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Ground State Preparation via Qubitization
Charles Marteau
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We describe a protocol for preparing the ground state of a Hamiltonian H on a quantum computer. This is done by designing a quantum algorithm that implements the imaginary time evolution operator: e^(-τH). The method relies on the so-called ``qubitization'' procedure of Low and Chuang which, assuming the existence of a unitary encoding of the Hamiltonian H = ⟨G| U_H |G⟩, produces a new operator W_H whose moments are the Chebyshev polynomials of H when projected on |G⟩. Using this result and the expansion of e^(-τH) in terms of Chebyshev polynomials we construct a circuit that implements an approximation of the imaginary time evolution operator which, at large time, projects any state on the ground state, provided a non-trivial initial overlap between the two. We illustrate our method on two models: the transverse field Ising model and a single qubit toy model.
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