How is data encoded in a quantum neural network?

There are many possible ways to encode data into a quantum neural network (QNN). In one of the first papers to suggest the use of variational circuits to classify data [1], the authors suggest the following general architecture for a QNN:

The circuit starts with the $|0rangle$ state, encodes a data point $textbf{x}$ using a circuit $S_textbf{x}$, and transforms it using a parametrized unitary $U(theta)$ (using rotations to encode parameters). The result of the classification is then given by the measurement of one of the qubit.

Now, what does the circuit $S_textbf{x}$ look like? You have several possibilities, discussed and compared in a recent paper [2]. The two main ones are amplitude encoding and angle encoding.

Amplitude encoding (also called wavefunction encoding) consists in the following transformation:

$$ S_textbf{x}|0rangle=frac{1}{||textbf{x}||}sum_{i=1}^{2^n} x_i |irangle $$

where each $x_i$ is a feature (component) of your data point $textbf{x}$, and ${|irangle}$ is a basis of your $n$-qubit space (like $|0..00rangle, |0..01rangle,...,|1..11rangle$). The advantage of this encoding is that you can store $2^n$ features using only $n$ qubits (so in the IBM case, 32 features). The disadvantage is that in general this circuit $S_{textbf{x}}$ will have a depth of $O(2^n)$ and be very hard to construct.

Angle encoding (also called qubit encoding) consists in the following transformation:

$$ S_{textbf{x}} |0rangle=bigotimes_{i=1}^n cos(x_i)|0rangle + sin(x_i)|1rangle. $$ It can be constructed using a single rotation with angle $x_i$ (normalized to be in $[-pi,pi]$) for each qubit, and can therefore encode $n$ features with $n$ qubits (so in your case only $5$ features). But it can be very easily constructed and has a depth of only 1. Note that there is a slight variant of this encoding, referred as dense angle encoding, that can encode $2n$ features in $n$ qubits $$ S_{textbf{x}} |0rangle=bigotimes_{i=1}^n cos(x_{2i-1})|0rangle + e^{ix_{2i}}sin(x_{2i-1})|1rangle. $$ by using a phase gate after each rotation.

The question of what encoding you should use and which one can provide a quantum advantage is still an open research problem, since for the moment, there's no proof or empirical evidence that QNN are useful at all for machine learning tasks.

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[1] Maria Schuld, Alex Bocharov, Krysta Svore and Nathan Wiebe, Circuit-centric quantum classifiers, 2018

[2] Ryan LaRose and Brian Coyle, Robust data encodings for quantum classifiers, 2020