1.3 Time Evolution

1.3 Time Evolution#

Overview#

How do quantum states change in time? This section establishes that time evolution must preserve inner products (unitarity). This requirement leads to the Schrödinger equation and identifies the Hamiltonian as the generator of dynamics. Two complementary perspectives — Schrödinger (states change) and Heisenberg (operators change) — illuminate connections to conservation laws, symmetries, and Lie groups extending beyond single qubits.

Topics#

Topic

Title

Core Question

1.3.1

Unitary Evolution

Why must time evolution be unitary, and how does the Schrödinger equation follow?

1.3.2

Schrödinger Picture

How do quantum states evolve under a Hamiltonian, and what are Rabi oscillations?

1.3.3

Heisenberg Picture

How do observables evolve, and what do conservation laws reveal about symmetry?

Key Concepts#

  • Unitary operator: \(\hat{U}^\dagger\hat{U} = \hat{I}\); preserves inner products, normalization, and measurement probabilities.

  • Hermitian generator: Every unitary \(\hat{U}(\theta) = \mathrm{e}^{\mathrm{i}\hat{G}\theta}\) derives from Hermitian \(\hat{G}\); for time evolution, \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar}\).

  • Schrödinger equation: \(\mathrm{i}\hbar\partial_t\vert\psi\rangle = \hat{H}\vert\psi\rangle\) governs quantum state dynamics.

  • Heisenberg equation: \(\mathrm{d}\hat{O}/\mathrm{d}t = (\mathrm{i}/\hbar)[\hat{H},\hat{O}]\) governs observable evolution; conserved quantities commute with \(\hat{H}\).

  • Larmor precession and Rabi oscillations: Static magnetic field causes spin precession; resonant driving induces oscillations between energy levels.

  • Lie groups: \(U(1)\) and \(SU(2)\) emerge naturally as qubit symmetry groups, prefiguring gauge groups of fundamental interactions.

Learning Objectives#

  • Derive the Schrödinger equation from unitarity requirements and Hermitian generators.

  • Solve time-dependent Schrödinger equations for qubit Hamiltonians.

  • Convert between Schrödinger and Heisenberg pictures using equations of motion.

  • Connect conservation laws to symmetry generators and identify \(U(1)\) and \(SU(2)\) Lie groups.

Project#

Project: Pulse Shaping and GRAPE Optimization for Quantum Gate Design#

Objective: Design optimal control techniques (GRAPE: Gradient Ascent Pulse Engineering) for implementing robust quantum gates under realistic Hamiltonian constraints and decoherence.

Hamiltonian Model: \(\hat{H} = \frac{\omega_0}{2}\hat{\sigma}^z + \Omega(t)\hat{\sigma}^x\) with dephasing rate \(\gamma_\phi\) and spontaneous emission \(\gamma_1\).

Suggested Approach:

  • Literature survey of optimal control theory for quantum systems

  • Derive GRAPE algorithm; parametrize pulses using basis functions

  • Solve Liouville–von Neumann equations numerically

  • Optimize fidelity through gradient-based methods

Expected Deliverable: Research report (6–8 pages) covering optimal control theory, GRAPE algorithm derivation, numerical results for 1–2 qubit gates, fidelity-time trade-offs, robustness analysis, and experimental implementation discussion.