Qubits and the Bloch Sphere
A qubit is the quantum analogue of a classical bit. Whereas a classical bit is either 0 or 1, a qubit exists in a superposition of both:
\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1\]| The computational basis states $ | 0\rangle = \begin{pmatrix}1\0\end{pmatrix}$ and $ | 1\rangle = \begin{pmatrix}0\1\end{pmatrix}$ span the Hilbert space $\mathcal{H} \cong \mathbb{C}^2$. |
Modulo global phase, any pure single-qubit state can be written as:
\[|\psi\rangle = \cos\!\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\!\frac{\theta}{2}|1\rangle\]where $\theta \in [0, \pi]$ and $\phi \in [0, 2\pi)$. This parameterisation places every single-qubit pure state on the surface of the Bloch sphere: $\hat{n} = (\sin\theta\cos\phi,\, \sin\theta\sin\phi,\, \cos\theta)$.
Key Bloch sphere points:
| State | $\theta$ | $\phi$ | Bloch vector |
|---|---|---|---|
| $|0\rangle$ | 0 | — | $(0,0,1)$ |
| $|1\rangle$ | $\pi$ | — | $(0,0,-1)$ |
| $|{+}\rangle = \tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$ | $\pi/2$ | 0 | $(1,0,0)$ |
| $|{-}\rangle = \tfrac{1}{\sqrt2}(|0\rangle-|1\rangle)$ | $\pi/2$ | $\pi$ | $(-1,0,0)$ |
| $|{+i}\rangle = \tfrac{1}{\sqrt2}(|0\rangle+i|1\rangle)$ | $\pi/2$ | $\pi/2$ | $(0,1,0)$ |
A single-qubit gate is a rotation of this sphere: $U = e^{-i\hat{n}\cdot\vec{\sigma}\,\theta/2}$ for some axis $\hat{n}$ and angle $\theta$.
Unitary Operators and the Pauli Matrices
The requirement that quantum evolution is norm-preserving forces every gate to be a unitary operator: $UU^\dagger = U^\dagger U = I$. The set of $2\times 2$ unitary matrices forms the Lie group $U(2)$; discarding the global phase gives $SU(2)$, which double-covers the rotation group $SO(3)$.
A basis for all $2\times 2$ Hermitian operators — and hence a generating set for $SU(2)$ — is given by the Pauli matrices:
\[X = \begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad Y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}, \quad Z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\]These satisfy the algebra $[X, Y] = 2iZ$ (and cyclic permutations) and $X^2 = Y^2 = Z^2 = I$. As gates:
-
$X$ (NOT gate): $X 0\rangle = 1\rangle$, $X 1\rangle = 0\rangle$. Bit-flip. -
$Y$: $Y 0\rangle = i 1\rangle$, $Y 1\rangle = -i 0\rangle$. Bit-and-phase flip. -
$Z$ (phase flip): $Z 0\rangle = 0\rangle$, $Z 1\rangle = - 1\rangle$.
General rotation operators:
\[R_x(\theta) = e^{-i\theta X/2} = \cos\!\frac{\theta}{2}\, I - i\sin\!\frac{\theta}{2}\, X\] \[R_y(\theta) = e^{-i\theta Y/2} = \cos\!\frac{\theta}{2}\, I - i\sin\!\frac{\theta}{2}\, Y\] \[R_z(\theta) = e^{-i\theta Z/2} = \begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}\]Hadamard, Phase, and T Gates
The Hadamard gate maps the computational basis to the Hadamard (Fourier) basis:
\[H = \frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\] \[H|0\rangle = |{+}\rangle, \quad H|1\rangle = |{-}\rangle\]$H$ is both Hermitian and unitary ($H = H^\dagger$, $H^2 = I$), making it its own inverse. Geometrically it is a $\pi$ rotation about the $(\hat{x}+\hat{z})/\sqrt{2}$ axis on the Bloch sphere.
The phase gate $S$ and $T$ gate introduce relative phases:
\[S = \begin{pmatrix}1&0\\0&i\end{pmatrix} = R_z\!\left(\frac{\pi}{2}\right), \qquad T = \begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix} = R_z\!\left(\frac{\pi}{4}\right)\]Note $S = T^2$ and $Z = S^2 = T^4$. The $T$ gate (also called the $\pi/8$ gate) is the gate most expensive to implement fault-tolerantly; its cost dominates resource estimates for practical algorithms.
Key identities connecting these gates:
\(HXH = Z, \quad HZH = X, \quad HYH = -Y\) \(HTH = R_x\!\left(\frac{\pi}{4}\right) \cdot (\text{phase})\)
CNOT and the Toffoli Gate
| Multi-qubit gates entangle qubits and enable universal computation. The controlled-NOT (CNOT) gate flips the target qubit if and only if the control qubit is $ | 1\rangle$: |
| in the basis ${ | 00\rangle, | 01\rangle, | 10\rangle, | 11\rangle}$. Acting on a superposition: |
This is a Bell state — a maximally entangled state — produced by $H$ on the control followed by CNOT.
| The Toffoli gate (CCNOT) has two controls and one target; it flips the target iff both controls are $ | 1\rangle$: |
The Toffoli gate is universal for reversible classical computation (it can simulate NAND), and combined with $H$ it becomes universal for quantum computation.
Universality and the Solovay-Kitaev Theorem
A gate set is universal if any unitary $U \in U(2^n)$ can be approximated to within $\varepsilon$ using gates from that set. Two landmark results:
Universality of ${H, T, \text{CNOT}}$: Any $n$-qubit unitary can be decomposed into a sequence of single-qubit gates and CNOTs. Furthermore, adding the $T$ gate to ${H, \text{CNOT}}$ achieves density in $U(2^n)$; the set is universal over $\mathbb{C}$.
Solovay-Kitaev theorem: Let $\mathcal{G}$ be a universal gate set closed under inverses and dense in $SU(2)$. Then any $U \in SU(2)$ can be approximated to within $\varepsilon$ using
\[O\!\left(\log^c \frac{1}{\varepsilon}\right) \text{ gates}, \quad c \approx 2\]In practice $c \approx 3.97$. This polynomial overhead in $\log(1/\varepsilon)$ guarantees that increasing precision is efficient. The constructive proof uses a divide-and-conquer group-commutator decomposition: at each level, $U \approx VWV^\dagger W^\dagger$ where $V, W$ are recursively approximated gates.
Gate Fidelity and Circuit Noise
Physical gates are imperfect. The fidelity between an ideal gate $U$ and a realised channel $\mathcal{E}$ is:
\[F(U, \mathcal{E}) = \int \langle\psi|U^\dagger \mathcal{E}(|\psi\rangle\langle\psi|)U|\psi\rangle\, d\psi\]where the integral is over the Haar measure on the qubit Hilbert space. For a depolarising channel with error rate $p$:
\[\mathcal{E}(\rho) = (1-p)\rho + \frac{p}{3}(X\rho X + Y\rho Y + Z\rho Z)\]the average gate fidelity is $F = 1 - p$.
Circuit depth and gate count directly control accumulated noise. Typical near-term hardware targets:
| Metric | Superconducting | Trapped Ion |
|---|---|---|
| Single-qubit gate fidelity | $99.9\%$ | $99.99\%$ |
| Two-qubit gate fidelity | $99.5\%$ | $99.8\%$ |
| Gate time (single-qubit) | $\sim 20\,\text{ns}$ | $\sim 10\,\mu\text{s}$ |
| Coherence time $T_2$ | $\sim 100\,\mu\text{s}$ | $\sim 1\,\text{s}$ |
The ratio $T_2 / t_{\text{gate}}$ sets an upper bound on the number of gates executable before decoherence dominates, motivating both faster gates and improved coherence times.