Nonperturbative embedding for highly nonlocal Hamiltonians

The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with at most two-body interactions. Although valid for arbitrary $k$-body interactions, their use is limited to small $k$ because the strength of interaction is $k\mathrm{th}$ order in perturbation theory. In this paper we develop a nonperturbative technique for obtaining effective $k$-body interactions using Hamiltonians consisting of at most $l$-body interactions with $l<k$. The idea is to first apply a unitary transformation on the system plus ancilla qubits (possibly using a gate-based device), then evolve with a new Hamiltonian which is more local than the original one (using an analog device), and finally reverse the unitary transformation. The net effect of this procedure is shown to be equivalent to evolving the system with the original nonlocal Hamiltonian. This technique does not suffer from the aforementioned shortcoming of perturbative methods and requires only one ancilla qubit for each $k$-body interaction irrespective of the value of $k$. It works best for Hamiltonians with a few many-body interactions involving a large number of qubits and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme.

Figure 1

Commutation relations of Pauli operators making up the example Hamiltonian (36). Circles represent terms in the Hamiltonian, and a line connecting two circles indicates that those terms anticommute.

Figure 2

A schematic description of the state-preparation protocol described in Sec. 6a. The top qubit is the ancilla, and the rest are system qubits. $H$ acting on the ancilla stands for the Hadamard gate, not to be confused with the Hamiltonian.

Figure 3

A schematic description of the state-preparation protocol applicable only to the special case described in Sec. 6c. The top qubit is the ancilla, and the rest are system qubits. $H$ acting on the ancilla stands for the Hadamard gate, not to be confused with the Hamiltonian. After the Hamiltonian dynamics, the ancilla qubit is measured, and conditional on the outcome, the system qubits are acted on by a different unitary.