Controlled NOT gate

From Wikipedia, the free encyclopedia

Jump to: navigation, search

The classical analog of the CNOT gate is the XOR gate.

How CNOT gate can be used (with Hadamard gates) in a computation.

It is that can represent CNOT gate.

Answer on output depending of input and CNOT function.

first qubit flips only if second qubit is 1.

The Controlled NOT gate (also C-NOT or CNOT) is a quantum gate that is an essential component in the construction of a quantum computer. It can be used to disentangle EPR states. Specifically, any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.

[edit] Operation

The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is 1.

Before		After
Control	Target	Control	Target
0	0	0	0
0	1	0	1
1	0	1	1
1	1	1	0

The resulting value of the second qubit corresponds to the result of a classical XOR gate.

The CNOT gate can be represented by the matrix:

$\operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}.$

The first experimental realization of a CNOT gate was accomplished in 1995. Here, a single Beryllium ion in a trap was used. The two qubits were encoded into an optical state and into the vibrational state of the ion within the trap. At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.

In addition to a regular Controlled NOT gate, one could construct a Function-Controlled NOT gate, which accepts an arbitrary number n+1 of qubits as input, where n+1 is greater than or equal to 2 (a quantum register). This gate flips the last qubit of the register if and only if a built-in function, with the first n qubits as input, returns a 1. The Function-Controlled NOT gate is an essential element of the Deutsch-Jozsa algorithm.

[edit] Proof of operation

Let $\left\{ |0\rangle =\! \begin{bmatrix} 1 \\ 0 \end{bmatrix}, |1\rangle =\! \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}$ be the orthonormal basis (using Bra-ket notation).

Let $| \psi \rangle = a | 0 \rangle + b | 1 \rangle =\! \begin{bmatrix} a \\ b \end{bmatrix}$ .

Let $| \phi \rangle = b | 0 \rangle + a | 1 \rangle =\! \begin{bmatrix} b \\ a \end{bmatrix}$ be the flip qubit of $| \psi \rangle$ .

[edit] When control qubit is 0

First, we shall prove that $\operatorname{CNOT}\ |0,\psi\rangle = |0,\psi\rangle$ :

It's not difficult to verify that $|0,\psi\rangle = a|0\rangle |0\rangle + b|0\rangle |1\rangle = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix}$ .

Then $\operatorname{CNOT}\ |0,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \\ 0 \\ 0 \end{bmatrix} = |0,\psi\rangle$ .

Therefore CNOT doesn't change the $|\psi\rangle$ qubit if the first qubit is 0.

[edit] When control qubit is 1

Now, we shall prove that $\operatorname{CNOT}\ |1,\psi\rangle = |1,\phi\rangle$ , which means that the CNOT gate flips the $|\psi\rangle$ qubit.

Similarly to the first demonstration, we have $|1,\psi\rangle = \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix}$ .

Then $\operatorname{CNOT}\ |1,\psi\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ a \\ b \end{bmatrix} = a \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$

As we can see that $|1,1\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$ and $|1,0\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ , using these on the equation above gives

$\begin{align} \operatorname{CNOT}\ |1,\psi\rangle & = a |1,1\rangle + b |1,0\rangle \\ & = |1\rangle \left( a |1\rangle + b |0\rangle \right) \\ & = |1,\phi\rangle \\ \end{align}$

Therefore the CNOT gate flips the $|\psi\rangle$ qubit into $|\phi\rangle$ if the control qubit is set to 1. A simple way to observe this is to multiply the CNOT matrix by a column vector, noticing that the operation on the first bit is identity, and a NOT gate on the second bit.

[edit] References

Nielsen, Michael A. & Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 0-521-63235-8.

Monroe, C. & Meekhof, D. & King, B. & Itano, W. & Wineland, D. (1995). "Demonstration of a Fundamental Quantum Logic Gate". Physical Review Letters 75 (75): 4714–4717. doi:10.1103/PhysRevLett.75.4714. [1]

This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.

Controlled NOT gate

From Wikipedia, the free encyclopedia

Contents

[edit] Operation

[edit] Proof of operation

[edit] When control qubit is 0

[edit] When control qubit is 1

[edit] References

Views

Personal tools

Navigation

Search

Interaction

Toolbox

Languages