The Schrödinger equation is the equation of motion for a quantum state. In the qubit picture the state is a complex vector $\lvert\psi\rangle$ in a Hilbert space, and the equation is a first-order linear ODE on that vector. The generator of the evolution is a Hermitian operator $H$ called the Hamiltonian, which encodes the total energy of the system.

$$i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\lvert\psi\rangle = H\lvert\psi\rangle$$

The central requirement on quantum time evolution is that it preserves the norm of $\lvert\psi\rangle$ — probabilities must always sum to 1. Any norm-preserving linear map is unitary, so the state at time $t + dt$ must be related to the state at time $t$ by a unitary operator:

$$\lvert\psi(t+dt)\rangle = U(t+dt,\,t)\,\lvert\psi(t)\rangle$$

For infinitesimal $dt$, continuity forces $U(t,t) = I$, so the operator can only differ from the identity by a first-order term. Writing $U(t+dt, t) = I - \frac{i}{\hbar}H\,dt$, the unitarity condition $U^\dagger U = I$ gives:

$$\left(I + \tfrac{i}{\hbar}H^\dagger\,dt\right)\!\left(I - \tfrac{i}{\hbar}H\,dt\right) = I + \tfrac{i}{\hbar}(H^\dagger - H)\,dt + O(dt^2) = I$$

which requires $H^\dagger = H$ — the generator must be Hermitian. Rearranging the finite-difference definition of the derivative then yields the Schrödinger equation.

For a time-independent Hamiltonian, the Schrödinger equation has an exact closed-form solution:

$$\lvert\psi(t)\rangle = U(t)\,\lvert\psi(0)\rangle, \qquad U(t) = e^{-iHt/\hbar}$$

The matrix exponential is defined by its power series $e^A = \sum_{k=0}^\infty A^k / k!$. Because $H$ is Hermitian, $-iHt/\hbar$ is skew-Hermitian (satisfies $A^\dagger = -A$), and the exponential of any skew-Hermitian matrix is unitary — so $U(t)$ is guaranteed to be unitary for all $t$, consistent with the derivation above.

The family of operators $\{U(t) : t \in \RR\}$ forms a one-parameter subgroup of the Lie group $\mathrm{U}(N)$ of $N\times N$ unitary matrices. The Hamiltonian $-iH$ (or $-iH/\hbar$ in SI units) belongs to the corresponding Lie algebra $\mathfrak{u}(N)$ of skew-Hermitian matrices. The exponential map $\exp : \mathfrak{u}(N) \to \mathrm{U}(N)$ is exactly the matrix exponential above — it converts an element of the algebra (a Hamiltonian) into an element of the group (a gate).

This correspondence means that specifying a Hermitian operator $H$ is equivalent to specifying a continuous family of unitary transformations $e^{-iHt}$ parametrized by time (or by a rotation angle). Every single-qubit and multi-qubit gate can be written in this form for some $H$.

When the Hamiltonian satisfies $H^2 = I$ — which holds for all Pauli matrices $X$, $Y$, $Z$ — the power series collapses to a two-term identity via Euler's formula:

$$e^{-i\theta H} = \cos\theta\,I - i\sin\theta\,H$$

The single-qubit rotation gates are defined exactly this way:

$$R_x(\theta) = e^{-i\theta X/2}, \quad R_y(\theta) = e^{-i\theta Y/2}, \quad R_z(\theta) = e^{-i\theta Z/2}$$

The factor of $1/2$ means a full rotation of the Bloch sphere requires $\theta = 2\pi$ rather than $\pi$, which is why a $180°$ physical rotation maps to a $\pi$ gate argument. The Hadamard gate can be written as $H = e^{i\pi(X+Z)/(2\sqrt{2})}$, placing it in the same framework.

• Schrödinger equation (QuTiP)