Using Many Quantum Bits

To use more than two, say five, qubits, we can just start with the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangl... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ and apply gates to any one or two of these qubits. For example, ${\mathbf{cnot}}^{({\mathsf {DB}})}$ applies the $\mathbf{cnot}$ operation from qubit $D$ to qubit $B$. Note that the order of $\mathsf {D}$ and $\mathsf {B}$ in the label for the $\mathbf{cnot}$ operation matters. In the bra-ket notation, we simply multiply the state with the bra-ket form of ${\mathbf{cnot}}^{({\mathsf {DB}})}$ from the left. One can express everything in terms of matrices and vectors, but now the vectors have length $2^5=32$, and the Kronecker product expression for ${\mathbf{cnot}}^{({\mathsf {DB}})}$ requires some reordering to enable inserting the operation so as to act on the intended qubits. Nevertheless, to analyze the properties of all reversible (that is, unitary) operations on these qubits, it is helpful to think of the matrices, because a lot of useful properties about unitary matrices are known. One important result from this analysis is that every matrix that represents a reversible operation on quantum states can be expressed as a product of the one- and two-qubit gates introduced so far. We say that this set of gates is ``universal''.

For general purpose computation, it is necessary to have access to arbitrarily many qubits. Instead of assuming that there are infinitely many from the start, it is convenient to have an operation to add a new qubit, namely, $\mathbf{add}$. To add a new qubit labeled $\mathsf {X}$ in the state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {X}}}}$, apply ${\mathbf{add}}^{({\mathsf {X}})}$ to the current state. This operation can only be used if there is not already a qubit labeled $\mathsf {X}$. In the bra-ket notation, we implement the ${\mathbf{add}}^{({\mathsf {X}})}$ operation by multiplying the ket expression for the current state by $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {X}}}}$.