Quantum Computing

Qubit states, quantum gates, algorithms and information measures for quantum processors

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

General state of a single qubit as a superposition of the computational basis states.

|\alpha|^2 + |\beta|^2 = 1

The squared amplitudes of a qubit must sum to 1, ensuring total probability is 100 %.

H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

Puts a qubit into an equal superposition — the most common single-qubit gate.

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Flips |0⟩ to |1⟩ and vice versa — the quantum equivalent of a classical NOT gate.

CNOT = \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \\ 0&0&1&0 \end{pmatrix}

Two-qubit gate that flips the target qubit only when the control qubit is |1⟩.

|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)

Maximally entangled two-qubit state — measuring one qubit instantly determines the other.

P(|k\rangle) = |\langle k|\psi\rangle|^2

Probability of measuring outcome |k⟩ from a quantum state |ψ⟩.

O(\sqrt{N})

Number of queries needed by Grover's algorithm to find an item in an unsorted database of N entries.

O((\log N)^3)

Time complexity for Shor's algorithm to factor an integer N on a quantum computer.

\text{States} = 2^n

Number of basis states an n-qubit register can be in superposition of simultaneously.

F(\rho, \sigma) = \left(\text{tr}\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\right)^2

Measures how close two quantum states are — 1 means identical, 0 means orthogonal.

S(\rho) = -\text{tr}(\rho \log_2 \rho)

Quantum analogue of Shannon entropy — measures the uncertainty in a quantum state.