Hamiltonian

In the physics used by these algorithms, a Hamiltonian is a Hermitian operator. Its eigenvalues are real. For a molecule they are energies. For a combinatorial problem encoded as a diagonal operator, an eigenvalue on a bit-string basis state is the cost of that string.

VQE estimates the lowest eigenvalue by preparing a state and measuring the expectation value. McClean, Romero, Babbush, and Aspuru-Guzik write a general Hermitian operator as a sum of Pauli terms so that the expectation becomes a weighted sum of measurement outcomes. QAOA uses two Hamiltonians: a diagonal cost operator and a mixer that generates transitions between strings. Quantum phase estimation reads an eigenvalue when the state is close to an eigenvector and the evolution can be run coherently. That evolution is Hamiltonian simulation: approximate the unitary generated by H, then hand the unitary to the routine that wanted it. Lloyd's 1996 paper is the first explicit algorithm this library cites for local systems. The step counts live on the simulation page. VQE exists because a long coherent evolution is the hard condition on present devices. HHL treats a sparse Hermitian matrix as a Hamiltonian, simulates it, and applies the reciprocal of its eigenvalues to a state. The cost tracks the condition number. The algorithm page is where the runtime and the output state are stated.