Computational Phase Transition Signature in Gibbs Sampling

by   H. Philathong, et al.

Gibbs sampling is fundamental to a wide range of computer algorithms. Such algorithms are set to be replaced by physics based processors-be it quantum or stochastic annealing devices-which embed problem instances and evolve a physical system into an ensemble to recover a probability distribution. At a critical constraint to variable ratio, decision problems-such as propositional satisfiability-appear to statistically exhibit an abrupt transition in required computational resources. This so called, algorithmic or computational phase transition signature, has yet-to-be observed in contemporary physics based processors. We found that the computational phase transition admits a signature in Gibbs' distributions and hence we predict and prescribe the physical observation of this effect. We simulate such an experiment, that when realized experimentally, we believe would represent a milestone in the physical theory of computation.



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Appendix A Ising Spin Embedding

Finding ground energy and ground state configurations of physical systems such as spin glasses is NP-hard barahona1982computational . 2-SAT and 3-SAT instances can be directly mapped onto the Hamiltonian minimization problem, see e.g. lucas2014ising ; biamonte2008nonperturbative . The standard procedure is as follows.

We convert logical bits to spins


The penalty Hamiltonian is constructed by real linear extension of by means of the following invertible mapping between Boolean variables and projectors,




where and acting on the spin.

By this construction, clauses in a SAT instance are mapped onto projectors . Hence, the Hamiltonian can be constructed by summing over all the clauses in an instance,


where assigns the value of corresponding to the clause. The ground state space is spanned by solutions of the SAT problem, and the ground energy is equal to minimal number of unsatisfiable clauses.

Substituting the projectors in Eq. (10), where is the Pauli matrix along the quantization axis, the Hamiltonian can be written in the form of the generalized Ising Hamiltonian with k-body interaction. In case of MAX 2-SAT, the Hamiltonian is a sum over projectors onto -SAT clauses


The coefficient indicates the local field of the spin and the couplings encode the 2-body interaction between the spin and the spin.

This method provides a way to physically realize SAT instances as spin Hamiltonians. If such a physical system is built, cooling the system into its ground state is equivalent to solving the SAT problems.