What are Variational Quantum Algorithms?

Overview

Variational quantum algorithms exist because near-term quantum hardware is powerful enough to run small circuits, but too noisy to run very deep, fully fault-tolerant algorithms.

A variational algorithm reframes “solve a hard problem” as:

• choose a parameterized quantum circuit (a controllable family of circuits)
• measure a quantity that tells you how good the current parameters are
• use a classical optimizer to update parameters
• repeat

This hybrid approach is a practical fit for NISQ devices because it can use:

• relatively shallow circuits
• repeated sampling (shots)
• classical computation to steer the search

Intuition

Variational means “optimization,” not magic

“Variational” here means we search over a family of candidate solutions.

• The quantum computer generates candidates (via a parameterized circuit).
• The classical computer decides how to change the parameters.

It’s similar in spirit to tuning a control knob:

• turn knobs → run experiment → observe outcome → adjust knobs → repeat

The hybrid loop

A useful mental model is a feedback loop:

1. Quantum step: prepare a state using parameters $\theta$
2. Measurement step: estimate a score (cost) from repeated measurements
3. Classical step: update $\theta$ to improve the score

The quantum computer is not “doing the optimization.” It is producing measurement data that the classical optimizer uses.

Difference from gate-based “one-shot” algorithms

Many textbook quantum algorithms (like the Deutsch family or Grover) are:

• a fixed sequence of gates
• one final measurement
• a direct interpretation of the output

Variational algorithms are different:

• they are iterative
• they rely on noisy estimates (shot noise)
• the final output is often an estimated quantity or best-found parameter set

Formal Description

A variational algorithm can be described by three ingredients.

1) An ansatz (a parameterized circuit)

You choose a circuit family $U(\theta)$. Running $U(\theta)$ on a simple initial state (often $|0\cdots 0\rangle$) produces a state that depends on parameters $\theta$.

The ansatz is your “search space.”

2) A cost function

You define a real-valued objective $C(\theta)$ that you want to minimize (or maximize). In variational quantum settings, $C(\theta)$ is often defined via expectation values measured from the circuit.

3) A classical optimizer

A classical algorithm proposes parameter updates:

• choose $\theta_0$
• repeat: measure $C(\theta)$ and propose $\theta \leftarrow \theta'$

The key constraints in NISQ:

• you only observe $C(\theta)$ through noisy sampling
• each evaluation of $C(\theta)$ costs circuit executions

Worked Example

Imagine you want a circuit to produce a state that makes measuring 1 on a particular qubit likely. Define a simple cost:

• $C(\theta) = 1 - P_\theta(\text{measure }1)$

If $P_\theta(1)$ is high, the cost is low.

Workflow:

1. pick an initial $\theta$
2. run the circuit many shots to estimate $P_\theta(1)$
3. compute $C(\theta)$
4. update $\theta$ to try to reduce $C$

This toy example captures the variational pattern without needing a specific named algorithm.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Next we zoom in on the quantum side of the loop: parameterized quantum circuits (PQCs). We’ll discuss what the parameters mean, why ansatz choice matters, and the tradeoff between expressivity and trainability.