Abstract
Mappings of classical computation onto statistical mechanics models have led to remarkable successes in addressing some complex computational problems. However, such mappings display thermodynamic phase transitions that may prevent reaching solution even for easy problems known to be solvable in polynomial time. Here we map universal reversible classical computations onto a planar vertex model that exhibits no bulk classical thermodynamic phase transition, independent of the computational circuit. Within our approach the solution of the computation is encoded in the ground state of the vertex model and its complexity is reflected in the dynamics of the relaxation of the system to its ground state. We use thermal annealing with and without ‘learning’ to explore typical computational problems. We also construct a mapping of the vertex model into the Chimera architecture of the D-Wave machine, initiating an approach to reversible classical computation based on state-of-the-art implementations of quantum annealing.
Highlights
Mappings of classical computation onto statistical mechanics models have led to remarkable successes in addressing some complex computational problems
This paper aims at bringing a new class of problems to the physics–computer science interface by introducing a twodimensional (2D) representation of a generic reversible classical computation, the result of which is encoded in the ground state of a statistical mechanics vertex model with appropriate boundary conditions
Our work emphasizes that the dynamics of relaxation to the ground state rather than the thermodynamics of the model is essential for understanding the complexity of ground-state computation
Summary
Mappings of classical computation onto statistical mechanics models have led to remarkable successes in addressing some complex computational problems. This paper aims at bringing a new class of problems to the physics–computer science interface by introducing a twodimensional (2D) representation of a generic reversible classical computation, the result of which is encoded in the ground state of a statistical mechanics vertex model with appropriate boundary conditions. The corresponding gate constraints are implemented through short-ranged one- and two-body interactions involving the spins of the vertex (as we show, this construction can be realized in physical programmable machines, such as the D-Wave machine.) One direction of the lattice represents ‘computational (rather than real) time’, as introduced by Feynman in the history representation of quantum computation[10], but here used for classical reversible circuits. Our work emphasizes that the dynamics of relaxation to the ground state rather than the thermodynamics of the model is essential for understanding the complexity of ground-state computation
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