Questions: Variational Quantum Eigensolver (VQE)

Question 1

The variational principle guarantees that for any trial state |psi>, the expectation value <psi|H|psi> is:

• A. Equal to the ground-state energy E_0
• B. Less than or equal to E_0
• C. Greater than or equal to E_0
• D. Unrelated to E_0

Answer: (C)

The variational principle states that for any normalized state |psi>, <psi|H|psi> >= E_0 where E_0 is the ground-state energy. Equality holds only when |psi> is the ground state. This means the measured energy always provides an upper bound, and minimizing over the parameters theta approaches E_0 from above. This is what makes VQE a rigorous method: even with noise and limited optimization, the result is a valid upper bound.

Question 2

VQE uses only the quantum computer for optimization — the classical computer is not involved in the parameter update.

• T. True
• F. False

Answer: False

VQE is explicitly a hybrid quantum-classical algorithm. The quantum computer prepares states and measures expectation values — tasks that are hard classically for large systems. The classical computer runs the optimization loop (gradient descent, COBYLA, SPSA, etc.) that updates the circuit parameters based on the measured energies. Neither component alone suffices: the quantum computer cannot optimize, and the classical computer cannot efficiently compute expectation values for large quantum systems.

Question 3

What is the 'ansatz' in VQE, and why is its choice critical to the algorithm's success?

Model answer: The ansatz is the parameterized quantum circuit that defines the family of trial states |psi(theta)>. Its choice is critical because it determines the expressiveness of the search space. If the ansatz cannot represent the true ground state (or a good approximation), the optimization will converge to a local minimum far above the true energy. However, overly expressive ansatze with many parameters suffer from barren plateaus (vanishing gradients) and require more measurements. The best ansatze are problem-inspired — for chemistry, the Unitary Coupled Cluster ansatz captures electron correlation structure while remaining relatively compact.

The ansatz design is the art of VQE. Hardware-efficient ansatze use gates native to the device, minimizing circuit depth, but may lack physical motivation and suffer from optimization difficulties. Chemically-motivated ansatze like UCCSD capture the right physics but may require deeper circuits. The barren plateau phenomenon — where gradients vanish exponentially with system size for random ansatze — is a fundamental challenge that constrains the practical scalability of VQE.