The Hamiltonian: where gates come from

Every gate in this IDE is a snapshot of a physical system evolving for a short time. The object that decides how a system evolves is the Hamiltonian. Here is who invented it, what it means, and how it turns into the matrices in the Gate Lab.

1833 · William Rowan Hamilton

An Irish mathematician, Hamilton rewrote Newton's mechanics in terms of a single function H(q, p): the total energy of a system expressed through positions q and momenta p. His equations ẋ = ∂H/∂p, ṗ = −∂H/∂q describe planets, pendulums and, a century later, electrons. He also invented quaternions, carving i² = j² = k² = ijk = −1 into Broom Bridge in Dublin in 1843; the Pauli matrices X, Y, Z are quaternions in disguise.

1926 · Schrödinger makes H an operator

i ℏ \frac{d}{d t} ∣ ψ (t)⟩ = H ∣ ψ (t)⟩

The Schrödinger equation. H is now a Hermitian matrix (H = H†) whose eigenvalues are the allowed energies. It tells the state how to move.

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

Solving it for a constant H gives a unitary U(t). A gate is simply U for the right duration t. This is the bridge between physics and the matrices you see in the Gate Lab.

Gates as time evolution

R_{X} (θ) = e^{- i θ X /2} \Leftarrow H = \frac{ℏΩ}{2} X,; θ = Ω t

Drive a transmon with a microwave pulse on resonance and its Hamiltonian is proportional to X (or Y, depending on the pulse phase). Pulse for time t and you get RX(Ωt). A π pulse is the X gate; a π/2 pulse is √X: that is why √X is native on IBM.

R_{Z} (φ) = e^{- i φZ /2} \Leftarrow H = \frac{ℏ ω}{2} Z

The qubit's own energy splitting is a Z Hamiltonian: left alone, a qubit rotates about Z forever. IBM implements RZ by shifting the phase of all later pulses instead of waiting, which is why RZ is free and error-less.

R_{Z Z} (θ) = e^{- i θ Z \otimes Z /2}, CNOT \sim e^{- i \frac{π}{4} (I - Z) \otimes (I - X)}

Two-qubit gates come from coupling Hamiltonians: a ZZ or XX interaction between neighbours. Cross-resonance (Eagle) and tunable couplers (Heron) are two ways of switching such a coupling on for a few hundred nanoseconds. Try RZZ and ECR in the Gate Lab.

Hamiltonians as problems to solve

H_{MaxCut} = \frac{1}{2} (i, j) \in E \sum (I - Z_{i} Z_{j}), H_{H_{2}} = k \sum c_{k} P_{k}

The same word is used for the thing you want to compute. QAOA encodes a graph problem as a cost Hamiltonian of ZZ terms (the QAOA example); VQE writes a molecule's energy as a sum of Pauli strings and minimizes ⟨ψ(θ)|H|ψ(θ)⟩ (the VQE example). The cutting tab reconstructs exactly such ⟨Pauli⟩ expectation values.

Three faces of one idea: Hamilton's energy function (1833) → Schrödinger's operator that moves the state (1926) → today's gates, which are e^{-iHt} for engineered H, and today's problems, which are H whose ground state we want.

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