VQE – Variational Quantum Eigensolver: A Foundational Hybrid Quantum Algorithm

Table of Contents

1. Introduction
2. What Is VQE?
3. The Variational Principle in Quantum Mechanics
4. VQE Workflow Overview
5. Hamiltonian Representation
6. Parameterized Quantum Circuits (Ansatz)
7. Measuring Expectation Values
8. Classical Optimization Loop
9. Cost Function and Objective
10. Choosing the Ansatz
11. Hardware-Efficient Ansatz
12. Chemistry-Inspired Ansatz (UCCSD)
13. Optimizer Strategies for VQE
14. Noise Resilience and Error Mitigation
15. Simulation vs Hardware Execution
16. Qiskit Implementation of VQE
17. PennyLane Implementation of VQE
18. Common Challenges in VQE
19. Real-World Applications
20. Conclusion

1. Introduction

The Variational Quantum Eigensolver (VQE) is a leading hybrid quantum-classical algorithm designed to find the ground state energy of a Hamiltonian. It is especially suited for today’s NISQ-era hardware.

2. What Is VQE?

VQE leverages a parameterized quantum circuit to prepare trial quantum states and uses a classical optimizer to minimize the expectation value of the system Hamiltonian.

3. The Variational Principle in Quantum Mechanics

The principle states that for any state \( |\psi( heta)
angle \), the expectation value:
\[
E( heta) = \langle \psi( heta) | H | \psi( heta)
angle
\]
is always greater than or equal to the ground state energy of \( H \). VQE exploits this to approximate the lowest eigenvalue.

4. VQE Workflow Overview

1. Choose a Hamiltonian (e.g., molecular or spin model)
2. Define a parameterized ansatz
3. Measure expectation value for given parameters
4. Minimize energy via classical optimizer

5. Hamiltonian Representation

The Hamiltonian is expressed as a weighted sum of Pauli strings:
\[
H = \sum_i c_i P_i, \quad P_i \in {I, X, Y, Z}^{\otimes n}
\]
For example:
\[
H = -1.05 Z_0 – 0.39 Z_1 + 0.01 X_0 X_1
\]

6. Parameterized Quantum Circuits (Ansatz)

These are trainable circuits:

qml.RY(theta[0], wires=0)
qml.CNOT(wires=[0, 1])
qml.RY(theta[1], wires=1)

7. Measuring Expectation Values

Each Pauli term in the Hamiltonian is measured separately, and their weighted sum gives the energy estimate.

8. Classical Optimization Loop

The optimizer receives energy from quantum circuit evaluations and proposes new parameters to minimize the objective.

9. Cost Function and Objective

The cost is:
\[
E( heta) = \sum_i c_i \langle \psi( heta) | P_i | \psi( heta)
angle
\]

10. Choosing the Ansatz

Ansatz must be expressive enough to capture the ground state but shallow enough for NISQ execution.

11. Hardware-Efficient Ansatz

• Shallow depth
• Local rotations + entangling layers
• Native to backend gate set

12. Chemistry-Inspired Ansatz (UCCSD)

• Unitary Coupled Cluster with Single and Double excitations
• Requires Trotterization and deeper circuits

13. Optimizer Strategies for VQE

• Gradient-free: COBYLA, Nelder-Mead, SPSA
• Gradient-based: Adam, L-BFGS-B
• SPSA is robust against noisy evaluations

14. Noise Resilience and Error Mitigation

• Use readout error mitigation
• Zero-noise extrapolation
• Measurement averaging

15. Simulation vs Hardware Execution

Simulators allow exact expectation values, whereas real hardware introduces:

• Sampling noise
• Decoherence
• Gate errors

16. Qiskit Implementation of VQE

from qiskit.algorithms import VQE
from qiskit.opflow import Z, I
from qiskit.circuit.library import TwoLocal
from qiskit.algorithms.optimizers import COBYLA

hamiltonian = (Z ^ I) + (I ^ Z)
ansatz = TwoLocal(2, "ry", "cx", reps=2)
vqe = VQE(ansatz, optimizer=COBYLA())
result = vqe.compute_minimum_eigenvalue(hamiltonian)

17. PennyLane Implementation of VQE

import pennylane as qml
dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def circuit(params):
    qml.RY(params[0], wires=0)
    qml.CNOT(wires=[0, 1])
    qml.RY(params[1], wires=1)
    return qml.expval(qml.Hamiltonian(coeffs, obs))

opt = qml.GradientDescentOptimizer()
params = opt.step(circuit, init_params)

18. Common Challenges in VQE

• Barren plateaus (flat gradients)
• Noise accumulation in deep circuits
• Ansätze that do not cover full Hilbert space

19. Real-World Applications

• Finding electronic ground state energies (LiH, BeH₂)
• Optimization over discrete binary functions
• Modeling magnetism and lattice systems

20. Conclusion

VQE is one of the most successful NISQ-compatible quantum algorithms. Its hybrid nature allows exploration of quantum chemistry, optimization, and materials problems using today’s limited quantum hardware with classical support.