L12: Variational Quantum Eigensolvers (Ground States of Molecules & Materials on NISQ Hardware) - Quantum Computing & Quantum Information — Two-Semester Calendar

Learning Objectives¶

State the variational principle and the VQE hybrid loop.
Build hardware-efficient and chemistry-inspired (UCCSD) ansätze.
Estimate $\langle H\rangle$ from shots; budget measurements per iteration.
Recognize failure modes: barren plateaus, local minima, noise floors.

The Variational Principle¶

For any parameterized state $|\psi(\boldsymbol\theta)\rangle$ :

E(\boldsymbol\theta) = \langle \psi(\boldsymbol\theta)| H |\psi(\boldsymbol\theta)\rangle \ge E_0.

Minimizing over $\boldsymbol\theta$ upper-bounds — and ideally reaches — the ground-state energy.

Eureka! VQE splits the work: the QPU prepares and measures a state classical computers cannot store; the CPU does the optimization it is good at.

The VQE Loop¶

Map problem to qubit Hamiltonian $H=\sum_k h_k P_k$ (L11: Jordan-Wigner).
Prepare ansatz $|\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\ket{\text{ref}}$ (e.g. Hartree-Fock reference).
Measure each Pauli group; assemble $E(\boldsymbol\theta)$ .
Classical optimizer updates $\boldsymbol\theta$ (COBYLA, SPSA, gradient via parameter-shift).
Repeat to convergence; report energy ± shot-noise error bar.

Ansatz Design¶

Hardware-efficient: layers of single-qubit rotations + entangling gates matching the device coupling map. Cheap, expressive, but physics-blind.
UCCSD: $U = e^{T-T^\dagger}$ with single/double excitations, Trotterized. Chemically motivated; deeper circuits.
ADAPT-VQE: grow the ansatz operator-by-operator using gradient screening.
Symmetry helps: conserve particle number and spin to shrink the search space.

Eureka! Ansatz choice is the modeling step — the same judgment call as choosing a trial wavefunction in classical variational Monte Carlo.

Gradients & Measurement Budget¶

Parameter-shift rule (exact for gates $e^{-i\theta P/2}$ ):

\partial_\theta E = \tfrac{1}{2}\left[E(\theta+\tfrac{\pi}{2}) - E(\theta-\tfrac{\pi}{2})\right].

Shot noise: estimating $E$ to precision $\varepsilon$ costs $\mathcal{O}(\sum_k |h_k|)^2/\varepsilon^2$ shots.

Eureka! Chemical accuracy is $1.6\,$ mHa $\approx 0.043\,$ eV — the shot budget, not the circuit, is often the bottleneck.

Failure Modes¶

Barren plateaus: for deep random ansätze, gradients vanish exponentially in $n$ . Mitigate: shallow/local ansätze, good initialization (HF), layerwise training.
Local minima in a nonconvex landscape; use restarts / SPSA.
Noise floor: decoherence biases $E(\boldsymbol\theta)$ upward — motivation for error mitigation (L17, Semester 2).

Materials-Science Targets¶

Dissociation curves (H $_2$ , LiH): the “hello world” of quantum chemistry.
Hubbard model plaquettes: correlated-electron physics beyond mean field.
Defect centers (NV, divacancies): small active spaces embedded in DFT — quantum embedding.
Related variational cousins: QAOA for combinatorial problems (alloy/structure search).

Eureka! The realistic near-term role: quantum solves a small strongly-correlated active space; classical DFT/DMFT handles the rest. Know both sides.

Mini-Lab¶

VQE for H $_2$ at 5 bond lengths (2-qubit reduced Hamiltonian); plot dissociation curve vs. exact diagonalization.
Compare hardware-efficient vs. UCCSD ansatz: iterations, depth, final error.
Repeat with a depolarizing noise model; observe the noise floor.

Takeaways¶

VQE = variational principle + hybrid quantum-classical optimization.
Ansatz and measurement budget are the real design decisions.
Materials workflow: active space on the QPU, environment on the CPU.