Classical ML vs. Quantum ML: The Hybrid Loop

How a neural network and a variational quantum model learn, side by side, and an honest look at when QML helps.

Advanced12 min read6 sections · 2 questions+45

By the end you will be able to

01Describe the hybrid loop: encoding, parameterised circuit U(θ), measurement, classical optimiser.
02Map each part of a neural network to its counterpart in a quantum model.
03Say where QML is a reasonable research bet and where classical ML is the better tool today.

A neural network learns by adjusting numbers. The input x is multiplied by a weight matrix W, passed through a non-linear activation, compared with the right answer by a loss function, and the weights are nudged downhill along the gradient. A quantum model keeps exactly this loop and replaces the model in the middle with a circuit.

Step	Classical neural network	Hybrid quantum model
Input	a vector x of numbers	x written into rotation angles (encoding, feature map)
Model	layers: W·x, then an activation	parameterised circuit U(θ), the ansatz
Output	numbers from the last layer	expectation values such as ⟨Z⟩, estimated from shots
Loss	computed on the CPU or GPU	computed on the CPU from the measured values
Update	backpropagation through all layers	gradients from extra circuit runs, or a gradient-free optimiser such as COBYLA
Hardware	CPU or GPU	QPU for the circuit, CPU for everything else

\frac{\partial ⟨ Z ⟩}{\partial θ} = \frac{1}{2} (⟨ Z ⟩_{θ + π /2} - ⟨ Z ⟩_{θ - π /2})

The parameter-shift rule for rx, ry and rz: run the circuit twice with θ shifted by ±π/2 and subtract. Two extra runs per parameter, each with its own shot noise, is why training on hardware is slow.

01One turn of the hybrid loop

1. The CPU chooses the parameters θ.
2. The QPU runs encoding + U(θ) for many shots.
3. The CPU turns the counts into expectation values and a loss.
4. The optimiser updates θ and the loop starts again until the loss stops falling.

Hands-on

Be the optimiser

The first ry is the data point x = 0.8, the second is the trainable weight θ = 1.2. Simulate and compute ⟨Z⟩ = P(0) − P(1) from the histogram (about −0.42). Here ⟨Z⟩ = cos(0.8 + θ): change θ to −0.8 and every shot reads 0; change it to 2.34 and almost every shot reads 1. Each edit is one turn of the loop done by hand.

OpenQASM 3

OPENQASM 3.0;
include "stdgates.inc";
qubit[1] q;
bit[1] c;
ry(0.8) q[0];
ry(1.2) q[0];
c[0] = measure q[0];

Open in the IDE Open in the Playground

02When does QML help? An honest answer

Images, text, tables of business data and large language models: classical ML is the right tool. Today's chips have few qubits, slow data loading and shot noise, and no advantage has been shown on such data.
Loading many classical numbers into a circuit is itself expensive, and that cost can cancel a speed-up that looks good on paper.
Promising directions: data that is already quantum (states from a quantum sensor or simulation, molecules and materials) and quantum kernels, which compare data points in a very large feature space. Whether they beat classical methods on practical problems is still an open research question.
Training has its own limits: in deep, randomly initialised circuits the gradients can vanish exponentially with the number of qubits (barren plateaus).

Knowledge check

2 questions · one try each counts

Question 1

In the hybrid loop, which part runs on the quantum processor?

Question 2

Which statement about QML today is the most honest?

Step

Classical neural network

Hybrid quantum model

Input

a vector x of numbers

x written into rotation angles (encoding, feature map)

Model

layers: W·x, then an activation

parameterised circuit U(θ), the ansatz

Output

numbers from the last layer

expectation values such as ⟨Z⟩, estimated from shots

Loss

computed on the CPU or GPU

computed on the CPU from the measured values

Update

backpropagation through all layers

gradients from extra circuit runs, or a gradient-free optimiser such as COBYLA

Hardware

CPU or GPU

QPU for the circuit, CPU for everything else

02When does QML help? An honest answer

Images, text, tables of business data and large language models: classical ML is the right tool. Today's chips have few qubits, slow data loading and shot noise, and no advantage has been shown on such data.
Loading many classical numbers into a circuit is itself expensive, and that cost can cancel a speed-up that looks good on paper.
Promising directions: data that is already quantum (states from a quantum sensor or simulation, molecules and materials) and quantum kernels, which compare data points in a very large feature space. Whether they beat classical methods on practical problems is still an open research question.
Training has its own limits: in deep, randomly initialised circuits the gradients can vanish exponentially with the number of qubits (barren plateaus).