently simulated there are far-reaching consequences for the field of complexity theory. Given recent advances in photon sources ^{13} and detectors ^{14} , we discuss the requirements for experimentally realising such devices.

[1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982).

[2] B. P. Lanyon, J. D. Whitfield, G. G. Gillet, et al., Nature Chemistry 2, 106 (2010).

[3] T. Kitagawa, M. A. Broome, et al., Nature Communications 3, 882 (2012).

[4] X. L. Qi and S. C. Zhang, Reviews of Modern Physics 83, 1057-1110 (2011).

[5] R. Jackiw and C. Rebbi, Physical Review D 13, 3398 (1976).

[6] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772 (2009).

[7] A. S. Sørensen, E. Demler, and M. D. Lukin, Physical Revew Letters 94, 086803 (2005).

[8] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer and G. D. Scholes, Nature 463, 644, (2010). [9] E. Harel and G. S. Engel, Proceedings of the National Academy of Sciences 109, 706 (2012).

[10] L. A. Pachon and P. Brumer, Journal of Physical Chemistry Letters 2, 2728 (2011).

[11] J. O. Owens, M. A. Broome, D. N. Biggerstaff, et al., New Journal of Physics 13, 075003 (2011).

[12] S. Aaronson and A. Arkhipov, STOC'11 Proceedings of the 43rd annual ACM symposium on Theory of comput- ing, 333-342 (2011). doi:10.1145/1993636.1993682 [13] A. Dousse, et al., Nature 466, 217 (2010).

[14] D. H. Smith, G. G. Gillett, et al. Nature Communica- tions 3, 625 (2012).

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