Qubit System Representations
Single Qubit Recap
Previously, we saw that a single qubit can be represented as a pair of amplitudes. $$A=\begin{bmatrix} v_0 \newline v_1 \end{bmatrix}$$ $$\text{OR}$$ $$A=v_0|0\rangle + v_1|1\rangle$$ Where $v_0$ represents $A$’s amplitude on $0$ and $v_1$ represents $A$’s amplitude on $1$.
Extending to Multi-Qubit States
In $q$-qubit systems, the representing vectors are $2^q$-dimensional where each component’s magnitude represents the amplitude on a specific value. The mapping of axis to value follows from the tensor product order.
Suppose $AB$ had the following state. $$AB=v_{00}|00\rangle + v_{01}|01\rangle + v_{10}|10\rangle + v_{11}|11\rangle$$ The corresponding vector would be the following. $$AB=\begin{bmatrix} v_{00} \newline v_{01} \newline v_{10} \newline v_{11} \end{bmatrix}$$
Classical Quantum Gates
With the exception of qubit initialization, quantum computing operations are all bijective modifications on their input qubits with no outputs.
The reason for this strict rule is for computational reversibility which is essential for taking out the garbage (to be explained in a later post).
This is why qubit value assignment isn’t possible. It is not possible to reverse the assigned value $o$ since every possible input $i$ maps to $o$.
Definitions
Initialize
$\texttt{INIT}(A)$
Creates a new qubit $A$ with full amplitude on value $0$.
$$\begin{bmatrix} 1 \newline 0 \end{bmatrix}$$
Not
$\texttt{NOT}(A)$
Negates $A$. In other words, adds $1$ to qubit $A$ ($\bmod\thickspace2$).
$$\begin{bmatrix} 0 & 1 \newline 1 & 0 \end{bmatrix}$$
Controlled Not
$\texttt{CNOT}(AB)$
If $A=1$ then negates $B$. In other words, adds $A$ to qubit $B$ ($\bmod\thickspace2$).
$$\begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 0 & 1 \newline 0 & 0 & 1 & 0\end{bmatrix}$$
$$ \begin{array}{c|c} AB & \texttt{CNOT}(AB) \newline 00 & 00 \newline 01 & 01 \newline \boxed{10} & 11 \newline \boxed{11} & 10 \end{array} $$
Controlled Controlled Not
$\texttt{CCNOT}(ABC)$
If $A=1$ and $B=1$ then negates $C$. In other words, adds $A\And B$ to qubit $C$ ($\bmod\thickspace 2$).
$$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \newline 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \newline 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \newline 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \newline 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \newline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \newline 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$
$$ \begin{array}{c|c} ABC & \texttt{CCNOT}(ABC) \newline 000 & 000 \newline 001 & 001 \newline 010 & 010 \newline 011 & 011 \newline 100 & 100 \newline 101 & 101 \newline \boxed{110} & 111 \newline \boxed{111} & 110 \end{array} $$
Syntactic Sugar
Sometimes we will define/use a subroutine $F(X_1X_2…X_n)$ that outputs a value which we later execute in some pseudocode fashion (i.e. $\text{If }F(AB)\text{ Then}…$). But technically returning a value isn’t allowed.
This is just syntactic sugar for creating a temporary qubit $T$ and applying $F$’s logic to modify $T$ as the output. $$\texttt{INIT}(T)\newline …\newline \text{// Apply $F$ logic onto $T$}\newline … \newline\text{If }T\text{ Then}…$$
Also note that since quantum computation must be reversible, branching must be done intelligently. For example, $\text{If}$ branches are syntactic sugar for controlled operations ($\texttt{CTRUEBRANCH}(T…)$).
Example for negation of adding $\texttt{CCNOT}$ to input. $$ \begin{array}{c|c} \text{compiled} & \text{syntactic sugar} \newline\newline \texttt{INIT}(A) & \texttt{INIT}(A) \newline \texttt{INIT}(B) & \texttt{INIT}(B) \newline \texttt{INIT}(C) & \texttt{INIT}(C) \newline \texttt{INIT}(D) & \texttt{INIT}(D) \newline \texttt{INIT}(T_1) & \newline \texttt{INIT}(T_2) & \newline & \texttt{def F}(X_1X_2)\lbrace \newline & \texttt{INIT}(T) \newline & \texttt{NOT}(X_1) \newline & \texttt{NOT}(X_2) \newline & \texttt{CCNOT}(X_1X_2T) \newline & \texttt{NOT}(X_1) \newline & \texttt{NOT}(X_2) \newline & \texttt{Output }T \newline & \rbrace\quad\quad\quad\quad\quad\quad \newline & \newline \texttt{NOT}(A) & \texttt{If F}(AB)\texttt{ Then}\lbrace \newline \texttt{NOT}(B) & \newline \texttt{CCNOT}(ABT_1) & \newline \texttt{NOT}(A) & \newline \texttt{NOT}(B) & \newline \texttt{CNOT}(T_1A)& \texttt{NOT}(A) \newline \texttt{CNOT}(T_1B)& \texttt{NOT}(B) \newline & \rbrace\quad\quad\quad\quad\quad\quad\quad \newline & \newline \texttt{NOT}(C) & \texttt{If F}(CD)\texttt{ Then}\lbrace \newline \texttt{NOT}(D) & \newline \texttt{CCNOT}(CDT_2) & \newline \texttt{NOT}(C) & \newline \texttt{NOT}(D) & \newline \texttt{CCNOT}(T_2CD)& \texttt{CNOT}(CD) \newline & \rbrace\quad\quad\quad\quad\quad\quad\quad \newline \end{array} $$