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It is well known that physical phenomena may be of great help in computingsome difficult problems efficiently. A typical example is prime factorizationthat may be solved in polynomial time by exploiting quantum entanglement on aquantum computer. There are, however, other types of (non-quantum) physicalproperties that one may leverage to compute efficiently a wide range of hardproblems. In this perspective we discuss how to employ one such property,memory (time non-locality), in a novel physics-based approach to computation:Memcomputing. In particular, we focus on digital memcomputing machines (DMMs)that are scalable. DMMs can be realized with non-linear dynamical systems withmemory. The latter property allows the realization of a new type of Booleanlogic, one that is self-organizing. Self-organizing logic gates are"terminal-agnostic", namely they do not distinguish between input and outputterminals. When appropriately assembled to represent a givencombinatorial/optimization problem, the corresponding self-organizing circuitconverges to the equilibrium points that express the solutions of the problemat hand. In doing so, DMMs take advantage of the long-range order that developsduring the transient dynamics. This collective dynamical behavior, reminiscentof a phase transition, or even the "edge of chaos", is mediated by families ofclassical trajectories (instantons) that connect critical points of increasingstability in the system's phase space. The topological character of thesolution search renders DMMs robust against noise and structural disorder.Since DMMs are non-quantum systems described by ordinary differentialequations, not only can they be built in hardware with available technology,they can also be simulated efficiently on modern classical computers. As anexample, we will show the polynomial-time solution of the subset-sum problemfor the worst...

Memcomputing: Leveraging memory and physics to compute efficiently