Mapping Binary Functions to a Practical Adiabatic Quantum Computer
David Rosenbaum, Marek Perkowski
2010 40th IEEE International Symposium on Multiple-Valued Logic
Efficiently mapping binary functions to adiabatic quantum computers is an important problem because the resulting circuits can be used as oracles in Grover's algorithm. This paper presents a method for mapping binary functions to a two-dimensional grid of qubits with nearest neighbor interactions which is used in a prototype from D-Wave Systems. This is done by writing the binary function in a special form. This allows the binary function to be implemented by converting each gate into a 3-local
... Hamiltonian. These 3-local Hamiltonians are then converted into twolocal Hamiltonians which are mapped to the grid of qubits. problem. In the circuit model of quantum computation, the oracle takes the form of a unitary matrix that flips the sign of the phase of each basis state that corresponds to a solution. This oracle can be constructed as a permutative circuit that inverts an ancilla qubit for every solution state. Initializing the ancilla qubit to the state 1 √ 2 |0 − 1 √ 2 |1 then results in a unitary operator that will flip the sign of the desired basis states due to phase kick-back. The adiabatic quantum version of Grover's algorithm takes the form of a Hamiltonian in which the solution states correspond to low energy levels. This oracle Hamiltonian can then be used to construct a Hamiltonian that will evolve to one of the solution states. Because the oracle is necessary for Grover's algorithm, implementing the Hamiltonian for the oracle is an important problem. Biamonte  developed a method for mapping a binary circuit represented as a planar tree of two-input single-output gates into 3-local Hamiltonians. This method is illustrated by mapping the tree of gates shown in figure 1(a) onto the two-dimensional grid of qubits shown in figure 1(b) . The operation I used in nodes with exactly one input indicates that the value of the node is copied from the input. In this paper, the grid utilized in D-Wave's prototype is used where each node has eight neighbors. These 3-local Hamiltonians are then decomposed into 2-local Hamiltonians which allows the desired circuit to be implemented on an adiabatic quantum computer. The main problem with this method  is that it is not algorithmic and no proof is provided that it is always even possible to map the tree of gates to the two-dimensional grid of qubits. Furthermore, no evidence is provided that this can be done efficiently. Backtracking is also required when this method is used because it is possible to reach a partial solution from which the final solution cannot be reached. This will cause any algorithm that uses this method to run slowly for large problems without the use of sophisticated heuristics. In this paper, a mapping method is shown which is capable of mapping a binary function to the two-dimensional grid of qubits while avoiding the difficulties with layout that result from using the method proposed by Biamonte . Furthermore, the method shown in this paper is completely algorithmic and also does not require backtracking. Mapping is performed by first writing the desired binary function in a special form which can be implemented efficiently by decomposing it to the Hamiltonians given by Biamonte  .