| CNOT gate | |
|---|---|
| Name | Controlled-NOT gate |
| Type | Two-qubit entangling gate |
| Matrix | 4×4 |
| Qubits | control, target |
| Introduced | 1990s (widespread in quantum computing implementations) |
CNOT gate
The CNOT gate (controlled-NOT) is a two-qubit quantum logic gate that flips the state of a target qubit conditional on the state of a control qubit. It is a fundamental entangling operation in Quantum computing and Quantum information that underpins universal gate sets, quantum error correction, and many quantum algorithms. Because it creates and manipulates quantum entanglement, CNOT is central to experimental platforms such as IBM Quantum, Google Quantum AI, and ion-trap systems developed at institutions like National Institute of Standards and Technology and University of Innsbruck.
The CNOT gate acts on a pair of qubits labeled control and target. In the computational basis {|00>, |01>, |10>, |11>}, the gate leaves the target unchanged when the control is |0> and applies a Pauli-X operation to the target when the control is |1>. The operation is logically reversible and is the quantum analogue of the classical exclusive or (XOR) operation when applied to basis states. In circuit diagrams the gate is typically drawn with a filled control dot connected to a ⊕ symbol on the target qubit. The CNOT's ability to conditionally correlate qubits makes it essential for preparing entangled states and implementing conditional dynamics in architectures from superconducting qubits to trapped ion platforms.
The matrix representation of CNOT in the computational basis is a 4×4 unitary:
\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix}
This maps basis states as: |00>→|00>, |01>→|01>, |10>→|11>, |11>→|10>. The corresponding truth table parallels the classical controlled-NOT: control 0 leaves target unchanged; control 1 flips target. The gate can also be expressed using tensor products and Pauli operators as: CNOT = |0⟩⟨0|⊗I + |1⟩⟨1|⊗X, linking it to the algebra of Pauli matrices and unitary operator formalism in quantum mechanics.
CNOT has been implemented across multiple hardware platforms. In superconducting qubits, cross-resonance and tunable coupler techniques produce effective CNOT or CNOT-equivalent gates; notable programs include IBM Quantum and Rigetti Computing. In trapped ion systems (e.g., at Max Planck Institute for Quantum Optics and NIST), Molmer–Sørensen and Cirac–Zoller approaches realize entangling interactions convertible to CNOT. Photonic implementations use linear optics and measurement-based schemes derived from Knill–Laflamme–Milburn protocols; companies such as Xanadu and research groups at University of Bristol have demonstrated photonic two-qubit gates. Solid-state spins in nitrogen-vacancy center systems and semiconductor quantum dots have realized controlled operations via exchange interaction or optical control. Many implementations implement a CNOT up to single-qubit rotations, requiring calibration against native two-qubit interactions such as iSWAP or CZ.
CNOT is a building block for quantum circuits and appears in constructions of quantum Fourier transform, Grover's algorithm, and Shor's algorithm. It enables classical reversible logic embedding (e.g., reversible adder circuits) and is used in multi-qubit controlled operations through decomposition techniques. In fault-tolerant quantum error correction codes such as the surface code and Steane code, CNOT gates perform syndrome extraction and entangling operations between data and ancilla qubits. Quantum compilers and platforms (e.g., Qiskit, Cirq) optimize use of CNOT relative to hardware-native gates to minimize error and circuit depth.
A canonical use of CNOT is generation of maximally entangled two-qubit states (Bell states). Starting from |00>, applying a single-qubit Hadamard gate H on the control followed by CNOT yields the state (|00>+|11>)/√2, a Bell state used for protocols like quantum teleportation and superdense coding. CNOT thus serves both as an entanglement generator and as a primitive for entanglement swapping and entanglement purification protocols employed in quantum communication experiments at institutions such as Delft University of Technology and University of Science and Technology of China.
Together with arbitrary single-qubit rotations, CNOT forms a universal set for quantum computation: any multi-qubit unitary can be decomposed into sequences of single-qubit gates and CNOTs. Standard decompositions include expressing controlled-U gates via CNOTs and single-qubit controls, and decomposing multi-controlled Toffoli gates into CNOT and single-qubit gates with ancillae. Research by Barenco et al. and others provided systematic synthesis methods; quantum compiler toolchains in IBM Qiskit and Google Cirq implement these decompositions, often translating to hardware-native entanglers like CZ or iSWAP which are locally equivalent to CNOT via single-qubit rotations.
Two-qubit gates such as CNOT commonly dominate error budgets due to decoherence, control pulse imperfections, crosstalk, and calibration drift. In superconducting devices, frequency crowding and residual ZZ coupling impair CNOT fidelity; in trapped ions, motional-mode heating and laser phase noise limit performance. Error mitigation strategies include randomized compiling, dynamical decoupling, composite pulse shaping, and active calibration; fault-tolerant thresholds drive engineering toward two-qubit fidelities compatible with surface code thresholds. Benchmarking protocols such as randomized benchmarking and gate set tomography (GST) quantify CNOT performance in experimental platforms developed at labs like Sandia National Laboratories and Lawrence Berkeley National Laboratory.
Category:Quantum gates Category:Quantum computing