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Quantum Phase Estimation (QPE)

Quantum Phase Estimation is a central building block in many quantum algroithm, commonly utilized as a spectral analysis tool for Hermitian matrices. Examples include Trotterized Hamiltonian simulation and qubitization-based block-encoding techniques, highlighting how eigenvalues can be extracted from controlled unitary dynamics. The following implementations demonstrate these techniques and showcase the core primitives underlying quantum chemistry, Hamiltonian simulation, and advanced linear-algebraic quantum algorithms.

  • QPE for a matrix - Quantum Phase Estimation (QPE) is a fundamental quantum algorithm and a common primitive in many algorithms, allowing one to estimate the eigenphase of a unitary matrix, \(e^{i M t}\), where \(M\) is a Hermitian matrix. By initializing the system in a random initial state, repeated execution of the QPE algorithm, utilizing a built-in Trotter propagator, suzuki_trotter, leads to an estimation of the phases, \(\{e^{i\theta_i}\}\). These phases are then directly related to the eigenvalues of \(M\).
  • QPE with qubitization - Given a block-encoding of the Hermitian matrix of interest, we construct the Szegedy quantum walk operator and utilize it within a quantum phase estimation procedure to estimate the eigenvalues of a Hydrogen molecule.