Single-Qubit Gates

In quantum computing, a gate is a physical operation that evolves the state of a qubit. Mathematically, every quantum gate is represented by a unitary operator $\hat{U}$.

Unitarity is essential because it preserves the norm of the state vector, ensuring that total probability remains equal to $1$. It also preserves inner products between states, maintaining the distinguishability of quantum states over time.

• • A unitary operator satisfies $\hat{U}^\dagger \hat{U} = \hat{U} \hat{U}^\dagger = I$.
• • Unitary evolution is reversible: $\hat{U}^{-1} = \hat{U}^\dagger$.
• • In the active picture, the state vector evolves as $|\psi\rangle \mapsto \hat{U}|\psi\rangle$.
• • In the passive picture, the operator evolves while the state remains fixed.

The Pauli gates $\hat{X}$, $\hat{Y}$, and $\hat{Z}$ are the most fundamental single-qubit operations. They correspond to $\pi$ rotations about the $x$, $y$, and $z$ axes of the Bloch sphere, respectively.

The $\hat{X}$ gate is the quantum analogue of the classical NOT gate, swapping $|0\rangle$ and $|1\rangle$. The $\hat{Z}$ gate introduces a relative phase of $-1$ on the $|1\rangle$ component.

• • $\hat{X}|0\rangle = |1\rangle$ and $\hat{X}|1\rangle = |0\rangle$.
• • $\hat{Z}|+\rangle = |-\rangle$ and $\hat{Z}|-\rangle = |+\rangle$, where $|\pm\rangle = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}}$.
• • Each Pauli gate has eigenvalues $\pm 1$.
• • Pauli gates are Hermitian and unitary: $\hat{X}^2 = \hat{Y}^2 = \hat{Z}^2 = I$.

The Hadamard gate $\hat{H}$ is indispensable for creating superposition. It maps the computational basis states into the $\hat{X}$ eigenbasis, and vice versa.

Geometrically, $\hat{H}$ corresponds to a $\pi$ rotation about the axis $(\hat{x} + \hat{z})/\sqrt{2}$ on the Bloch sphere. A key property is that $\hat{H}^2 = I$, meaning two consecutive Hadamard gates return the qubit to its original state.

• • $\hat{H}|0\rangle = |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$.
• • $\hat{H}|1\rangle = |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$.
• • $\hat{H}^2 = I$.
• • Hadamard is its own inverse: $\hat{H}^\dagger = \hat{H}$.

Phase gates rotate the qubit state around the $z$-axis of the Bloch sphere, adding a relative phase to the $|1\rangle$ component without changing its amplitude. The $\hat{S}$ and $\hat{T}$ gates are the most common examples.

Phase kickback is a subtle but crucial effect: when a controlled-phase gate is applied to a control qubit in the $|+\rangle$ state and a target qubit in $|1\rangle$, the phase is transferred back to the control qubit.

• • $\hat{S} = \sqrt{\hat{Z}} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$.
• • $\hat{T} = \sqrt{\hat{S}} = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$.
• • General phase rotation: $\hat{R}_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$.
• • Phase kickback: $\text{CZ}(|+\rangle \otimes |1\rangle) = |-\rangle \otimes |1\rangle$.

Any single-qubit unitary can be decomposed into a sequence of rotations about two orthogonal axes. The most common form is the Euler decomposition using $\hat{R}_z$ and $\hat{R}_y$.

Alternatively, any single-qubit gate can be viewed as a rotation by an angle $\theta$ about an arbitrary axis $\hat{n}$ on the Bloch sphere, combined with a global phase.

• • Euler decomposition: $\hat{U} = e^{i\alpha} \hat{R}_z(\beta) \hat{R}_y(\gamma) \hat{R}_z(\delta)$.
• • Rotation about an arbitrary axis $\hat{n} = (n_x, n_y, n_z)$.
• • Any unitary is equivalent to a rotation up to a global phase.

It is impossible to implement arbitrary single-qubit rotations exactly using a finite discrete gate set. However, the Solovay-Kitaev theorem guarantees that any single-qubit unitary can be approximated to arbitrary precision by a finite sequence of gates from a universal set.

The Hadamard $\hat{H}$ and $\hat{T}$ gates form a particularly important universal set for single-qubit computation. Any single-qubit gate can be approximated to within error $\epsilon$ using $\mathcal{O}(\log^c(1/\epsilon))$ gates from this set.

• • Exact synthesis is generally impossible with discrete gates.
• • $\{\hat{H}, \hat{T}\}$ is universal for single-qubit unitaries.
• • The Solovay-Kitaev theorem provides an efficient classical algorithm for finding such approximations.
• • This universality underpins fault-tolerant quantum computation.