Quantum Computing

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

Quantum Computing formulas describe how qubits behave, how quantum gates transform them, and how key algorithms and information measures are calculated. This category helps students, exam candidates, and practitioners working with quantum hardware or algorithms understand concepts like superposition, entanglement, measurement, and complexity. These formulas matter because they are the language of quantum processors, from basic state preparation to powerful speedups in search and factoring.

Qubit Superposition State

NEWIntermediate

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

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

Normalisation Constraint

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

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

Hadamard Gate

NEWIntermediate

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.

Pauli-X Gate (Quantum NOT)

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 Gate

NEWIntermediate

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⟩.

Bell State (EPR Pair)

NEWIntermediate

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

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

Born Rule (Measurement Probability)

NEWIntermediate

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

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

Grover's Search Complexity

O(\sqrt{N})

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

Shor's Factoring Complexity

O((\log N)^3)

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

Quantum State Space

\text{States} = 2^n

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

Quantum Fidelity

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.

Von Neumann Entropy

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

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