Fig. 20: Quantum oracle as a key component of the Grover step as part of DH-QMF.

The oracle marks every state whose energy is strictly smaller than the threshold value E(y), which is computed given the latest threshold index y. The result is recorded in a single-qubit flag: given its input state \(\left\vert z\right\rangle\) (where z ∈ {0, 1}), the oracle outputs \(\left\vert z\oplus f(x)\right\rangle\), where f(x) = 1 if, and only if, E(x) < E(y), and f(x) = 0 otherwise. a The circuit consists of several queries to the energy oracle O_E, which reversibly computes the energy corresponding to a given input state, and applications of a unitary module called “Compare”, which compares the values held by two registers and records the result (0 or 1) in a single-qubit ancilla. To infer if E(x) < E(y) for a given input \(\left\vert x\right\rangle\), we prepare the quantum state \(\left\vert y\right\rangle\) corresponding to the known threshold index y, then independently compute E(x) and E(y) by separately employing O_E, respectively, and compare their values using “Compare”. The computational registers for holding the energy values are initialized in \(\left\vert {\tilde{E}}_{0}\right\rangle\), where \({\tilde{E}}_{0}\) is a constant energy shift chosen so as to avoid negative energies. If E(x) < E(y) is TRUE, a 1 is recorded in an ancilla qubit that was initialized in \(\left\vert 0\right\rangle\); the ancilla remains unaltered otherwise. Using a CNOT gate, we copy out the result of the comparison to the single-qubit flag and reverse the whole circuit producing this result. b O_E is implemented by serially executing the shown circuit template for every vertex pair (i, ℓ). Depending on whether vertex[i] and vertex[ℓ] carry the same or different values, we respectively subtract or add the value J_iℓ in the data(H) register.