Extended Data Fig. 4: Construction of the quantum walk operator at the heart of state-of-the-art quantum chemistry algorithms.

(a) The quantum walk operator \({\mathcal{W}}\) can be decomposed using the LCU technique into two oracles, PREPARE and SELECT, which encode the details of the molecule. This operator is repeated many times, so its cost dominates the overall algorithm. (b) Construction of the SELECT oracle. This figure is adapted from Fig. 5 on page 15 of ref. ⁵³. We compile and optimize the part in the dashed box, along with its constituent components, for a range of parameters (up to 10 bits of precision per rotation, and for systems of up to 10 spin-orbitals). The registers μ and ν are the output of the PREPARE oracle, and contain the coefficients of the linear combination of \({\mathcal{O}}({M}^{2})\) unitaries in superposition, hence these registers use \({n}_{M}\approx {\log }_{2}(M)\) qubits each, where M is a parameter controlling the accuracy of the implementation of the Hamiltonian. These are used in the data:rot_μ,ν operations, which index into a quantum read-only-memory (as constructed in ref. ⁴⁰) to load a sequence of rotation angles corresponding to each unitary, in superposition. These angles are used by Givens rotations circuits (denoted R and R^†, following the method of ref. ⁵⁴) to diagonalize each of the unitaries we may wish to apply (in this case, each unitary represents a product of two Majorana operators). The diagonalized unitaries can be applied to the data registers \(\left\vert {\psi }_{\downarrow }\right\rangle\) and \(\left\vert {\psi }_{\uparrow }\right\rangle\) (which correspond to the electron spin-orbitals with spin down and spin up, respectively) with a CZ operation (denoted Z₁), conditioned on a register \(\left\vert {\rm{succ}}\right\rangle\) which encodes whether the PREPARE oracle was successful. These steps are repeated independently for μ and ν, with some extra controls based on the special case of ν = M + 1, used to encode the unitaries corresponding to single-electron rather than two-electron terms in the Hamiltonian.