Variational Algorithms

Variational Quantum Eigensolver (VQE) From Scratch

VQE finds ground-state energies of quantum Hamiltonians using a hybrid classical-quantum loop. This tutorial derives the variational principle, explains Jordan-Wigner fermion encoding, builds an H₂ ground-state computation end-to-end in Qiskit, and honestly discusses barren plateaus and why the ansatz choice makes or breaks the algorithm.

intermediate · ~26 min · prereq: Algorithms track (Tutorials 8-12)

QAOA for Combinatorial Optimization

QAOA encodes a combinatorial problem as a cost Hamiltonian, prepares a variational state by alternating cost and mixer evolutions, and uses a classical optimizer to find approximate solutions. This tutorial derives the MaxCut case from scratch, runs it in Qiskit, and honestly compares QAOA to classical baselines like Goemans-Williamson.

intermediate · ~23 min · prereq: Tutorial 13: Variational Quantum Eigensolver

Barren Plateaus: Why Most Variational Quantum Algorithms Fail at Scale

Barren plateaus are the dominant theoretical limit on variational quantum algorithms. Past a modest qubit count, the gradients of typical parameterized quantum circuits vanish exponentially in the system size, making training infeasible. This tutorial covers the McClean 2018 result, the cost-function-dependent and noise-induced extensions, and the four mitigation strategies that have moved the field — and gives an honest verdict on whether variational quantum computing has a future at scale.

advanced · ~20 min · prereq: Tutorial 13: Variational Quantum Eigensolver, Tutorial 17: Is QML Worth It? A Skeptic's Benchmark

ADAPT-VQE: Building the Ansatz One Operator at a Time

ADAPT-VQE is the most-cited barren-plateau mitigation strategy in quantum chemistry. Instead of a fixed ansatz, ADAPT grows the ansatz adaptively, adding one operator at a time from a problem-defined pool, picking the operator with the largest gradient. The resulting ansatz is shorter than UCCSD, more accurate at modest qubit counts, and structurally easier to train. This tutorial covers the algorithm, the operator-pool design, the qubit-ADAPT variant, and the regimes where ADAPT wins versus where it doesn't.

advanced · ~18 min · prereq: Tutorial 13: Variational Quantum Eigensolver, Tutorial 37: Barren Plateaus

The Parameter-Shift Rule: Computing Exact Quantum Gradients on Real Hardware

The parameter-shift rule is the standard exact-gradient method for variational quantum algorithms. Unlike finite differences, the rule produces unbiased gradient estimates with no truncation error, on real hardware, using only two extra circuit evaluations per parameter. This tutorial derives the rule from first principles, covers the generalized and stochastic variants for non-Pauli generators, and gives a decision rule for when parameter shift is the right gradient method.

intermediate · ~16 min · prereq: Tutorial 13: Variational Quantum Eigensolver, Tutorial 38: ADAPT-VQE

Quantum Natural Gradient: Geometry-Aware Optimization for Variational Quantum Algorithms

Standard gradient descent ignores the geometry of the parameter space. Quantum natural gradient (Stokes 2020) uses the quantum Fisher information matrix to rescale parameter updates by the local curvature, reaching minima with fewer iterations and partially mitigating some barren-plateau-adjacent training pathologies. This tutorial covers the math, the block-diagonal approximation that makes it tractable, and a decision rule for when QNG is worth the per-step overhead.

advanced · ~16 min · prereq: Tutorial 37: Barren Plateaus, Tutorial 39: The Parameter-Shift Rule

Imaginary-Time Evolution: How Quantum Algorithms Find Ground States

Imaginary-time evolution is a classical numerical technique for finding ground states: replace t with -i*t in the Schrödinger equation, and the wavefunction projects onto the ground state exponentially fast. Quantum analogs — variational imaginary-time evolution (VITE), quantum imaginary time evolution (QITE), and the deep connection to quantum natural gradient (tutorial 40) — are now central to ground-state quantum algorithms. This tutorial covers the math, the algorithms, and the regimes where each variant shines.

advanced · ~14 min · prereq: Tutorial 13: Variational Quantum Eigensolver, Tutorial 40: Quantum Natural Gradient

Hamiltonian Variational Ansatz: How to Build Trainable Ansätze from the Problem Itself

The Hamiltonian variational ansatz (HVA) builds variational circuits directly from the structure of the target Hamiltonian. Unlike generic hardware-efficient ansätze, HVA inherits problem symmetries, often avoids barren plateaus, and naturally connects to adiabatic quantum computing. This tutorial covers HVA construction, the connection to the quantum approximate optimization algorithm (QAOA, tutorial 14), and the design principles that make problem-tailored ansätze the production choice for variational chemistry and optimization.

advanced · ~14 min · prereq: Tutorial 14: QAOA for Combinatorial Optimization, Tutorial 37: Barren Plateaus, Tutorial 65: Imaginary-Time Evolution

Warm-Start Strategies: Initializing Variational Quantum Algorithms in the Right Region

Random initialization of variational parameters typically lands in the barren-plateau region of the cost landscape. Warm-start strategies — initializing from classical solutions, adiabatic schedules, parameter transfer from smaller systems, or other principled choices — sidestep this. The 2024-2025 evidence shows warm-started VQE and QAOA routinely achieve 10-100× faster convergence than random initialization, and reach better local minima. This tutorial covers the main strategies and the regimes where each wins.

intermediate · ~13 min · prereq: Tutorial 37: Barren Plateaus, Tutorial 66: Hamiltonian Variational Ansatz