Chapter 3 - Linear Algebra

Last updated Jun 24, 2022

• Type note
• Status good
• On quantum computing
• On linear algebra

Linear Algebra is to Quantum Computing as Boolean Algebra is to Classical Computing. Although we have to learn a new tool, it makes calculations much easier.

# Quantum States

• We write $\ket 0$ and $\ket 1$ as column vectors: $$\ket 0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \ket 1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$
• It is easier to write superpositions this way. A generic qubit would be: $$\begin{aligned}
  \ket \psi &= \alpha \ket 0 + \beta \ket 1 \newline
  &= \alpha \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 0 \\ 1 \end{pmatrix} \newline
  &= \begin{pmatrix} \alpha \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ \beta \end{pmatrix} \newline
  &= \begin{pmatrix} \alpha \\ \beta \end{pmatrix}.
  \end{aligned}$$

• The transpose of a Column vector is obtained by rewriting it as a row vector, and it is denoted by $ᵀ$. $$\begin{pmatrix} \alpha \\ \beta \end{pmatrix}^T = \begin{pmatrix} \alpha & \beta \end{pmatrix}.$$
• In quantum computing, we typically use the conjugate transpose, which is obtained by taking the Complex Conjugate of each component of the transpose. It is denoted by $†$. $$\begin{pmatrix} \alpha \\ \beta \end{pmatrix}^\dagger = \begin{pmatrix} \alpha^* & \beta^* \end{pmatrix}.$$
• A bra $\bra \psi$ is the conjugate transpose of a ket, and conversely, a ket is the conjugate transpose of a bra. $$\bra \psi = \ket{\psi}^\dagger, \qquad \ket \psi = \bra \psi^\dagger.$$

• The inner product of $\ket \psi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ and $\ket \phi = \begin{pmatrix} \gamma \\ \delta \end{pmatrix}$ is defined as $\braket{\psi | \phi}$, which is called a bra-ket or bracket: $$\displaylines{
  \braket{\phi | \psi} = \begin{pmatrix} \alpha^* & \beta^* \end{pmatrix}\begin{pmatrix} \gamma \\ \delta \end{pmatrix} \newline
  \braket{\phi | \psi} = \alpha^\gamma + \beta^\delta.
  }$$ which is a scalar value. That’s why an inner product is also known as scalar product.
• The inner product of $\ket \psi$ and $\ket phi$ is just the Complex Conjugate of the inner product of $\ket \psi$ and $\ket \phi$: $$\braket{\psi | \phi} = \braket{\phi | \psi}^*.$$

• The inner product of $\ket \psi$ with itself, denoted $\braket{\psi | \psi}$, is just the total probability, i.e. $|\alpha|^2 + |\beta|^2$, and if it is $1$, then the state $\ket \psi$ is normalized.
• Any two states on opposite sides of the Bloch sphere have zero inner product, and the states with zero inner product are called orthogonal states.
• Orthonormal states are those states that are both normalized and orthogonal to each other. e.g. ${\ket 0, \ket 1}$ are orthonormal, so are ${\ket +, \ket -}$, and ${\ket i, \ket {-i}}$.

• For an orthonormal basis ${\ket a, \ket b}$, the state of a qubit can be written as $$\ket \psi = \alpha \ket a + \beta \ket b,$$ where $\alpha = \braket{a | \psi}$ and $\beta = \braket{b | \psi}$. Here $\braket{a | \psi}$ is the amplitude of $\ket \psi$ in $\ket a$, i.e. the amount of $\ket \psi$ that is in $\ket a$, or the amount of overlap between $\ket \psi$ and $\ket a$, which in mathematical terms is called the projection of $\ket \psi$ onto $\ket a$.
• Inner products can be used to find the amplitudes and orthonormality of a state in a certain basis, which provides a convenient way to change basis states, and perform calculations using different computer algebra systems.

• Quantum Gate are matrices that keep the total probability equal to $1$.

• The previously introduced common One-Qubit Quantum Gates can be represented as matrices:

• Using linear algebra, we can compute the effect of a sequence of quantum gates.
• For example: $HSTH\ket 0$ can be computed by simplifying/multiplying the matrices together: $$
  HSTH\ket 0 =
  \frac{1}{\sqrt 2} \begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix}
  \begin{pmatrix}1 & 0 \\ 0 & i \end{pmatrix}
  \begin{pmatrix}1 & 0 \\ 0 & e^{i\pi / 4} \end{pmatrix}
  \frac{1}{\sqrt 2} \begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix}
  \begin{pmatrix}1 \\ 0 \end{pmatrix}.
  $$

• We can prove different circuit identities using Linear Algebra, e.g. $HXH = Z$ can be proven by multiplying the matrices together, either manually or, in most cases, using a computer program.

• Quantum Gate are unitary matrices, and unitary matrices are quantum gates.

• A quantum gate is always reversible, and its inverse is $U^†$.

• As opposed to inner products $\braket{\psi | \phi}$ which are scalar, an outer product $\ket \psi \bra \phi$ is always a Matrix: $$\ket{\psi}\bra{\phi} = \begin{pmatrix}\alpha \\ \beta\end{pmatrix}\begin{pmatrix}\gamma^* & \delta^* \end{pmatrix}.$$
• We can add outer products together to construct various Quantum Gate.
• The outer product of $\ket \phi$ and $\ket \psi$ is just the conjugate transpose of the outer product of $\ket \psi$ and $\ket \phi$: $$\ket \phi \bra \psi = \ket \psi \bra \phi ^ \dagger.$$

• A complete orthonormal basis ${\ket a, \ket b}$ satisfies the completeness relation $$\ket a \bra a + \ket b \bra b = I.$$

• The mathematical language of Quantum Computing is Linear Algebra.
• Quantum states are represented by column vectors called kets, and the conjugate transpose of a ket is a bra.
• Multiplying a bra and a ket is an inner product that yields the projection or amplitudes of the states onto each other.
• States with zero inner product are orthogonal, and a state whose inner product with itself is $1$ is normalized.
• All quantum gates are unitary matrices.
• A quantum gate is always reversible.
• Multiplying a ket and a bra is an outer product, which is a Matrix.
• A complete orthonormal basis satisfies the completeness relation, meaning the sum of the outer products of each basis vector with itself equals the Identity matrix.