Project 6: Variational Quantum Eigensolver (VQE) - Hybrid Power

Build a VQE workflow to estimate the ground state energy of a small Hamiltonian.

Project Overview

Attribute Value
Difficulty Level 3: Advanced
Time Estimate 2-3 weeks
Main Language Python
Alternative Languages Julia, Q#
Knowledge Area Hybrid optimization
Tools Qiskit or Cirq
Main Book “Quantum Computation and Quantum Information” by Nielsen & Chuang

What you’ll build: A hybrid loop that uses a parameterized circuit and a classical optimizer to minimize energy.

Why it teaches quantum: VQE shows how near-term quantum hardware is used today.

Core challenges you’ll face:

  • Building a parameterized ansatz
  • Measuring expectation values
  • Optimizing parameters with a classical loop

Real World Outcome

You will estimate the ground state energy of a simple Hamiltonian and compare to the exact solution.

Example Output:

$ python vqe.py --hamiltonian h2
Estimated energy: -1.13
Exact energy: -1.14

Verification steps:

  • Compare results to known eigenvalues
  • Track convergence over iterations

The Core Question You’re Answering

“How can a noisy quantum device help solve eigenvalue problems?”

VQE is the flagship hybrid algorithm for NISQ devices.

Concepts You Must Understand First

Stop and research these before coding:

  1. Variational principle
    • Why does minimizing energy find the ground state?
    • Book Reference: Nielsen & Chuang, Ch. 4
  2. Parameterized circuits
    • How do gate parameters define an ansatz?
    • Book Reference: “Quantum Computing: An Applied Approach” by Jack Hidary, Ch. 8
  3. Expectation values
    • How do you estimate Hamiltonian terms by measurement?
    • Book Reference: Nielsen & Chuang, Ch. 4

Questions to Guide Your Design

  1. Ansatz choice
    • How complex should the ansatz be for a small Hamiltonian?
    • How will you prevent over-parameterization?
  2. Optimizer selection
    • Will you use gradient-free methods or gradients?
    • How will you handle noisy objective values?

Thinking Exercise

Simple Hamiltonian

Given a 1-qubit Hamiltonian H = Z, what is the ground state and energy?

Questions while working:

  • How does measurement in the Z basis reveal energy?
  • Why does the ansatz need to reach 1>?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “What is VQE and why is it hybrid?”
  2. “What is a variational ansatz?”
  3. “How do you compute expectation values?”
  4. “Why is optimization difficult with noisy measurements?”
  5. “What problems is VQE good for?”

Hints in Layers

Hint 1: Starting Point Start with a very small Hamiltonian (1-2 qubits).

Hint 2: Next Level Measure each Pauli term separately.

Hint 3: Technical Details Use a simple optimizer and track convergence.

Hint 4: Tools/Debugging Compare against exact diagonalization to validate.

Books That Will Help

Topic Book Chapter
Variational principle Nielsen & Chuang Ch. 4
Ansatz design “Quantum Computing: An Applied Approach” by Jack Hidary Ch. 8
Expectation values Nielsen & Chuang Ch. 4

Implementation Hints

  • Keep measurement circuits minimal.
  • Use a fixed seed for reproducibility.
  • Plot energy vs iteration to show convergence.

Learning Milestones

  1. First milestone: You can build a parameterized circuit.
  2. Second milestone: You can compute expectation values.
  3. Final milestone: You can converge to a ground-state estimate.