Frontier physics

Qubits and quantum gates

A qubit is a superposition of two basis states; unitary gates act on it, and circuits combine gates with measurement.

A classical bit is 0 or 1; a pure qubit is a normalized vector $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ . Complex amplitudes set measurement probabilities, while relative phase enables interference. One measurement cannot reveal both amplitudes.

|\alpha|^2+|\beta|^2=1,\qquad P(0)=|\alpha|^2,\quad P(1)=|\beta|^2

Definition: Quantum gate

A unitary transformation on a state space; a final measurement yields classical outcomes. For example, $H|0\rangle=(|0\rangle+|1\rangle)/\sqrt2$ , while phase gates rotate relative amplitudes.

The Bloch sphere represents pure single-qubit states; global phase is not shown.

Operations and circuits

The $X,Y,Z,H$ and phase gates form universal operations with suitable combinations. Two-qubit gates such as CNOT can create entanglement; single-qubit gates alone cannot turn a product state into an entangled one. Noise yields mixed states described by density matrices.

Example: Measurement probability

For $|\psi\rangle=(|0\rangle+i|1\rangle)/\sqrt2$ , what is the probability of outcome 1 in the computational basis?

Solution

$\beta=i/\sqrt2$ , so $P(1)=|\beta|^2=1/2$ . The phase $i$ does not change this measurement probability.

On the Bloch sphere, a pure state is $|\psi\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle$ , up to global phase. The $X,Y,Z$ gates are Pauli rotations, while Hadamard changes basis and creates equal superposition; two-qubit gates such as CNOT can entangle. Closed-system evolution preserves norm and is represented by a unitary matrix; measurement converts amplitudes into classical outcomes.

An isolated qubit subject to decoherence is no longer pure and is described by a density matrix $\rho$ , with measurement probability $p_i=\mathrm{Tr}(\rho |i\rangle\langle i|)$ . Real circuits have imperfect gates, finite coherence times, and nonideal readout. Gate fidelity, calibration, and connectivity are therefore as important as the ideal mathematical representation of a qubit.

The expectation values $\langle\sigma_x,\sigma_y,\sigma_z\rangle$ define a Bloch vector for a pure state, while mixed states lie inside the sphere. A one-qubit gate rotates this vector, and a two-qubit gate can create nonseparable correlations. This geometric picture helps track gates and noise.

For |ψ⟩=α|0⟩+β|1⟩, what is the probability of measuring 1?

Which operation can create entanglement from an initial product state?

References

Michael A. Nielsen, Isaac L. Chuang (2010). Quantum Computation and Quantum Information