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Appendix A Ising Spin Embedding
Finding ground energy and ground state configurations of physical systems such as spin glasses is NPhard barahona1982computational . 2SAT and 3SAT 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
(7) 
The penalty Hamiltonian is constructed by real linear extension of by means of the following invertible mapping between Boolean variables and projectors,
(8) 
and
(9) 
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,
(10) 
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 kbody interaction. In case of MAX 2SAT, the Hamiltonian is a sum over projectors onto SAT clauses
(12)  
The coefficient indicates the local field of the spin and the couplings encode the 2body 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.
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