Variational Quantum Algorithms

The NISQ Era Challenge

Today's quantum computers are Noisy Intermediate-Scale Quantum (NISQ) devices: 50-1000 qubits with high error rates (~0.1-1%) and limited coherence times (microseconds to milliseconds). They can't run deep error-corrected circuits. Variational algorithms work around these limitations by using shallow circuits that finish before decoherence destroys the quantum state.

The Variational Principle

From quantum mechanics: any quantum state's energy expectation is always greater than or equal to the true ground state energy. Mathematically: E[θ] = ⟨ψ(θ)|H|ψ(θ)⟩ ≥ E₀. This guarantees that minimizing the measured energy pushes us toward the ground state, even with imperfect circuits.

Why It Works: Division of Labor

Three Flavors of Variational Algorithms

What is an Ansatz?

An ansatz is a parameterized quantum circuit that prepares trial states |ψ(θ)⟩. Think of it as a template with adjustable knobs(rotation angles θ). The goal: design a template flexible enough to reach the solution, but structured enough to train efficiently. Too simple = can't express the answer. Too complex = impossible to optimize.

The Expressibility-Trainability Tradeoff

Barren Plateaus: The Training Nightmare

In barren plateaus, gradients become exponentially small across most of parameter space. Random initialization lands you in a flat region where all directions look identical. Classical optimizer receives ~10⁻¹⁵ gradient signals swamped by ~10⁻³ shot noise - optimization stalls completely.

Hardware-Efficient Ansatzes

Hardware-efficient ansatzes use only gates native to the quantum processor, avoiding costly gate decompositions that amplify errors. For superconducting qubits: single-qubit rotations (Rx, Ry, Rz) + CNOT. For trapped ions: Rx, Ry + Mølmer-Sørensen gates.

Chemistry-Inspired Ansatzes (UCCSD Example)

For molecular problems, Unitary Coupled Cluster (UCC) ansatzes borrow from classical quantum chemistry. UCCSD (Singles + Doubles excitations) promotes electrons from occupied to virtual orbitals:

The Quantum-Classical Optimization Loop

Variational algorithms require minimizing a cost function C(θ)that can only be evaluated on quantum hardware. Each evaluation costs 1000s of quantum circuit executions (shots) to beat statistical noise. The classical optimizer must find the minimum efficiently despite expensive, noisy function evaluations.

Parameter-Shift Rule: Exact Quantum Gradients

Classical neural networks use backpropagation to compute gradients. Quantum circuits can't backpropagate through measurements. Instead, the parameter-shift rule computes exact partial derivatives by evaluating the circuit at shifted parameter values.

Gradient-Free Optimizers: When Gradients Fail

In barren plateaus or high-noise regimes, gradients become unreliable. Gradient-free methods navigate by exploring the cost landscape directly, no derivatives needed.

Cost Landscape Topology: Why Optimization Is Hard

Advanced: Natural Gradient and Quantum Fisher Information

Standard gradient descent treats all parameters equally. But in quantum circuits, some parameters θᵢ have stronger effects than others. Natural gradient descentuses the quantum Fisher information matrix F to rescale gradients:

The Electronic Structure Problem

In quantum chemistry, we want to solve the time-independent Schrödinger equation: Ĥ|ψ⟩ = E|ψ⟩. The electronic Hamiltonian Ĥ describes electron-electron repulsion, electron-nuclear attraction, and kinetic energy. Finding the ground state energy E₀ (lowest eigenvalue) tells us molecular stability, bond lengths, reaction barriers, and spectroscopic properties.

Step 1: Second Quantization

Classical chemistry uses orbitals φᵢ(r). Second quantization replaces wavefunctions with creation/annihilation operators acting on Fock space (occupation number representation).

Step 2: Fermion-to-Qubit Mapping

Fermions anticommute ({aᵢ,aⱼ} = 0), but qubits are bosonic (Pauli operators commute when i≠j). We need a transformation that preserves fermionic statistics using qubit operations.

Step 3: Chemical Accuracy Requirement

To be useful for drug discovery and materials design, quantum simulations must achieve chemical accuracy: energy errors within 1 kcal/mol = 0.0016 Hartree. This precision determines reaction feasibility and molecular stability predictions.

Practical Example: H₂ Molecule with VQE

Beyond Ground States: What Else Can VQE Do?