Controlled NOT gate
Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.
[1][2] The gate is sometimes named after Richard Feynman who developed an early notation for quantum gate diagrams in 1986.
The CNOT gate is also used in classical reversible computing.
The CNOT gate operates on a quantum register consisting of 2 qubits.
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.
More generally, the inputs are allowed to be a linear superposition of
The CNOT gate transforms the quantum state:
The action of the CNOT gate can be represented by the matrix (permutation matrix form): The first experimental realization of a CNOT gate was accomplished in 1995.
At the time of the experiment, the reliability of the CNOT-operation was measured to be on the order of 90%.
[6] 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).
The function-controlled NOT gate is an essential element of the Deutsch–Jozsa algorithm.
, the behaviour of the CNOT appears to be like the equivalent classical gate.
However, the simplicity of labelling one qubit the control and the other the target does not reflect the complexity of what happens for most input values of both qubits.
Insight can be gained by expressing the CNOT gate with respect to a Hadamard transformed basis
is the eigenbasis for the spin in the Z-direction, whereas the Hadamard basis
Switching X and Z and qubits 1 and 2, then, recovers the original transformation.
"[8] This expresses a fundamental symmetry of the CNOT gate.
The observation that both qubits are (equally) affected in a CNOT interaction is of importance when considering information flow in entangled quantum systems.
Working through each of the Hadamard basis states, the results on the right column show that the first qubit flips between
: 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 therefore its own inverse.
The tensor product of two Hadamard transforms operating (independently) on two qubits is labelled H2.
A common application of the CNOT gate is to maximally entangle two qubits into the
Bell state; this forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms.
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 output state after applying the CNOT gate is
[10][11] Trapped ion quantum computers: 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.
[12] 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.
