represented by "kets", from a notation known as bra-ket. The vector representation of a single qubit is Here, and are the complex probability amplitudes of the qubit. These values determine the probability of measuring a 0 or a 1, when measuring the state of the qubit. See measurement below for details. The value zero is represented by the ket and the value one is represented by the ket The tensor product (or Kronecker product) is used to combine quantum states. The combined state for a qubit register is the tensor product of the constituent qubits. The tensor product is denoted by the symbol The vector representation of two qubits is: The action of the gate on a specific quantum state is found by multiplying the vector which represents the state by the matrix representing the gate. The result is a new quantum state Notable examples There exists an uncountably infinite number of gates. Some of them have been named by various authors, and below follow some of those most often used in the literature. Identity gate The identity gate is the identity matrix, usually written as I, and is defined for a single qubit as where I is basis independent and does not modify the quantum state. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits. Pauli gates (X,Y,Z) The Pauli gates are the three Pauli matrices and act on a single qubit. The Pauli X, Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by radians. The Pauli-X gate is the quantum equivalent of the NOT gate for classical computers with respect to the standard basis which distinguishes the z axis on the Bloch sphere. It is sometimes called a bit-flip as it maps to and to . Similarly, the Pauli-Y maps to and to . Pauli Z leaves the basis state unchanged and maps to Due to this nature, Pauli Z is sometimes called phase-flip. These matrices are usually represented as The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the identity matrix. The Pauli matrices also anti-commute, for example The matrix exponential of a Pauli matrix is a rotation operator, often written as Controlled gates Controlled gates act on 2 or more qubits, where one or more qubits act as a