Controlled NOT gate

The CNOT gate operates on a quantum register consisting of 2 qubits. The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is |1\rangle. If \{|0\rangle,|1\rangle\} are the only allowed input values for both qubits, then the TARGET output of the CNOT gate corresponds to the result of a classical XOR gate. Fixing CONTROL as |1\rangle, the TARGET output of the CNOT gate yields the result of a classical NOT gate. More generally, the inputs are allowed to be a linear superposition of \{|0\rangle,|1\rangle\}. The CNOT gate transforms the quantum state: a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle into: a|00\rangle + b|01\rangle + c|11\rangle + d|10\rangle The action of the CNOT gate can be represented by the matrix (permutation matrix form): : \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. == Behaviour in the Hadamard transformed basis ==

When viewed only in the computational basis \ A quantum circuit that performs a Hadamard transform followed by CNOT then another Hadamard transform, can be described as performing the CNOT gate in the Hadamard basis (i.e. a change of basis): The single-qubit Hadamard transform, H1, is Hermitian and its own inverse. The tensor product of two Hadamard transforms operating (independently) on two qubits is labelled H2. We can therefore write the matrices as: When multiplied out, this yields a matrix that swaps the |01\rangle and |11\rangle terms over, while leaving the |00\rangle and |10\rangle terms alone. This is equivalent to a CNOT gate where qubit 2 is the control qubit and qubit 1 is the target qubit:{{efn|That is, H_2 \cdot \operatorname{CNOT} \cdot H_2 = \operatorname{SWAP} \cdot \operatorname{CNOT} \cdot \operatorname{SWAP} where \operatorname{SWAP} is the SWAP gate.}} \frac{1}{2} \begin{bmatrix}\begin{array}{rrrr} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{array}\end{bmatrix} . \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{bmatrix} . \frac{1}{2} \begin{bmatrix}\begin{array}{rrrr} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{array}\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 \end{bmatrix} ==Constructing a Bell state==

A common application of the CNOT gate is to maximally entangle two qubits into the |\Phi^+\rangle Bell state; this forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms. To construct |\Phi^+\rangle, the inputs A (control) and B (target) to the CNOT gate are : \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_A and |0\rangle_B. After applying CNOT, the resulting Bell state \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) has the property that the individual qubits can be measured using any basis and will always present a 50/50 chance of resolving to each state. In effect, the individual qubits are in an undefined state. The correlation between the two qubits is the complete description of the state of the two qubits; if we both choose the same basis to measure both qubits and compare notes, the measurements will perfectly correlate. When viewed in the computational basis, it appears that qubit A is affecting qubit B. Changing our viewpoint to the Hadamard basis demonstrates that, in a symmetrical way, qubit B is affecting qubit A. The input state can alternately be viewed as : |+\rangle_A and \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)_B. In the Hadamard view, the control and target qubits have conceptually swapped and qubit A is inverted when qubit B is |-\rangle_B. The output state after applying the CNOT gate is \tfrac{1}{\sqrt{2}}(|++\rangle + |--\rangle), which can be shown as follows: : = \frac{1}{\sqrt{2}}(|+\rangle_A|+\rangle_B + |-\rangle_A|-\rangle_B) : = \frac{1}{2\sqrt{2}}((|0\rangle_A + |1\rangle_A)(|0\rangle_B + |1\rangle_B) + (|0\rangle_A - |1\rangle_A)(|0\rangle_B - |1\rangle_B)) : = \frac{1}{2\sqrt{2}}((|00\rangle + |01\rangle + |10\rangle + |11\rangle) + (|00\rangle - |01\rangle - |10\rangle + |11\rangle)) : = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle). ==C-ROT gate==

The C-ROT gate (controlled Rabi rotation) is equivalent to a C-NOT gate except for a \pi /2 rotation of the nuclear spin around the z axis. == Implementations ==

Trapped ion quantum computers: • Cirac–Zoller controlled-NOT gate • Mølmer–Sørensen gate == Regulation ==

In May, 2024, Canada implemented export restrictions on the sale of quantum computers containing more than 34 qubits and error rates below a certain CNOT error threshold, along with restrictions for quantum computers with more qubits and higher error rates. The same restrictions quickly popped up in the UK, France, Spain and the Netherlands. They offered few explanations for this action, but all of them are Wassenaar Arrangement states, and the restrictions seem related to national security concerns potentially including quantum cryptography or protection from competition. ==See also==