4. Variational algorithms

Deutsch, Bernstein-Vazirani, and Grover assume a perfect, deep circuit. Devices available now are noisy and shallow. Variational algorithms move the difficulty into a classical loop: a short quantum circuit has free angles, a measurement scores it, and an ordinary optimizer proposes better angles.

The two named versions are VQE (variational quantum eigensolver), for the lowest energy of a physical system, and QAOA (quantum approximate optimization algorithm), for combinatorial problems. Both are the same loop.

The problem

A Hamiltonian $H$ is a matrix of energies. Its lowest eigenvalue $E_0$ is the ground-state energy: the energy of the most stable state. For a molecule, that number decides bond lengths and reaction barriers. Computing it exactly for a large system is what the classical calculation spends its time on.

The variational principle says you do not need the exact ground state to get an upper bound. For any guess $|\psi\rangle$,

with equality only at the true ground state. So: propose $|\psi(\theta)\rangle$ with a circuit, estimate the left-hand side, and change $\theta$ until the estimate stops falling.

The principle

The circuit is a guess, not the answer. The measurement is a score.

On one qubit, take $H = Z$. Its eigenvalues are $+1$ (state $|0\rangle$) and $-1$ (state $|1\rangle$), so $E_0 = -1$. A guess prepared by RY(θ) has

when the circuit starts from $|0\rangle$. The score is read from the probabilities. No new simulator feature is required.

The optimizer’s job is to drive $\langle Z \rangle$ down toward $-1$. At $\theta = \pi$ the score is $-1$ and the bound is tight: the guess is the ground state.

From the simple case to the harder one

One qubit, one parameter. The worked example. You can see the whole curve. The minimum is obvious, which is the point of a first example.

A sum of terms. A molecular Hamiltonian is not one $Z$. It is a long sum of Pauli strings, $H = \sum_k c_k P_k$, with hundreds or thousands of terms even for a small molecule. You estimate each $\langle P_k \rangle$ with its own measurement setting and add them. The circuit does not change. The number of experiments does.

Many parameters. A useful guess for a molecule is a layered circuit: a block of rotations and CNOTs, repeated. Twenty parameters is a normal small instance. The score is a bumpy function of those parameters. Gradient-free methods and the parameter-shift rule are how the classical optimizer moves. Local minima are the usual failure, not a wrong simulator.

QAOA, the combinatorial cousin. Replace the energy with a cost, such as the number of violated edges in a scheduling or cutting problem. Alternate two layers: a cost layer that phases each bad assignment, and a mixer layer of $X$ rotations that proposes new assignments. More layers means a better approximation and a deeper circuit. One layer is already the same loop as VQE: prepare, score, update the angles.

A worked example

import math
import numpy as np
from fqkit import QuantumCircuit, RY, Parameter, bind_parameters, run
 
np.set_printoptions(precision=4, suppress=True)
 
def energy(angle):
    theta = Parameter("theta")
    qc = QuantumCircuit(1)
    qc.add_gate(RY(theta), [0])
    bind_parameters(qc, {"theta": angle})
    probabilities = np.abs(run(qc)) ** 2
    return float(probabilities[0] - probabilities[1])   # <Z>
 
for angle in (0.0, math.pi / 2, math.pi):
    score = round(energy(angle), 4)
    print(f"theta = {angle:.4f}   <Z> = {score}")

theta = 0.0000   <Z> = 1.0
theta = 1.5708   <Z> = 0.0
theta = 3.1416   <Z> = -1.0

The ground-state energy is $-1$. The last angle attains it. The first angle is the worst possible guess.

Problems

1. Print $P(0)$ and $P(1)$ next to $\langle Z \rangle$ for $\theta = \pi/3$. Check that the score really is the difference of the probabilities, and that it equals $\cos(\pi/3)$.
2. Change the ansatz from RY to RX. At which of the three angles, if any, do you still reach $-1$? Explain using what RX does to $|0\rangle$.
3. Suppose $H = -Z$ instead of $Z$. What is $E_0$, and which of the three angles in the example attains it? You should be able to answer before editing the code.

Where it is used

These algorithms are research tools. They have published hardware results. They are not yet a replacement for a classical chemistry code or a classical solver.

• VQE, chemistry and materials. The target is a ground-state energy. Demonstrations have covered the hydrogen molecule and a few other very small systems, as a check that the loop can be run on a device. The industrial targets people cite (catalysts, battery electrolytes, drug-like molecules) need more qubits and quieter gates than current machines give. The algorithm is how those calculations are being attempted.
• QAOA, combinatorial optimization. The same loop with a cost instead of a Hamiltonian. Prototype problems include Max-Cut, portfolio selection, and shift scheduling. For those sizes, a classical solver is still faster and exact. QAOA is studied because the circuit shape matches the hardware, and because the approximation can improve as more layers fit.
• What “used” should not be taken to mean. A variational run on a simulator, including the one-qubit example above, is a numerical method. It becomes a quantum-hardware method only when the expectation values are estimated from shots on a device rather than from a state vector.

That closes the algorithm path: one exact query, a hidden string, an unstructured search, and a scored circuit you tune.