allow for destructive wave interference. When a qubit is measured in the standard basis, the result is a classical bit. The Born rule describes the norm-squared correspondence between amplitudes and probabilitieswhen measuring a qubit , the state collapses to with probability 2, or to with probability 2. Any valid qubit state has coefficients α and β such that . As an example, measuring the qubit would produce either or with equal probability. Each additional qubit doubles the dimension of the state space. As an example, the vector represents a two-qubit state, a tensor product of the qubit with the qubit . This vector inhabits a four-dimensional vector space spanned by the basis vectors , , , and . The Bell state is impossible to decompose into the tensor product of two individual qubitsthe two qubits are entangled because their probability amplitudes are correlated. In general, the vector space for an n-qubit system is 2n-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2100 classical values. Unitary operators The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus and . The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are The CNOT gate can then be represented using the following matrix: As a mathematical consequence of this definition, , , , and . In other words, the CNOT applies a NOT gate ( from before) to the second qubit if and only if the first qubit is in the state . If the first qubit is , nothing is done to either qubit. In summary, a quantum computation can be