Variational Quantum Algorithm

A class of hybrid quantum-classical algorithms using parameterized circuits optimized by classical computers.

Variational Quantum Algorithms (VQAs) are the primary approach for near-term (NISQ) quantum computing. They combine quantum circuit evaluation with classical optimization.

The Structure

┌─────────────────────────────────────────────────────┐
│                Classical Computer                    │
│  ┌─────────────┐                    ┌────────────┐  │
│  │  Optimizer  │◄──── Cost ◄───────│   Measure  │  │
│  │   (θ→θ')    │                    │ expectation│  │
│  └──────┬──────┘                    └─────▲──────┘  │
│         │ New θ                           │         │
│         ▼                                 │         │
│  ┌──────────────────────────────────────────────┐  │
│  │              Quantum Computer                 │  │
│  │  |0⟩ ──[Ansatz U(θ)]──── Measure             │  │
│  └──────────────────────────────────────────────┘  │
└─────────────────────────────────────────────────────┘

1. Ansatz: Parameterized quantum circuit $U(\theta )$
2. Evaluate: Measure expectation value $⟨O{⟩}_{\theta }$
3. Optimize: Classical optimizer updates $\theta$
4. Repeat: Until convergence

Key Components

Ansatz (Circuit Structure)

Parameterized circuit that can represent the solution:

Cost Function

What you’re minimizing:

• Energy (for VQE)
• Objective value (for QAOA)
• Classification loss (for QML)

Classical Optimizer

Updates parameters based on cost:

• Gradient-based: ADAM, L-BFGS
• Gradient-free: COBYLA, Nelder-Mead
• Quantum-aware: SPSA, QNG

Popular VQAs

Why Variational?

NISQ-Friendly

• Short circuits (low depth)
• Resilient to some noise
• Flexible structure

Trainable

• Classical optimizer handles complexity
• Adapts to problem instance
• No deep understanding needed

Challenges

Outlook

VQAs are pragmatic but limited:

• Good for exploring NISQ capabilities
• Unclear when they outperform classical
• May not scale to large problems

Bridge to fault-tolerant era or dead end? Jury’s out.

See also: VQE, QAOA, NISQ