  Quantum classifiers are trainable quantum circuits used as machine learning
models. The first part of the circuit implements a quantum feature map that
encodes classical inputs into quantum states, embedding the data in a
high-dimensional Hilbert space; the second part of the circuit executes a
quantum measurement interpreted as the output of the model. Usually, the
measurement is trained to distinguish quantum-embedded data. We propose to
instead train the first part of the circuit -- the embedding -- with the
objective of maximally separating data classes in Hilbert space, a strategy we
call quantum metric learning. As a result, the measurement minimizing a linear
classification loss is already known and depends on the metric used: for
embeddings separating data using the l1 or trace distance, this is the Helstrom
measurement, while for the l2 or Hilbert-Schmidt distance, it is a simple
overlap measurement. This approach provides a powerful analytic framework for
quantum machine learning and eliminates a major component in current models,
freeing up more precious resources to best leverage the capabilities of
near-term quantum information processors.
