Concept 3 of 7 · How we find it

The VQE loop

VQE is a thermostat for energy — a quantum circuit prepares a guess at the ground state, a classical optimizer measures how good the guess is and adjusts the dials, and the loop keeps going until the energy stops dropping.

VQE is a hybrid loop. The quantum chip does what it is good at — state preparation and measurement — while the classical optimizer does the numerical search, repeating until the energy plateaus.

What it actually means

VQE — Variational Quantum Eigensolver — is a hybrid algorithm. Quantum computers and classical computers each do part of the work.

The quantum part is a parametrized circuit called the ansatz. It has a fixed structure but adjustable knobs — typically tens to hundreds of rotation angles. For any setting of the knobs θ, the circuit produces a specific quantum state |ψ(θ)⟩. By choosing the structure carefully, you make sure that somewhere in the space of possible θ values there is a state close to the molecule’s true ground state.

The quantum chip’s job is to:

Run the circuit with the current θ.
Measure the expectation value of the Hamiltonian on the resulting state.
Report back a number: E(θ).

That number is the energy of the current guess. The lower it is, the closer the guess is to the ground state. (This is guaranteed by the variational principle — the energy of any state is always ≥ the true ground-state energy.)

The classical part is a numerical optimizer — the kind that has been solving “minimize f(x)” problems for decades. It takes the current E(θ), proposes a new θ that should give lower energy, and asks the quantum chip to evaluate it. After many iterations the energy stops dropping, and the θ at that point is your VQE-approximation to the ground state.

Why hybrid? Because today’s quantum computers are too noisy to run very deep circuits. VQE keeps the circuit shallow (good for noise tolerance) and puts all the iterative work on the classical side. It is pragmatic.

The elegant alternative

The more elegant approach — Quantum Phase Estimation (QPE) — gives ground-state energies to arbitrarily high precision (within the chosen basis and active space) but needs fault-tolerant hardware to sustain the deep, repeated controlled-evolution circuits it requires. That hardware doesn’t exist yet.

Why it matters for our problem

Every notebook in this repo, except the Bell state demo, is a VQE.

Notebook	Circuit qubits	Ansatz	Result
`02_h2_binding.ipynb`	2	small `EfficientSU2`	H₂ ground state to nanohartree precision, in seconds
`03_wh_binding.ipynb`	6	`EfficientSU2(reps=4)`, ~60 parameters	WH⁻ ground state to ~11 mHa accuracy — roughly 300 meV off, workable for a demonstration but a long way from chemical accuracy

The ansatz is the pivotal choice

The ansatz choice is the most consequential decision in VQE. We use EfficientSU2 — a hardware-efficient ansatz that maps cleanly onto real QPU gate sets — because it is robust and runs fast. The chemistry-aware alternative, UCCSD, would give better accuracy but takes ~24 minutes per cost evaluation at our problem size, which is impractical for a demonstration.

This trade-off (accuracy vs. tractability) is exactly the kind of choice that gets cheaper when fault-tolerant hardware arrives: QPE or trotterized UCCSD become practical, and within the chosen basis and active space (and with a suitable initial state preparation) you can drive the algorithmic error arbitrarily low.

One nuance to carry forward

The basis-set and active-space errors don’t go away — those are physics choices, not algorithm choices. Concept 4 picks up active spaces: why 6 qubits is enough for WH⁻, and why it isn’t enough for the real fusion problem.