Advanced Linear Algebra for Quantum States

Alright, let's solidify our understanding of the mathematical stage where quantum computations play out, especially for Quantum Machine Learning (QML). While you likely have a background in linear algebra, we need to be precise about the structures used to describe the multi-qubit systems central to QML algorithms. This involves moving beyond single-qubit descriptions into the realm of composite systems.

Hilbert Spaces: The Foundation

Quantum states live in complex vector spaces equipped with an inner product, known as Hilbert spaces. For systems composed of $n$ qubits, the relevant Hilbert space is typically the $2^n$-dimensional complex space, denoted as $\mathcal{H}_n = (\mathbb{C}^2)^{\otimes n} \cong \mathbb{C}^{2^n}$.

A single qubit's state space is $\mathcal{H}_1 = \mathbb{C}^2$, spanned by the computational basis vectors $|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $|1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Any pure state of a single qubit can be written as a complex linear combination (superposition) of these basis states: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $\alpha, \beta \in \mathbb{C}$ are complex amplitudes satisfying the normalization condition $|\alpha|^2 + |\beta|^2 = 1$. This normalization ensures that the probabilities of measuring the qubit in the $|0\rangle$ or $|1\rangle$ state sum to one.

The inner product (or scalar product) between two states $|\phi\rangle = \gamma|0\rangle + \delta|1\rangle$ and $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is defined as: $\langle \phi | \psi \rangle = (\gamma^* \quad \delta^*) \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \gamma^*\alpha + \delta^*\beta$ Here, $\langle \phi |$ represents the conjugate transpose (Hermitian conjugate) of the column vector $|\phi\rangle$, often called a "bra" vector corresponding to the "ket" vector $|\psi\rangle$. The inner product provides a way to calculate the probability amplitude of projecting one state onto another. The probability of measuring state $|\psi\rangle$ in state $|\phi\rangle$ is given by $|\langle \phi | \psi \rangle|^2$.

Tensor Products: Combining Systems

When dealing with multiple qubits, as is standard in QML circuits, we need a way to describe the combined state space. This is achieved using the tensor product ($\otimes$). If we have two qubits, one in state $|\psi_A\rangle \in \mathcal{H}_A = \mathbb{C}^2$ and the other in state $|\psi_B\rangle \in \mathcal{H}_B = \mathbb{C}^2$, the state of the composite system lives in the tensor product space $\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B \cong \mathbb{C}^4$.

The computational basis for this two-qubit system is formed by taking the tensor products of the single-qubit basis vectors:

• $|0\rangle \otimes |0\rangle = |00\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ 0 \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$
• $|0\rangle \otimes |1\rangle = |01\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix} \\ 0 \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$
• $|1\rangle \otimes |0\rangle = |10\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ 1 \cdot \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}$
• $|1\rangle \otimes |1\rangle = |11\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix} \\ 1 \cdot \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$

These four basis vectors, $|00\rangle, |01\rangle, |10\rangle, |11\rangle$, span the $\mathbb{C}^4$ Hilbert space. A general two-qubit state is a superposition: $|\Psi\rangle = c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle$ with normalization $\sum_{ij} |c_{ij}|^2 = 1$.

The tensor product combines the Hilbert spaces of individual qubits ($\mathbb{C}^2$) into a larger Hilbert space ($\mathbb{C}^4$ for two qubits) that describes the composite system.

This generalizes directly: for $n$ qubits, the Hilbert space is $\mathcal{H}_n = \mathbb{C}^{2^n}$, spanned by $2^n$ computational basis states like $|x_1 x_2 \dots x_n\rangle$, where $x_i \in \{0, 1\}$. This exponential growth in the dimension of the state space is both a source of potential computational power for quantum algorithms and a significant challenge for classical simulation.

Separable vs. Entangled States

A multi-qubit state $|\Psi\rangle \in \mathcal{H}_{AB}$ is called separable if it can be written as a simple tensor product of states from the individual subsystems, i.e., $|\Psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle$. For example: $|+\rangle \otimes |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) = \frac{1}{2}(|00\rangle - |01\rangle + |10\rangle - |11\rangle)$ This state is fully described by specifying the states of qubit A and qubit B independently.

However, not all states in $\mathcal{H}_{AB}$ can be written this way. States that cannot be factorized into a tensor product of individual qubit states are called entangled. The canonical examples are the Bell states, such as: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ There are no single-qubit states $|\psi_A\rangle = \alpha|0\rangle + \beta|1\rangle$ and $|\psi_B\rangle = \gamma|0\rangle + \delta|1\rangle$ such that $|\psi_A\rangle \otimes |\psi_B\rangle = |\Phi^+\rangle$. You can see this by writing out the product: $(\alpha|0\rangle + \beta|1\rangle) \otimes (\gamma|0\rangle + \delta|1\rangle) = \alpha\gamma|00\rangle + \alpha\delta|01\rangle + \beta\gamma|10\rangle + \beta\delta|11\rangle$ For this to equal $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, we would need $\alpha\delta = 0$ and $\beta\gamma = 0$, while $\alpha\gamma = 1/\sqrt{2}$ and $\beta\delta = 1/\sqrt{2}$. These conditions cannot be satisfied simultaneously. Entangled states exhibit correlations between the qubits that cannot be explained classically and are considered a fundamental resource in quantum computation and information, including many QML algorithms.

Operators in Tensor Product Spaces

Linear operators acting on these composite Hilbert spaces represent quantum gates or measurements applied to multiple qubits. If $U_A$ acts on $\mathcal{H}_A$ and $U_B$ acts on $\mathcal{H}_B$, their combined action on a separable state is: $(U_A \otimes U_B) (|\psi_A\rangle \otimes |\psi_B\rangle) = (U_A |\psi_A\rangle) \otimes (U_B |\psi_B\rangle)$ If only one qubit is acted upon, say by $U_A$, while the other is unaffected, the operator is $U_A \otimes I_B$, where $I_B$ is the identity operator on $\mathcal{H}_B$.

For example, the CNOT gate, a standard two-qubit gate, acts on the computational basis states as:

• CNOT $|00\rangle = |00\rangle$
• CNOT $|01\rangle = |01\rangle$
• CNOT $|10\rangle = |11\rangle$ (flips target if control is 1)
• CNOT $|11\rangle = |10\rangle$ (flips target if control is 1)

Its matrix representation in the $|00\rangle, |01\rangle, |10\rangle, |11\rangle$ basis is: $\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$ This operator cannot be written as a simple tensor product $U_A \otimes U_B$, signifying its ability to create entanglement.

Understanding how to represent states and manipulate them using operators within these high-dimensional tensor product spaces is essential. QML techniques heavily rely on this formalism. Quantum feature maps, for instance, encode classical data points $x$ into quantum states $|\phi(x)\rangle$ residing in these potentially vast Hilbert spaces. Quantum kernel methods compute inner products $K(x_i, x_j) = |\langle \phi(x_i) | \phi(x_j) \rangle|^2$ within this space. Variational algorithms use parameterized quantum circuits, which are sequences of operators (gates) applied to these multi-qubit states, to minimize a cost function. A firm grasp of the underlying linear algebraic structure is therefore indispensable as we move forward.