Quantum Embedding

Introduction

A quantum embedding represtnes classical data as quantum states in Hilbert Space via a quantum feature map. With the input $x$, we get $|x\rangle$ after quantum embedding, which is a set of gate parameters in a quantum circuit.

Let's consider classical input date with $m$ example with $N$ featuress each,

$$\mathcal{D} = x^{(1)}, \cdots, x^{(m)}, \cdots, x^{(M)},$$

where $x^{(1)}$ is the $N$-dimensional vector for $m = 1, \cdots, M$. To embed this data into $n$ quantum subsystems. We can use various type of embedding techniques like Basis Embedding and Amplitude Embedding we are going to talk about next.

Basis Embedding

Basis Embedding associates each input with a computational basis state of a qubit subsystem. The classical data has to be in the form of binary bit strings. For example, $x = 1101$ is represented by the 4-qubit quantum state $|1001\rangle$.

One bit of classical information is represented by one quantum subsystem.

Thus, $x^{(m)} = (b_{1},\cdots,b_{N})$ with $b_{i} \in [0,1]$ for $i=1,\cdots, N$. we have

$$x^{(m)} \mapsto |X^{(m)}\rangle.$$

The number of quantum subsystem $(n)$ most at least equals to $N$.

Amplitude Embedding

Data is encoded into amplitudes of quantum state. A normalized classical $N$-dimensional classical datapoint $x$ is represented by the amplitude of a $n$-qubit quantum state $|\psi_{x}\rangle$ as

$$|\psi_{x}\rangle = \sum_{i = 1}^{N}x_{i}|i\rangle,$$

where $N = 2^{n}$, $x_{i}$ is the i-th element of $x$ and $|i\rangle$ is the i-th computational basis state. $x_{i}$ can be either float or integer. For example, we have a normalized classical data

$$x_{\text{norm}} = \frac{1}{\sqrt{31.25}}(1,0.0,-5.5,0.0),$$

the corrosponding amplitude embedding is

$$\begin{array}{ll} |\psi_{x_{\text{norm}}}\rangle & = \frac{1}{\sqrt{31.25}}[1|00\rangle + 0|01\rangle - 5.5|10\rangle+0|11\rangle]\\ \ & = \frac{1}{\sqrt{31.25}}[1|00\rangle - 5.5|10\rangle] \end{array}$$

Let's consider the classical $\mathcal{D}$ above. Its amplitude embedding can be easily understood if we concatenate all the input example $x$ together into one vector

$$\alpha = C_{\text{norm}}x^{(1)}_{1},\cdots,x^{(1)}_{N},x^{(2)}_{1},\cdots,x^{(2)}_{N},\cdots,x^{(M)}_{1},\cdots,x^{(M)}_{N},$$

where $C_{\text{norm}}$ is the normalization constant, $|\alpha|^{2} = 1$. The input dataset can be represented as

$$|\mathcal{D}\rangle = \sum_{i=1}^{2^{n}}\alpha_{i}|i\rangle,$$

where $\alpha_{i}$ are the elements of the element vector $\alpha$ and $|i\rangle$ are computational basis states. The number of amplitude to be encoded is $N\times M$.

• $N$: N features for each $m \in 1,...,M$.
• $M$: M examples.

As a system of $n$ qubit provides $2^{n}$ amplitude, it requires $n \geq log_{2}(NM)$ qubits.

That is

$$\text{number of qubits} = n = \lceil log_{2}(NM) \rceil.$$

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More embedding methodologies from PENNYLANE - Embedding templates

• Angle Embedding
• Displacement Embedding
• IQPE Embedding
• QAOA Embedding
• Squeezing Embedding

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Reference

1. PENNYLANE - Quantum Embedding: https://pennylane.ai/qml/glossary/quantum_embedding