Sketches toward quantum AI

Qubit states in one line

A single qubit lives in a two-dimensional complex vector space. Any pure state can be written as

|ψ⟩ = α|0⟩ + β|1⟩

with the constraint ∣α∣² + ∣β∣² = 1. Multi-qubit systems are tensor products of these spaces, and gates are unitary operators on them.

Variational circuits

A variational quantum circuit is a gate sequence U(θ) with tunable parameters θ. The circuit prepares a state |ψ(θ)⟩ and we define a classical loss L(θ) based on measurement statistics.

L(θ) = ⟨ψ(θ)| H |ψ(θ)⟩

where H is an observable (for example a Hamiltonian or a task-specific operator). Gradients can be estimated via parameter-shift rules.

Hybrid training loop

A realistic near-term pattern is a hybrid loop: classical networks handle perception and post-processing, while a small quantum circuit is used as a trainable layer or sampler.

def step(x, params_classical, params_quantum):
    features = classical_encoder(x, params_classical)
    expectation = run_quantum_circuit(features, params_quantum)
    logits = classical_head(expectation, params_classical)

    loss = loss_fn(logits, target)
    loss.backward()
    optim.step()

    return loss

The quantum circuit is treated as a differentiable component with noisy gradient estimates; the rest of the stack looks like familiar deep learning.

Quantum AI mermaid schema

Systemically, the hybrid model can be drawn as: