The Schrodinger Equation and Gate Time

The Schrodinger equation governs how quantum states evolve; engineering a gate means running a chosen Hamiltonian for a chosen time.

The equation of motion

The time evolution of a closed quantum system is set by the Schrodinger equation: i hbar d|psi>/dt = H|psi>, where H is the Hamiltonian, the operator representing total energy. Everything a qubit does between measurements is a solution of this equation.

From Hamiltonian to unitary

For a time-independent H, the solution is |psi(t)> = exp(-iHt/hbar) |psi(0)>. The evolution operator U(t) = exp(-iHt/hbar) is unitary because H is Hermitian. So the abstract gates of the circuit model are physically realised by choosing a Hamiltonian and letting it act for a set duration.

Building a specific gate

To implement, say, an X rotation, hardware applies a control field whose Hamiltonian is proportional to the Pauli X, then holds it for the time that produces the desired rotation angle. A longer pulse rotates further. This is why gate time is set by the interaction strength: stronger coupling, faster gates.

The gate-time trade-off

Gate time competes directly with coherence. Faster gates fit more operations inside the coherence window, but stronger couplings that speed gates also tend to couple the qubit to noise, shortening coherence. Every platform balances gate speed against isolation, and the figure of merit is coherence time divided by gate time.

Simulating dynamics

Running the Schrodinger equation forward is also a computational task in itself. Simulating the evolution of a many-body quantum system — molecules, materials, or plasmas — is exponentially hard classically, which is the original motivation for quantum computers: a controllable quantum system can emulate another quantum system's Hamiltonian evolution directly.