12.7 The Hamiltonian

Time evolution of a quantum system, represented as a sequence of quantum states, occurs via a continuous and additive family of unitary matrices U_t, where we often consider 0 ≤ t ≤ 1. We define each family via Hamiltonian

for some Hermitian matrix H. Other constants that may appear in the numerator are absorbed into H. 189, Section 13.4.2

We call the Hermitian matrix H the Hamiltonian of the quantum system after mathematician and physicist William Rowan Hamilton (Figure 12.20). Being Hermitian, we can write any Hamiltonian as a Pauli sum. Hamilton, William Rowan

Given H in U_t = exp(–iHt), the process of finding an approximate and suitable unitary U_t and related circuit within a given error is known as Hamiltonian simulation. 60 63, Section 2.3 Hamiltonian$simulation simulation$Hamiltonian

Figure 12.20: William Rowan Hamilton (1805–1865)

At this point...