Which features make the brain such a powerful and energy-efficient computing machine? Can we reproduce them in the solid state, and if so, what type of computing paradigm would we obtain? I will show that a machine that uses memory (time non-locality) to both process and store information, like our brain, and is endowed with intrinsic parallelism and information overhead – namely takes advantage, via its collective state, of the network topology related to the problem – has a computational power far beyond our standard digital computers [1, 2]. We have named this novel computing paradigm “memcomputing” [1, 2, 3, 4]. As examples, I will show the polynomial-time solution of prime factorization, the search version of the subset-sum problem , and approximations to the Max-SAT beyond the inapproximability gap  using polynomial resources and self-organizing logic gates, namely gates that self-organize to satisfy their logical proposition . I will also show that these machines are described by a topological field theory, and they compute via an instantonic phase, implying that they are robust against noise and disorder . The digital memcomputing machines we propose can be efficiently simulated, are scalableand can be easily realized with available nanotechnology components. Work supported in part by CMRR and MemComputing, Inc. (http://memcpu.com/).
 M. Di Ventra and Y.V. Pershin, Computing: the Parallel Approach, Nature Physics 9, 200 (2013).
F. L. Traversa and M. Di Ventra, Universal Memcomputing Machines, IEEE Transactions on Neural Networks and Learning Systems 26, 2702 (2015).
 M. Di Ventra and Y.V. Pershin, Just add memory, Scientific American 312, 56 (2015).
 M. Di Ventra and F.L. Traversa, Memcomputing: leveraging memory and physics to compute efficiently, J. Appl. Phys. 123, 180901 (2018).
 F. L. Traversa and M. Di Ventra, Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines, Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 023107 (2017).
 F. L. Traversa, P. Cicotti, F. Sheldon, and M. Di Ventra, Evidence of an exponential speed-up in the solution of hard optimization problems,Complexity 2018, 7982851 (2018).
 M. Di Ventra, F. L. Traversa and I.V. Ovchinnikov, Topological field theory and computing with instantons, Annalen der Physik 1700123 (2017).