Self-Assessment Quiz: The Variational Quantum Eigensolver

Twenty questions on VQE: the variational principle, ansatz design, measurement cost, optimization, and the evidence for and against advantage. Aim for 16+.

Question 1

VQE finds:

A) The exact ground-state energy B) An upper bound on the ground-state energy, minimized over an ansatz C) All eigenvalues D) The time evolution

Question 2

The variational principle guarantees:

A) $\langle\psi|H|\psi\rangle \ge E_0$ for any normalized $|\psi\rangle$ B) $\langle\psi|H|\psi\rangle \le E_0$ C) Convergence to $E_0$ D) Exactness for any ansatz

Question 3

VQE's hybrid loop consists of:

A) Quantum state preparation and measurement, classical parameter optimization B) Purely quantum optimization C) Purely classical simulation D) Quantum optimization only

Question 4

The Hamiltonian is expressed as:

A) A single unitary B) A weighted sum of Pauli strings C) A classical matrix that must be diagonalized D) A circuit

Question 5

The number of Pauli terms for an $N$-orbital molecular Hamiltonian scales as:

A) $O(N)$ B) $O(N^2)$ C) $O(N^4)$ D) $O(2^N)$

Question 6

Each Pauli term generally requires:

A) Its own measurement basis (though commuting terms can be grouped) B) No measurement C) A separate quantum computer D) Error correction

Question 7

Estimating an energy to precision $\varepsilon$ requires shots scaling as:

A) $O(1/\varepsilon)$ B) $O(1/\varepsilon^2)$ C) $O(\log 1/\varepsilon)$ D) $O(1)$

Question 8

Chemical accuracy is 1 kcal/mol ≈ 1.6 mHartree. Reaching it typically requires:

A) Hundreds of shots B) Millions to billions of shots C) One shot D) No shots

Question 9

The UCCSD ansatz is:

A) Physically motivated but deep, often too deep for NISQ hardware B) Shallow and hardware-friendly C) Exact for all molecules D) Classical

Question 10

Hardware-efficient ansätze are:

A) Shallow and device-matched, but less physically motivated and prone to barren plateaus B) Always more accurate C) Deeper than UCCSD D) Exact

Question 11

A barren plateau causes:

A) Gradients that vanish exponentially in qubit count, stalling optimization B) Hardware failure C) Faster convergence D) Lower energy

Question 12

Barren plateaus are worsened by:

A) Deep random ansätze, global cost functions, and noise B) Shallow circuits C) Local cost functions D) Good initialization

Question 13

VQE's energy estimate is affected by hardware noise in that it:

A) Is variationally protected and always still an upper bound B) Can fall below the true ground-state energy, violating the variational bound C) Is unaffected D) Becomes exact

Question 14

Which optimizer is commonly used for noisy VQE landscapes?

A) Newton's method requiring exact Hessians B) SPSA or other gradient-free/stochastic methods C) Exact line search D) None

Question 15

The strongest current criticism of VQE is:

A) It cannot run on hardware B) Measurement cost and barren plateaus make it unlikely to scale to classically hard instances C) The variational principle is wrong D) It requires too few qubits

Question 16

True or false: VQE has demonstrated quantum advantage on a chemistry problem.

Question 17

True or false: A better ansatz always lowers the VQE energy.

Question 18

True or false: Grouping commuting Pauli terms can reduce measurement cost substantially.

Question 19

Short answer. Explain why hardware noise can push a VQE energy below the true ground state, and why that is diagnostically useful.

Question 20

Short answer. Estimate the shot count to reach chemical accuracy for a Hamiltonian with 500 Pauli terms, and comment on feasibility.

Answer Key

Q Ans Note
1 B It is a variational upper bound, exact only if the ansatz contains the true ground state.
2 A The Rayleigh–Ritz principle.
3 A The defining hybrid structure.
4 B $H = \sum_k c_k P_k$ with $P_k$ tensor products of Paulis.
5 C $O(N^4)$ from the two-electron integrals.
6 A Qubit-wise commuting groups can share a measurement basis, cutting the count several-fold.
7 B Standard shot-noise scaling — the dominant practical cost.
8 B With $O(N^4)$ terms each needing $1/\varepsilon^2$ shots, realistic molecules require $10^9$+ shots per energy evaluation, times hundreds of optimizer iterations.
9 A Unitary coupled cluster is chemically principled and typically too deep for current devices.
10 A The trade: shallow enough to run, but expressiveness and trainability suffer.
11 A Gradient magnitude decays as $O(2^{-n})$, so the optimizer sees a flat landscape.
12 A All three make plateaus worse; noise-induced plateaus are especially pernicious as they cannot be fixed by initialization.
13 B Noise breaks the variational guarantee — the measured value is no longer $\langle\psi|H|\psi\rangle$ for a valid $|\psi\rangle$.
14 B SPSA tolerates stochastic evaluations; exact-gradient methods assume precision the shot noise does not provide.
15 B Both are structural rather than engineering problems, which is why enthusiasm for VQE has cooled.
16 False Every VQE demonstration to date has been on molecules solvable classically, usually trivially.
17 False A more expressive ansatz has more parameters and a harder landscape; it may converge to a worse local minimum or fail to train. Expressiveness and trainability trade off.
18 True Qubit-wise commuting grouping typically reduces distinct measurement settings by an order of magnitude, and general commuting grouping does better still.
19 — The variational bound assumes you measure $\langle\psi|H|\psi\rangle$ for a genuine normalized state. Under noise the device prepares a mixed state and readout error biases the estimates, so the reported number is not a valid expectation value of $H$ for any state — it can land below $E_0$. Diagnostically this is useful: an energy below the known exact value is unambiguous proof of uncorrected systematic error, and it is one of the few self-checks available without a classical reference.
20 — Per term, $\varepsilon_{\text{term}} \approx 1.6$ mHa needs $\sim(\sigma/\varepsilon)^2$ shots; with $\sigma \sim 1$ Ha and terms combining in quadrature, a common estimate is $\sim(\sum_k |c_k|)^2/\varepsilon^2$. For 500 terms with coefficients summing to ~10 Ha, that is $(10/0.0016)^2 \approx 4\times10^{7}$ shots per energy evaluation. At ~200 optimizer iterations, $\sim10^{10}$ shots. At 10 kHz, roughly 11 days per molecule — for a molecule a laptop solves in seconds. Feasibility is the problem, not correctness.