Quantum Computing

Broadly stated, the term quantum computation comprieses of the following three elements:

• A register or a set of registers,
• A unitary matrix $U$, as an abstract representation of the quantum algorithm,
• Measurements to extract the information of interest.

As a mathematial abstraction, a quantum computation is the set $\{\mathcal{H},U,\{Mm\}\}$, where $H = \mathbb{C}^{2^{n}}$ is the Hilbert space of an $n-$qubit register, $U \in U\left(2^{n}\right)$ represents the quantum algorithm and $\{M_{m}\}$ is the set of measurement operators. The hardware circuitry along with equipment to control and manipulate the qubits is called a quantum computer.

Single qubit gates

These are the simplest set of gates which take one qubit as an input, act upon (change the state) and produce a qubit at the output.

A generic 1-qubit quantum gate corresponds to a $2 \times 2$ unitary matrix, which has the following form:

\[U = e^{\imath \theta} \begin{pmatrix} a & -b^{*} \\\ b & a^{*} \end{pmatrix} \]

where $a,b \in \mathbb{C}$ such that $\lvert a \rvert^{2}+ \lvert b \rvert^{2} = 1$, and $\alpha \in \mathbb{R}$ results in arbitrary rotation. This matrix is essentially a Givens rotation matrix Wikipedia.

Examples

$X$ gate.

• This gate flips the state of the qubit. In other words, it changes the state of the qubit from $a\vert 0 \rangle + b\vert 1 \rangle$ to $a\vert 1 \rangle+ b\vert 0 \rangle$.
• The matrix representation for$X$ gate is, $\sigma_{x}=\begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix}$. Multiplying the vector representing the qubit to the matrix is equivalent to the gate operation. i.e., $\begin{pmatrix} b \\\ a \end{pmatrix} =\begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix} \begin{pmatrix} a \\\ b \end{pmatrix}$

$Y$ gate.

• This gate perform two flips (a bit flip and a phase flip) on the state of the qubit. In other words, it changes the state of the qubit from $a\vert 0 \rangle + b\vert 1 \rangle$, to $b\vert 0 \rangle - a\vert 1 \rangle$.
• The matrix representation for$Y$ gate is, $\sigma_{y}=\begin{pmatrix} 0 & -1 \\\ 1 & 0 \end{pmatrix}$. Multiplying the vector representing the qubit to the matrix is equivalent to the gate operation.

$Z$ gate.

• This gate performs a sign flip on the state of the qubit. In other words, it changes the state of the qubit from $a\vert 0 \rangle + b\vert 1 \rangle$, to $a\vert 0 \rangle - \vert 1 \rangle$.
• The matrix representation for $X$ gate is, $\sigma_{z}=\begin{pmatrix} 1 & 0 \\\ 0 & -1 \end{pmatrix}$. Multiplying the vector representing the qubit to the matrix is equivalent to the gate operation.

The matrix representations $\sigma_{x}, \;\sigma_y\;and\; \sigma_z$ are known as Pauli's matrices.

Hadamard gate: $H$ gate

• This gate works as follows: $\vert 0 \rangle$ changes to $\frac{1}{\sqrt2}(\vert 0 \rangle +\; \vert 1 \rangle)$, and the $\vert 1 \rangle$ changes to $\frac{1}{\sqrt2}(\vert 0 \rangle - \vert 1 \rangle)$.
• For an example, $a\vert 0 \rangle + b\vert 1 \rangle,$ changes to, $\frac{a}{\sqrt2}(\vert 0 \rangle + \vert 1 \rangle) + \frac{b}{\sqrt2}(\vert 0 \rangle - \vert 1 \rangle)$.
• It can be simplified to give, $\frac{a+b}{\sqrt2}\vert 0 \rangle + \frac{a-b}{\sqrt2} \vert 1 \rangle$.

Mathematically, the following transformation captures the essence of $H$ gate.

\[\begin{pmatrix} \frac{a+b}{\sqrt2} \\\\ \frac{a-b}{\sqrt2} \end{pmatrix} = \frac{1}{\sqrt2} \begin{pmatrix} 1 & 1 \\\ 1 & -1 \end{pmatrix} \begin{pmatrix} a \\\ b \end{pmatrix}\]

Decomposition of Quantum Gates

Two level decomposition

A 2-level unitary correspond to unitary operation that non-trivially perform on only 2 of the states. Any controlled 1-qubit gate can be abstracted into a 2-level unitary matrix, e.g. for a single qubit gate $U = \begin{pmatrix} a & b \\\\ c & d \end{pmatrix}$.

Universal Decomposition

Any arbitrary unitary gate acting on n-qubit can be implemented as a cascade of single-qubit and controlled-NOT (CNOT) gates.

\[\textbf{U}|\psi \rangle = \begin{pmatrix} u_{11} & u_{12} & \ldots & u_{1,N} \\ u_{11} & u_{12} & \ldots & u_{1,N} \end{pmatrix}\]

CNOT gates

CNOT stands for Controlled-NOT, which is one of the key quantum logic gate. It is a two qubit gate. The gate flips the second qubit (called the target qubit) when the first gate (the control gate) is $\lvert 1 \rangle$, while the target qubit remain unchanged when the control gate is in state $\lvert 0 \rangle$.

\[\begin{aligned} U_{\text{CNOT}} &= \lvert 0 \rangle \otimes \langle 0 \rvert \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \lvert 1 \rangle \otimes \langle 1 \rvert \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\\ &= \lvert 0 \rangle \langle 0 \rvert \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \lvert 1 \rangle \langle 1 \rvert \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\\ &= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix} \end{aligned}\]

In terms of Gray matrix, we can also simply express this as,

\[\begin{aligned} \begin{pmatrix} \lvert \acute{\psi}_{1} \rangle \\ \lvert \acute{\psi}_{2} \rangle \end{pmatrix} &= G_{2} \begin{pmatrix} \lvert \psi_{1} \rangle \\ \lvert \psi_{2} \rangle \end{pmatrix} \\ &= \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \lvert \psi_{1} \rangle \\ \lvert \psi_{2} \rangle \end{pmatrix} \end{aligned}\]

The simplest CNOT gate is the single qubit controlled CNOT discussed above, which can be explicitly denoted as $C^{1}\text{NOT}$. Generalization of this to multi quibit controlled CNOT, denoted by $C^{n-1}\text{NOT}$.

In quantum circuit design, applying a rotation for which the binary representations of i − 1 and j − 1 differ in a single bit can be accomplished by a single fully-controlled one-qubit rotation (a particular Givens rotation) and hence costs a small number of gates. All other rotations require a permutation of data before the rotation is applied and thus should be avoided.

Generic U decomposion

Examples: 3 cubit generic quantum gate

julia> A1,A2,A3,A4=sequenceΓ(3);
julia> A4
3×28 adjoint(::Matrix{Int64}) with eltype Int64:
 6  8  8  7  7  7  3  3  3  3  4  4  4  4  4  2  2  2  2  2  2  1  1  1  1  1  1  1
 5  6  5  8  6  5  7  8  6  5  3  7  8  6  5  4  3  7  8  6  5  2  4  3  7  8  6  5
 6  8  6  7  8  6  3  7  8  6  4  3  7  8  6  2  4  3  7  8  6  1  2  4  3  7  8  6