together. For example, putting the Pauli X gate after the Pauli Y gate, both of which act on a single qubit, can be described as a single combined gate C: The product symbol () is often omitted. Exponents of quantum gates All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices. Positive integer exponents are equivalent to sequences of serially wired gates (e.g. and the real exponents is a generalization of the series circuit. For example, and are both valid quantum gates. for any unitary matrix . The identity matrix () behaves like a NOP and can be represented as bare wire in quantum circuits, or not shown at all. All gates are unitary matrices, so that and where is the conjugate transpose. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts: For example, some negative exponents of the phase shift gates are and Note that for a Hermitian matrix and because of unitarity, so for all Hermitian gates. They are involutory. Examples of Hermitian gates are the Pauli gates, Hadamard, CNOT, SWAP and Toffoli. Hermitian unitary matrices has the property where Parallel gates The tensor product (or Kronecker product) of two quantum gates is the gate that is equal to the two gates in parallel. If we, as in the picture, combine the Pauli-Y gate with the Pauli-X gate in parallel, then this can be written as: Both the Pauli-X and the Pauli-Y gate act on a single qubit. The resulting gate act on two qubits. Sometimes the tensor product symbol is omitted, and indexes are used for the operators instead. Hadamard transform The gate is the Hadamard gate applied in parallel on 2 qubits. It can be written as: This "two-qubit parallel Hadamard gate" will when applied to, for example, the two-qubit zero-vector create a quantum state that have equal probability of being observed in any of its four possible outcomes; and We can write this operation as: Here the amplitude for each measurable state is . The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases. See measurement for details. performs the Hadamard transform on two qubits. Similarly the gate performs a Hadamard transform