Editing Quantum computing (section)

=== Unitary operators<span class="anchor" id="gate-application"></span> ===
{{See also|Unitarity (physics)}}

The state of this one-qubit [[quantum memory]] can be manipulated by applying [[quantum logic gate]]s, analogous to how classical memory can be manipulated with [[Logic gate|classical logic gates]]. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a [[Matrix (mathematics)|matrix]]
<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>
Mathematically, the application of such a logic gate to a quantum state vector is modelled with [[matrix multiplication]]. Thus
: <math>X|0\rangle = |1\rangle</math> and <math>X|1\rangle = |0\rangle</math>.
The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are
<math display="block">
 |00\rangle := \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix};\quad
 |01\rangle := \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix};\quad
 |10\rangle := \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix};\quad
 |11\rangle := \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.
</math>
The [[Controlled NOT gate|controlled NOT (CNOT)]] gate can then be represented using the following matrix:
<math display="block">
 \operatorname{CNOT} :=
 \begin{pmatrix}
  1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 0 & 1 \\
   0 & 0 & 1 & 0
 \end{pmatrix}.
</math>
As a mathematical consequence of this definition, <math display="inline">\operatorname{CNOT}|00\rangle = |00\rangle</math>, <math display="inline">\operatorname{CNOT}|01\rangle = |01\rangle</math>, <math display="inline">\operatorname{CNOT}|10\rangle = |11\rangle</math>, and <math display="inline">\operatorname{CNOT}|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

In summary, quantum computation can be described as a network of quantum logic gates and measurements. However, any [[deferred measurement principle|measurement can be deferred]] to the end of quantum computation, though this deferment may come at a computational cost, so most [[quantum circuit]]s depict a network consisting only of quantum logic gates and no measurements.