Quantum Computing 2 -- Quantum gates (single qubit)

Quantum gates are the elementary operations for the qubits of a quantum computer. They are comparable to the logic gates for classical bits. In contrast to the Boolean operators, all quantum gates are reversible, so there is always an inverse operation that can undo all computation steps. There is also a much wider variety of possible operations for qubits.

A quantum gate for a single qubit is described by a unitary matrix: \[ U = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)\, \quad a,b,c,d \in \mathbb{C} \] The operation on a state \(| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle \) corresponds to the multiplication of the matrix \(U \) with the state vector: \[\begin{align*} U |\psi \rangle &= U (\alpha |0\rangle + \beta |1\rangle)\\ &= \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{c} \alpha \\ \beta \end{array} \right) \\ &= \left( \begin{array}{c} a \alpha + b \beta \\ c \alpha + d \beta \end{array} \right) \\ &= (a \alpha + b \beta) |0\rangle + (c \alpha + d \beta) |1\rangle \end{align*}\] Executing two operators \(U \) and \ (V ) one after the other is equivalent to an operator resulting from the matrix multiplication of \(U \) and \(V \): \[\begin{align*} V \cdot U &= \left( \begin{array}{cc} a' & b' \\ c' & d' \end{array} \right) \cdot \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \\ &= \left( \begin{array}{cc} a' a + b' c & a' b + b' d \\ c' a + d' c & c' b + d' d \end{array} \right) \end{align*}\] Since the operators are unitary matrices \(U \), the adjoint matrix \(U ^ \dagger \) corresponds to the inverse operator. (The quantum gate \(U ^ \dagger \) thus reverses the effect of the gate \(U \).) \[\begin{align*} U^\dagger &= \left( \begin{array}{cc} a^\ast & c^\ast \\ b^\ast & d^\ast \end{array} \right) \\ U^\dagger \cdot U &= I = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \end{align*}\] where \(I\) is the identity operator or unit matrix and \(z^\ast\) denotes the complex conjungate of \(z\).

Hadamard-Gate

The Hadamard gate is one of the most fundamental operations in quantum computing. The Hadamard gate for single qubit is described in the \(\{| 0 \rangle, | 1 \rangle \} \) basis by the following matrix:

\[\begin{align*} H &= \frac{1}{\sqrt{2}} \left( \begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array} \right) \\ H|0\rangle &= \frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle) \equiv |+\rangle\\ H|1\rangle &= \frac{1}{\sqrt{2}}(|0 \rangle - |1 \rangle) \equiv |-\rangle \end{align*}\] The basis states \(| 0 \rangle \) and \(| 1 \rangle \) are converted by the Hadamard gate into a superposition. You can also understand this operation as a transition between the bases \(\{| 0 \rangle, | 1 \rangle \} \) and \(\{| + \rangle, | - \rangle \} \). The Hadamard matrix is self-adjoint (\(H ^ \dagger = H \)), so that the two-time application of the Hadamard gate gives the identity (\(H \cdot H = H ^ \dagger \cdot H = I \) ).

Phase-Shift

The phase-shift \(R_ \phi \) is a whole family of single-qubit gates. It is described by the following matrix in the \(\{| 0 \rangle, | 1 \rangle \} \) basis:

\[\begin{align*} R_\phi &= \left( \begin{array}{cc} 1 & 0 \\ 0 & \exp(i \phi) \end{array} \right) \\ R_\phi|0\rangle &= |0\rangle \\ R_\phi|1\rangle &= \exp(i \phi) |1\rangle \end{align*}\] where \(\exp (i \phi) = \cos (\phi) + i \sin (\phi) \). This operation leaves the amplitudes of the state vector unchanged and only introduces a phase shift between the vector components by the angle \(\phi \in [- \pi, \pi] \). The adjoint matrix yields \(R_ \phi ^ \dagger = R _ {- \phi} \) such that \(R _ {- \phi} \cdot R_ \phi = I \).

Of particular importance are the phase shift gates \(T = R _ {\pi / 4} \), \(S = R _ {\pi / 2} \) and \(Z = R_ \pi \).

Pauli-Gates

The Pauli gates are a family of three operators with the matrices

\[ \begin{align*} X &= \left(\begin{array}{cc}0 & 1 \\ 1 & 0 \end{array}\right) \\ Y &= \left(\begin{array}{cc}0 & -i \\ i & 0 \end{array}\right) \\ Z &= \left(\begin{array}{cc}1 & 0 \\ 0 & -1 \end{array}\right) \end{align*} \] The \(X \) gate is the quantum mechanical equivalent to the classical NOT operator and interchanges the basis states so that \(X | 0 \rangle = | 1 \rangle \) and \(X | 1 \rangle = | 0 \rangle \) (bit-flip). The \(Z \) gate performs a special phase-shift operation around the angle \(\pi \) (phase-flip). The \(Y \) gate is a combination of bit and phase flip. The Pauli matrices are self-adjoint and obey the algebra \(X \cdot X = Y \cdot Y = Z \cdot Z = -i X \cdot Y \cdot Z = I \).

Note by Markus Michelmann
1 week, 5 days ago

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Thanks for posting these, as well as the problems. It's cool to realize that this stuff is actually somewhat accessible to non-specialists.

Steven Chase - 1 day, 14 hours ago

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