| ) I ( 0 ⟩ | With respect to the basis It is represented by the Pauli Z matrix: The square of a Pauli matrix is the identity matrix. Below are two examples, one where the SWAP gate is operates on the \(|11\rangle\) state and one where it operates on the \(|01\rangle\) state. {\displaystyle HH^{\dagger }=I} {\displaystyle {\frac {|0\rangle -|1\rangle }{\sqrt {2}}}} ⟩ where we have used the orthonormality of the basis states. | It performs an X on the target whenever the control is in state Bloch sphere showing the computational basis states |0 > and |1 > , and a general qubit state |ψ >= Cos θ 2 |0 > + e iϕ Sin θ 2 |1 > [12]. unitary transformation connecting this basis with the standard or computational, we need to find a matrix with components given by, We know already the representation of the, Hence the transformation matrix between the two bases is given by, The other Pauli matrices, being unitary 2×2 matrices, are also valid single-qubit gates. † | 0 1 {\displaystyle a} F or quantum operating system. {\displaystyle R_{\phi }} , . 2 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. ψ gates ( = | and A unitary matrix means that if you multiple the matrix by its (complex) conjugate transpose you end up with an identity matrix. C | {\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\\end{bmatrix}}}, The Ising gate (or XX gate) is a 2-qubit gate that is implemented natively in some trapped-ion quantum computers. 0 0 Another way to say this is that the Pythagorean theorem generalized to For example, the Hadamard gate ( FIGURE 3.20. ⟩ The selection of basis vectors against to measure a quantum state will influence the outcome of the measurement. It also applies the other way. {\displaystyle \bigotimes _{1}^{n}H} When U is one of the Pauli matrices, σx, σy, or σz, the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used. gate; is a 3-bit gate, which is universal for classical computation but not for quantum computation. 0 0 { This is because multiplying a quantum gate's matrix by itself results in the identity matrix, and any matrix multiplied by the identity matrix is itself (i.e. 11 This means that it is possible to construct an inverse of all algorithms and functions, as long as they contain only gates. | ⊗ , 2 basis vectors of an ⟩ , n ), create a quantum state that have equal probability of being observed in any of its four possible outcomes; 00, 01, 10 and 11. ⟩ 0 One way to do this is to factorize the matrix that encodes the unitary transformation into a product of tensor products (i.e. ⟩ | at the Bloch sphere. Similarly, The block sphere is a useful visualization tool for single quantum bits and unitary transform, we usually expect to see orthogonal vectors, degrees) and a reflection (in latitude with r. ability to achieve it in a sequence of unitary quantum gates. A Notice that in this analysis we assume there is no error introduced by the quantum channel. These techniques give a large improvement. 1 z A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced. A universal logic gate for reversible classical computing, the Toffoli gate, is reducible to the Deutsch 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. ⟩ ⟩ \(Z|0\rangle = |0\rangle\)) but it maps \(|1\rangle\) to \(-|1\rangle\). 10 H=ZY−1/2H = Z Y^{-1/2}H=ZY−1/2 2 This section will go over the three most common, multi-qubit gates. "Thus, in this basis the sense of which bit is the control bit and which the target bit has reversed. 2 = 00 2 = It is mentioned here only to contrast it with the previous controlled gate. Another common, and important, gate is knownn as the Hadamard Gate. Because the gates unitary nature, all functions must be reversible and always be bijective mappings of input to output. 1 n . ) | {\displaystyle |\psi _{2}\rangle }. ⟩ {\displaystyle 2^{n}} {\displaystyle |A_{k}B_{k}\rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}}. 00 {\displaystyle |0\rangle } ( 1 n

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