Hamiltonian

The operator describing a quantum system’s total energy. Central to quantum simulation and variational algorithms.

The Hamiltonian $H$ is the fundamental operator in quantum mechanics. It describes the energy of a system and governs how it evolves in time.

Role in Quantum Mechanics

$H ∣ ψ_{n} ⟩ = E_{n} ∣ ψ_{n} ⟩$

Eigenstates have definite energies. The ground state has the lowest energy $E_{0}$ .

$i ℏ \frac{d}{d t} ∣ ψ ⟩ = H ∣ ψ ⟩$

Solutions: $∣ ψ (t)⟩ = e^{- i H t /ℏ} ∣ ψ (0)⟩$

Hamiltonians appear everywhere:

Simulating Hamiltonian evolution: $U (t) = e^{- i H t}$

is a core quantum computing application.

VQE finds ground state energy: $E_{0} = ∣ ψ ⟩ min ⟨ ψ ∣ H ∣ ψ ⟩$

Evolve system under time-dependent Hamiltonian: $H (t) = (1 - s (t)) H_{0} + s (t) H_{P}$

On qubits, any Hamiltonian can be written as:

$H = i \sum c_{i} P_{i}$

where $P_{i}$ are Pauli strings (products of $I, X, Y, Z$ ).

Example (2 qubits): $H = 0.5 \cdot ZZ + 0.3 \cdot X I + 0.2 \cdot I Z$

$H = - J ⟨ i, j ⟩ \sum Z_{i} Z_{j} - h i \sum X_{i}$

Models magnetic systems, phase transitions.

$H = pq \sum h_{pq} a_{p}^{†} a_{q} + pq rs \sum h_{pq rs} a_{p}^{†} a_{q}^{†} a_{r} a_{s}$

Electronic structure of molecules.

$H = J ⟨ i, j ⟩ \sum (X_{i} X_{j} + Y_{i} Y_{j} + Z_{i} Z_{j})$

Spin-spin interactions.

Measuring $⟨ H ⟩$ requires measuring each Pauli term:

$⟨ H ⟩ = i \sum c_{i} ⟨ P_{i} ⟩$

Many measurements needed (measurement overhead).

Understanding Hamiltonians is essential for:

See also: VQE, Quantum Simulation