ψQMLQUANTUM MACHINE LEARNING
Chapter 01 / Concept 01 of 48

Linear algebra for QML

Complex vector spaces, inner products, unitary matrices, tensor products and spectral decomposition.

Input / assumptions→ Quantum or learning model→ Measurement / inference→ Validation

Core idea

Complex vector spaces, inner products, unitary matrices, tensor products and spectral decomposition. The central design question is what information the representation retains, which operations are physically implementable, and how prediction quality and total resource cost are measured. In QML, a mathematical formulation must be paired with an explicit data-access model and a reproducible evaluation protocol.

$$\langle\psi|\phi\rangle=\sum_j\psi_j^*\phi_j$$

Worked example

Normalize the vector $(1,i)$ to $|\psi\rangle=(|0\rangle+i|1\rangle)/\sqrt2$. Its norm squared is $1/2+1/2=1$.

How to study or implement it

Start with a minimal analytic case that can be checked by hand. Build a noiseless simulation, inspect intermediate states or predictions, and then repeat with finite sampling. Compare against a suitable classical reference using the same dataset, preprocessing, and evaluation split. Record circuit depth, qubit count, optimization steps, and measurement budget.

Interpretation and limitations

Complex conjugation matters: a plain dot product is not a quantum inner product. When evaluating experimental claims, separate theoretical asymptotic statements from measured end-to-end performance. Include preprocessing, state preparation, parameter tuning and output extraction in the comparison.

Further exploration

See the chapter overview, interactive lab, guided learning path, and curated sources.

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