Nonperturbative $\mathit{k}$-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins

An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows $\mathit{k}$-body interactions to be captured exactly using two-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a computational system into its ground state.

Figure 1

Illustrating the mapping between circuits (with Boolean variables $x_i$) and spins $\left(s_i\right)$ for the example given in Eq. (3). One can use any number of methods to embed logical networks [18] into the ground space of Hamiltonians.

Figure 2

Karnaugh maps: (a) Two-local (positive polarity) interactions circled (e.g., $q_1{x}_1{x}_2+q_2{x}_1\wedge z_{\star }+q_3{x}_2\wedge f^{\star }$) and (b) linear (positive polarity) fields circled (e.g., $l_1{x}_1+l_2{x}_2+l_3{x}_3$). The interactions in cubes (a) and (b) form a basis for the space of realizable ($\le 3$ qubit) positive-definite logical gadget Hamiltonians expressible as $H\left(x_1,x_2,z_{\star }\right)=k_0+l_1{x}_1+l_2{x}_2+l_3{x}_3+q_1{x}_1\wedge x_2+q_2{x}_1\wedge z_{\star }+q_3{x}_2\wedge z_{\star }$, where $\forall i$, $k_0,l_i,q_i\ge 0$. (c) A Karnaugh map illustrating (with ovals) the linear and quadratic terms needed to set the null space of the Hamiltonian (8) to be in $\text{span}\left\{\phantom{|}|x_1{x}_2⟩|y_{\star }⟩|y_{\star }=x_1\wedge x_2,\forall x_1,x_2\in \left\{0,1\right\}\right\}$.